diff --git a/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/2301.02867v1.pdf.txt b/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/2301.02867v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d0fa01fbbe23143475019c0430b3905fb03f269 --- /dev/null +++ b/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/2301.02867v1.pdf.txt @@ -0,0 +1,1356 @@ +arXiv:2301.02867v1 [hep-th] 7 Jan 2023 +Covariant calculation of the partition function +of the two-dimensional sigma model +on compact two-surfaces +O.D. Andreev, R.R. Metsaev, and A.A. Tseytlin +Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninski prospect 53, +Moscow 117924, USSR +(submitted 17 July 1989) +Abstract +Motivated by string theory connection, a covariant procedure for perturbative +calculation of the partition function Z of the two-dimensional generalized σ- +model is considered. The importance of a consistent regularization of the measure +in the path integral is emphasized. The partition function Z is computed for a +number of specific 2-manifolds: sphere, disk and torus. +Published in: +Yad.Fiz. 51 (1990) 564-576 [Sov.J.Nucl.Phys. 51 (1990) 359-366] +1 + +Contents +1 +Introduction +2 +2 +Calculation of partition function of σ-model on compact 2-surfaces +3 +3 +Partition function on specific 2-surfaces: sphere, disk and torus +9 +4 +Partition function of the N = 1 supersymmetric σ-model +13 +1 +Introduction +A promising approach to string theory is the so-called σ-model approach. It may help +elucidate the structure and first principles of string theory (see, e.g., Refs.[1, 2]). +A central role in the σ-model approach is played by the partition function Z of the +generalized two-dimensional σ-model. Z is closely related to the generation functional +for the string S-matrix and to the effective action of the string theory [2]. +The string partition function differs from the usual σ-model partition function by +a factor of the volume of the M¨obius group. In the theory of closed strings a possible +implementation of the operation of division by the M¨obius group volume is by taking +the derivative with respect to the log of the UV cutoff +∂ +∂ ln ε of the regularized partition +function ZR. The reason for this is the presence of a logarithmic divergence [3] in the +regularized volume ΩR of the M´obius group [4]. +In the theory of open strings the procedure of “division” by the M¨obius volume +reduces to a renormalization of power divergencies as the regularized volume of the +M¨obius group SL(2, R) contains only power divergences [5, 3]. The remaining loga- +rithmic two-dimensional UV divergences can be interpreted as being due to the mass- +less poles in the scattering amplitudes. As a result, the renormalized string partition +function coincides with the effective action S for the massless modes of the open string. +We shall perform the calculation of the partition function of the two-dimensional +σ-model on compact surfaces emphasizing the role of the measure in the functional +integral in the procedure of calculating the covariant expression for Z. In Sec.2 we +consider three possible ways of determining the regularized measure that lead to a +covariant answer. In Sec.3 we give examples of the calculation of the leading terms in +Z for some specific cases of 2-manifolds: the sphere, disk (hemisphere), and the torus. +Taking into account the procedure for dividing by the M¨obius volume, we obtain an +alternative to the S-matrix method of [6] for calculating the string effective action. In +Sec.4 we consider a generalization of this approach to the supersymmetric case. +Let us make a comment on the interpretation of infinities that are present in Z. +In addition to the already mentioned M¨obius and other two-dimensional UV infinities, +in the case of 2-surfaces of higher genera there exist the so-called modular infinities +corresponding to degeneration of the Riemann surfaces [9].1 The “modular’ correction +to the β-functions corresponds to the infinities associated with the degeneration of +trivial cycles [10]. The partition function Z should be renormalizable with respect +1In the framework of the σ-model approach, in the case of surfaces of higher genera it is necessary +to use the Schottky [7] or the branch-point type [8] parameterization for the moduli space in which +the on-shell scattering amplitudes have formal SL(2, C) invariance. +2 + +to all infinities (modular and local), i.e. it should be finite after the renormalization +corresponding to the complete β-function [11]. +2 +Calculation of partition function of σ-model on +compact 2-surfaces +We shall consider the bosonic σ-model (µ, ν = 1, ..., D) +Z = +� +[Dx] exp +� +− I(x) +� +, +(2.1) +I = +1 +4πα′ +� +d2σ√g +�α′ +ε2ϕ(x) + ∂axµ∂axνGµν(x) + α′R(2)φ(x) +� +, +(2.2) +defined on a compact closed two-dimensional surface. Here Gµν, φ, and ϕ are the bare +fields that depend on the two-dimensional cutoff ε. The renormalized value of ϕ will +be chosen to be zero. The theory is defined by the action I and the measure [Dx]. +Imposing the requirement of invariance under the general coordinate transformations +xµ → x′µ , +Gµν → G′ +µν = ∂xα +∂x′µ +∂xβ +∂x′ν Gαβ , +(i.e. that upon a transformation of xµ the “coupling constants” Gµν of the theory +are also transformed), below we shall consider three ways of calculating the partition +function (2.1) that are consistent with the requirement of this covariance. +1. Let us first choose the measure [Dx] to be trivial: +[Dx] = +� +σ +dDx(σ) . +To cancel the power divergences we make use of the bare tachyon field ϕ(x) (with the +renormalized value of ϕ set to zero). We separate xµ into a constant and a non-constant +parts, xµ = yµ + ηµ, inserting “one” into (2.1) (cf. [12]) +1 = +� +dDy +� � +dDη δ(D)(x(σ) − y − η) δ(D)(P µ[y, η]) Q[y, η] , +Q = det ∂P µ[y − a, η + a] +∂aν +��� +a=0 , +(2.3) +where P = 0 is a gauge condition and Q is the ghost determinant. One possible choice +is +P µ = +� +d2σ√g ηµ , +Q = V D , +V = +� +d2σ √g . +(2.4) +The condition P = 0 implies that η does not contain a zero mode of the Laplace +operator (a constant). We substitute x = y +η into the action and expand it in powers +of η: +I = +1 +4πα′ +� +d2σ√g +�α′ +ε2ϕ + ∂aηµ∂aην� +Gµν + ∂λGµνηλ + 1 +2∂λ∂ρGµνηληρ + . . . +� ++α′R(2)(φ + 1 +2∂µ∂νφηµην + . . . +�� +. +(2.5) +3 + +The leading (one-loop) contribution of the integral over η is +Z0 = [det ′(Gµν∆)]−1/2 = exp +� +− 1 +2N′ ln G − 1 +2D ln det ′∆ +� +, +(2.6) +where N′ is the regularized number of nonzero eigenmodes of the Laplace operator, +G = det Gµν and D is the dimensionality of space-time. +The number N′ can be expressed in terms of the heat kernel in a familiar way +N′ = +� +d2σ √g Kε − 1 = +V +4πε2 + 1 +6χ + O(ε2) , +(2.7) +Kε = +� +n +fn(σ)fn(σ′) exp(−λnε2) , +(2.8) +where fn(σ) and λn are, respectively, the eigenfunctions and eigenvalues of the Laplace +operator on the two-dimensional surface of Euler number χ = +1 +4π +� +d2σ√gR(2). +Taking (2.7) into account, we obtain for (2.6) +Z0 = Z0 exp +�� +− +V +4πε2 − 1 +6χ + 1 + O(ε2) +� +ln G +� +, +Z0 = exp +� +− 1 +2D ln det ′∆ +� +. +(2.9) +The dependence of Z on the dilaton field (to order α′2) is easily found from (2.5): +Z = +� +dDy Z0 e−χφ� +1 − α′πχ∂µ∂νφ Gµν D(σ, σ) + O(α′2) +� +, +(2.10) +where D is the regularized Green function of the Laplace operator +D(σ, σ′) = +� +λn̸=0 +fn(σ)fn(σ′) +λn +exp +� +− λnε2) . +(2.11) +For ε → 0 it has the form [13] +D(σ, σ) = − 1 +2π ln ε + O(1) . +(2.12) +To determine the dependence of Z on the graviton field Gµν it is necessary to consider +the two possible one-particle-irreducible two-loop diagrams. Their contribution to Z +is found to be +Z = +� +dDy Z0 e−χφ� +1 + c1GµνGλρ∂λ∂ρGµν + c2GµαGνβGρλ∂ρGµν∂λGαβ ++c3GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) +� +, +c1 = −1 +2πα′D(σ, σ)N′ , +c2 = 1 +2πα′ +� +d2σ d2σ′√g +� +g′ D(σ, σ′) ∂a∂b′D(σ, σ′) ∂a∂b′ D(σ, σ′) , +c3 = πα′ +� +d2σ d2σ′ √g +� +g′ ∂aD(σ, σ′) ∂a∂b′D(σ, σ′) ∂b′D(σ, σ′) . +(2.13) +4 + +We ensure the covariance of Z0 by means of the special choice of the bare fields +φ′ = φ + a ln +� +G(x) , +ϕ′ = ϕ + b ln +� +G(x) . +(2.14) +Substituting x = y +η and expanding ln +� +G(x) in powers of η we obtain the following +correction to the action in (2.2) +∆I = +1 +4πα′ +� +d2σ√g α′ � b +ε2 + aR(2)�� +ln +√ +G + 1 +4Gµν∂λ∂ρGµνηληρ +− 1 +4GµβGνα∂ρGµν∂λGαβηρηλ + . . . +� +. (2.15) +The values of a and b are calculated from the condition that Z0 has a required covariant +form (Z0 ∼ +√ +G) +a = −1 +6 , +b = −1 . +(2.16) +We now find the correction to Z from (2.15) taking into account (2.16) and the final +expression for Z0 +∆Z = Z0 +� +dDy +√ +G e−χφ� +1 + 1 +2πα′�1 +6χ + +V +4πε2 +� +D(σ, σ) +× +� +GµνGλρ∂λ∂ρGµν − GµαGνβGρλ∂ρGµν∂λGαβ + O(α′2) +�� +(2.17) +As a result, from (2.9), (2.10), (2.13), and (2.17), we obtain +Z = Z0 +� +dDy +√ +Ge−χφ� +1 − πα′χD(σ, σ)∂µ∂νφGµν + ˜c1GµνGλρ∂λ∂ρGµν ++ ˜c2GµαGνβGρλ∂ρGµν∂λGαβ + ˜c3GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) +� +, +˜c1 = c1 + 1 +2πα′(1 +6χ + +V +4πε2)D(σ, σ) , +˜c2 = c2 − 1 +2πα′(1 +6χ + +V +4πε2)D(σ, σ) , +˜c3 = c3 . +(2.18) +After the ci’s have been calculated using (2.7) and (2.12), the power divergences cancel +and the dependence of Z on ε takes the form +Z = Z0 +� +dDy +√ +G e−χφ � +1 + 1 +2α′χ(ln ε + O(1))∂µ∂νφ Gµν +− 1 +4α′(ln ε + O(1))GµνGλρ∂λ∂ρGµν ++ 1 +8α′(ln ε + O(1))GµαGνβGρλ∂ρGµν∂λGαβ ++ 1 +4α′(ln ε + O(1))GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) +� +(2.19) +Using in (2.19) the expression for the target space scalar curvature R in terms of Gµν +and integrating by parts we observe that we can rewrite Z in the manifestly covariant +form +Z = Z0 +� +dDy +√ +G e−χφ � +1 + 1 +2α′� +ln ε + O(1) +�� +R + χD2φ +� ++ O(α′2) +� +, +(2.20) +5 + +where Dµ in D2 is the covariant derivative. +2. Next, let us consider the manifestly covariant method of calculating Z based on +the expansion for the action and the measure in normal coordinates. Let us define the +measure [Dx] by the formal product +[Dx] = +� +σ +dDx(σ) +� +G(x(σ)) . +(2.21) +To preserve the general covariant invariance in the regularized theory it is necessary to +regularize the measure and the action in a consistent manner. We choose the regularized +expression for the measure (2.21) in the form +[Dx] = +� +σ +dDx(σ) eM +(2.22) +M = 1 +2 +� +d2σ√g ln G(x)Kε(σ, σ) . +(2.23) +Now let us set xµ = yµ + ηµ(y, ξ) where ξµ is the tangent vector to the geodesic joining +the points yµ and yµ + ηµ +ηµ = ξµ − 1 +2Γµ +αβξαξβ − 1 +6 +� +∂γΓµ +αβ − 2Γλ +γαΓµ +λβ +� +ξαξβξγ + . . . +(2.24) +The expansions of the action and measure in powers of ξ have the form [14] +I = +1 +4πα′ +� +d2σ√g +� +∂aξµ∂aξν� +Gµν + 1 +3Rµλρνξλξρ + 2 +45Rλµρ +γRανβγξλξρξαξβ + O(ξ5) +� ++ α′R(2)� +φ + Dµφ ξµ + 1 +2DµDνφ ξµξν + O(ξ3) +�� +, (2.25) +M = 1 +2 +� +d2σ√g Kε(σ, σ) +� +ln G − 1 +3Rµνξµξν + O(ξ3) +� +. +(2.26) +Since the kinetic term is invariant under a constant shift ξ → ξ + a and ξ may contain +a constant part under the condition (2.4), it is desirable to fix the symmetry y → +y − a, η → η + a by means of another gauge condition [12] +P µ = +� +d2σ√g ξµ . +(2.27) +In this case the ghost determinant in (2.3) is +Q = det +� � +d2σ√g λµ +ν +� +, +λµ +ν = ∂ξµ(y, η) +∂ην +− ∂ξν(y, η) +∂ηµ +. +(2.28) +Its covariant expression takes the form +Q = V D exp +� +− 1 +3V +� +d2σ√g Rµνξµξν + O(ξ3) +� +. +(2.29) +To determine the measure in the y integral, i.e. +� +dDy +√ +G, it is necessary to take +into account not only the one-loop contribution (2.6) but also (2.26). Using that the +regularized number of eigenvalues is +N = +� +d2σ√g Kε(σ, σ) , +(2.30) +6 + +and also (2.7), we arrive at the expression for the covariant measure +√ +G in the integral +over y. In fact, +√ +G is the contribution of the only (constant) zero mode of the Laplace +operator on the compact surface. The partition function Z then takes the form +Z = +� +dDy +√ +G e−χφ F(R, DR, Dφ) . +(2.31) +It is not difficult to calculate the first terms of the expansion of F in powers of α′. +From (2.25), (2.26), and (2.29), we obtain +Z = Z0 +� +dDy +√ +G e−χφ� +1 + α′(a1 + a2 + a3)R + α′b1D2φ + O(α′2) +� +. +(2.32) +The coefficients a1 and b1 correspond to contributions from the action (2.25), a2 arises +from the measure (2.26), and a3 from the ghost determinant (2.29). The expressions for +these coefficients in terms of the Green functions (2.11) have the following appearance +a1 = π +3 +� +d2σ√g ∂a∂′aD(σ, σ′)|σ=σ′D(σ, σ) − ∂aD(σ, σ′)|σ=σ′∂′aD(σ, σ′)|σ=σ′ , +a2 = −π +3 +� +d2σ√gKε(σ, σ)D(σ, σ) , +(2.33) +a3 = − 2π +3V +� +d2σ√gD(σ, σ) , +b1 = −1 +4 +� +d2σ√gR(2)D(σ, σ) . +Explicit calculations give +a1 = −1 +6N′ ln ε + ¯a1 , +a2 = 1 +6N ln ε + ¯a2 , +a3 = 1 +3 ln ε + ¯a3 , +b1 = 1 +2χ ln ε + ¯b1 , +a0 = a1 + a2 + a3 = 1 +2 ln ε + ¯a0 , +(2.34) +where the ¯ai and ¯bi are finite constants. It is easy to see that the power infinities cancel, +and the resulting expression for Z in (2.32) coincides with (2.20). +3. +Let us now consider one more method of calculating Z, which is explicitly +covariant and turns out to be simpler in practice. Here we define the measure [Dx] as +follows +[Dx] = J dDy [Dξ] , +(2.35) +where the factor J is fixed from the normalization condition +� +[Dδx] e−||δx||2 = +� +dDδy +� +[Dδξ] J e−||δx||2 = 1 , +||δx||2 = +1 +4πα′ +� +d2σ√g δxµδxν Gµν . +(2.36) +The expression for ||δx||2 expanded in normal coordinates has the form +||δx||2 = +1 +4πα′ +� +d2σ√g +� +Gµν + 1 +3Rµλ1ρ1νξλ1ξρ1 +− 2 +45Rµλ1ρ1 +γ1Rα1νβ1γ1ξλ1ξρ1ξα1ξβ1 + . . . +�� +δyµ + δξµ + 1 +3Rµ +λ2ρ2κξλ2ξρ2δyκ +− 1 +45Rµ +λ2ρ2γ2Rγ2α2β2κξλ2ξρ2ξα2ξβ2δyκ + . . . +�� +δyν + δξν + 1 +3Rν +λ3ρ3σξλ3ξρ3δyσ +− 1 +45Rν +λ3ρ3γ3Rγ3α3β3σξλ3ξρ3ξα3ξβ3δyσ + . . . +� +. +7 + +Integrating successively over δy and δξ, we find J. Taking into account the expression +(2.35) for the action, we have +Z = Z0 +� +dDy +√ +Ge−χφ� +exp +� � +d2σ√g +�πα′ +3 Rµλνρ∂aξµ∂aξνξλξρ − πα′ +3 K′ +ε(σ, σ)Rµνξµξν +− πα′ +V Rµνξµξν − 4π2α′2 +45 +Rλµρ +γRανβγ∂aξµ∂aξνξλξρξαξβ ++ π2α′2� 4 +45K′ +ε(σ, σ) + 2 +3V +� +Rµλρ +γRµ +αβγξλξρξαξβ + . . . +� +− +� +d2σd2σ′√g +� +g′π2α′2�1 +9K′ +ε +2(σ, σ′) + 2K′ +ε(σ, σ′) +9V ++ 1 +V 2 +� +(2.37) +× RµρλνRµ +αβ +νξλ(σ)ξρ(σ)ξα(σ′)ξβ(σ′) + O(α′3) +�� +. +We have redefined y → (2πα′)1/2y and ξ → (2πα′)1/2ξ, set φ to be constant for simplic- +ity, and took the one-loop contribution into account. Starting from (2.37), we easily +find the expression for the order α′ terms in Z. It is given by the first three terms in +the exponent in (2.37). The contribution of the second term cancels that of the first +one so that the coefficient of R turns out to be proportional to D(σ, σ), so that as in +(2.20) we get +Z = Z0 +� +dDy +√ +G e−χφ � +1 + 1 +2α′(ln ε + const)R + O(α′2) +� +. +(2.38) +The divergent parts of the coefficients of the R2 and R2 +µν terms are calculated in a +similar way. One gets for the R2 term +Z = Z0 +� +dDy +√ +Ge−χφ� +1 + . . . + 1 +2π2α′2D2(σ, σ)R2 + . . . +� +. +(2.39) +The divergent contribution to the coefficient of the R2 +µν term comes effectively only +from the vertex −π2α′2 +V +R2ξξξξ, i.e. +Z = Z0 +� +dDy +√ +Ge−χφ� +1 + . . . − π2α′2D2(σ, σ)RµνRµν + . . . +� +. +(2.40) +The methods of computing Z considered above admit a natural generalization to +the case of 2d surfaces with boundaries (with free open string or Neumann boundary +conditions). Then the Green function D is replaced by the Neumann function. There +are new (linear) power divergencies which can be canceled by a redefinition of the +values of the boundary analogs of the tachyon and dilaton couplings. The σ-model +action in this case has the form +I = +1 +4πα′ +� +d2σ√g +�α′ϕ +ε2 + ∂axµ∂axνGµν + α′R(2)φ +� ++ 1 +2π +� +ds +�ϕ′ +ε + Kφ′� +, +(2.41) +where K is the extrinsic curvature. It is necessary to set φ = φ′ to ensure that the +constant part of the dilaton couples to the Euler characteristic. +It should be emphasized that the above calculation of Z was done for surfaces of any +genus. However, we did not integrate over the moduli space of the Riemann surfaces +and, therefore, the logarithmic divergences found are only the ordinary local ones. +8 + +The expression Z is renormalizable with respect to these local infinities on a surface +of an arbitrary genus (ψi = (G, φ)) +dZ +d ln ε = +∂Z +∂ ln ε − βi ∂Z +∂ψi = 0 , +(2.42) +where βi = − +d +d ln εψi are the local β-functions of the σ-model (cf. (2.20)) +βG +µν = α′Rµν + O(α′2) , +βφ = 1 +6D − 1 +2α′D2φ + O(α′2) . +(2.43) +Assuming that Z is renormalizable also at the next order and using the known +expressions for the α′2 terms in the β-functions (2.43) [12, 15], we find the following +expression for the logarithmically divergent term in Z to order α′2 +Z = λ +� +dDy +√ +G e−χφ � +1 + 1 +2 ln ε +� +α′R + 1 +8(4 − χ) α′2RµαβνRµαβν� ++ . . . +� +. +(2.44) +We shall also confirm the coefficient of the RµαβνRµαβν term directly in the case of the +torus (χ = 0) in the next section. +Let us note also that the (ln ε)2 coefficients of R2 and R2 +µν that we found in (2.39) +and (2.40) are consistent with the renormalizability of Z. +3 +Partition function on specific 2-surfaces: sphere, +disk and torus +Let us now consider the calculation of Z for some simplest surfaces: the sphere, disk, +and torus. In these cases the coefficients of the leading terms in the α′ expansion of Z +can be found explicitly. +1. Let us start with the 2-sphere. In spherical coordinates the eigenfunctions and +eigenvalues of the Laplace operator have the form +fn,m = Yn,m(θ, φ) , +λn,m = n(n + 1) , +(3.1) +where the Yn,m are the orthonormal spherical functions. The regularized expression for +the Green’s function has the form +D(σ, σ′) = +� +n̸=0 +n +� +m=−n +1 +n(n + 1)e−n(n+1)ε2 Y ∗ +n,m(θ, ϕ)Yn,m(θ′, ϕ′) . +(3.2) +At coincident points, it becomes +D(σ, σ) = 1 +4π +� +n̸=0 +2n + 1 +n(n + 1)e−n(n+1)ε2 . +(3.3) +The leading terms in expansion in ε → 0 are easily calculated using the Euler-Maclaurin +resummation formula +D(σ, σ) = − 1 +2π ln ε + γ − 1 +4π ++ ε2 +6π + O(ε4) , +(3.4) +9 + +where γ is the Euler constant. We note that the ln ε and ε2 terms can be calculated +from (2.7) using integration over ε. Taking (3.4) into account, we can write (2.20) as +(here χ = 2) +Z = Z0 +� +dDy +√ +Ge−2φ� +1 + α′� +R + 2D2φ) +�1 +2ln ε + a + O(ε2) +� ++ O(α′2) +� +, +(3.5) +where a = 1 +4(γ − 1) is a scheme-dependent constant. +It is easy to see that Z is renormalizable, i.e. +making the replacement Gµν = +G(R) +µν −ln ε βG +µν and φ = φ(R)−ln ε βφ (cf. (2.43)) we get rid of the logarithmic divergences +and thus find +Z = Z0 +� +dDy +√ +Ge−2φ� +1 + aα′� +R + 2D2φ) + O(α′2) +� +. +(3.6) +Note that this expression is not the same as the closed string effective action obtained +using the S-matrix method. The reason is that the generating functional for the string +tree-level S-matrix is given by Ω−1Z, i.e Z divided by the volume of the group SL(2, C) +of M¨obius transformations. The presence of a logarithmic singularity in the regularized +volume of SL(2, C) suggest that one can think of +∂ +∂ ln ε as a possible realization of the +operation of division by Ω in the case of closed strings [4]. Indeed, as follows from +(3.5), +∂Z +∂ ln ε = 1 +2α′Z0 +� +dDy +√ +G e−2φ � +R + 2D2φ + O(α′) +� +, +(3.7) +which agrees with the effective action found from the tree-level closed string S-matrix. +2. The calculation of Z for disk topology (with a metric of half-sphere) almost +analogous to the case of the sphere. A new feature is that in view of the presence +of the boundary, we impose the Neumann boundary condition at the boundary of +half-sphere +∂θxµ�� +θ= π +2 = 0 . +(3.8) +The expansion of the fluctuation field η in eigenfunctions of the Laplace operator on +the disk has the form +η(θ, φ) = +� +n,m +an,mYn,m , +n + m = 2k , +n ̸= 0 . +(3.9) +and the expression for the regularized Neumann function at coincident points is (cf. +(3.3)) +D(σ, σ) = +∞ +� +n=1 +1 +2πn e−n(n+1)ε2 . +(3.10) +Using the Euler-Maclaurin formula, we obtain (cf. (3.4)) +D(σ, σ) = − 1 +2π ln ε + γ +4π + 1 +4ε − +5 +12πε2 + O(ε3) . +(3.11) +The expression for Z is the same as in (2.18), (2.20) with D(σ, σ) given by (3.11). +The power divergences ε−2 and ε−1 in Z on the disk are canceled by renormalizing the +tachyon fields ϕ and ϕ′, respectively (see (2.41)). +10 + +3. In the case of the 2-torus we shall depart from the scheme used above, which +was based on the heat kernel regularization. This is due to the technical difficulties +of calculating the sums with the spectral e−λnε2 regularization.2 We shall consider the +τ-parametrization, in which the torus is represented as a (τ, 1) parallelogram on the +complex z-plane. The string σ-model partition function on the torus has the form [17] +Z = +� +F +d2τ +4πτ 2 +2 +e4πτ2 +(2πτ2)12|f(e2πiτ)|−48 +� +Dx e−I , +(3.12) +� +Dx e−I0 = 1 , +I = I0 + Iint , +f(e2πiτ) = +∞ +� +n=1 +(1 − e2πinτ) , +where the fundamental region F is specified by the conditions +−1 +2 < τ1 ≤ 1 +2, +|τ| > 1 , +τ = τ1 + iτ2 . +We shall consider only the dependence of Z on the metric Gµν. +By studying the +dependence of Z on G|muν = δµν + hµν using the expansion in powers of hµν, we will +then restore the coefficients of the R, R2, R2 +µν and R2 +µαβν terms (assuming that the +scheme used for the regularization and renormalization preserves the covariance of Z). +Since the metric on the parallelogram is flat, it is possible to use the following +regularization prescription +D(z, z) = − 1 +2π ln ε , +δ(2)(z, z) = 0 , +corresponding to discarding of power divergences. This prescription ensures the co- +variance of Z without need for a nontrivial measure factor. The Green function on the +torus has the form [17] +D(z, z′) = − 1 +4π ln |θ(z, z′)|2 +|θ′(0)|2 ++ 1 +2τ2 +� +Im(z − z′) +�2 , +(3.13) +where θ(z, z′) is the theta function ϑ11(z, z′) [18]. +We redefine x → (2πα′)1/2x and expand the σ-model action I in (2.2) in powers of +η = x − y. Then +Z = ⟨Z⟩ , +Z = +� +Dη exp +� +− 1 +2 +� +d2σ√g∂aηµ∂aην� +δµν + hµν + (2πα′)1/2∂λhµνηλ ++πα′∂ρ∂λhµνηρηλ + . . . +�� +, +⟨. . .⟩ = +� +dDy +� +[dτ] . +(3.14) +The coefficient of R is found from the hµν□hµν term (R = +1 +4hµν□hµν + . . . ). As a +result, +Z ∼ 1 + 2πα′c0R + O(α′2) , +c0 = 4 +� +d2zd2z′� +∂z∂z′D∂¯zD∂¯z′D + ∂z∂¯z′D∂¯zD∂z′D ++∂¯z∂¯z′D∂zD∂z′D + ∂¯z∂z′D∂zD∂¯z′D +� +. +(3.15) +2Note that in [16] the authors used a regularization based on a cutoff on the upper limits of the +sums over eigenmodes of the Laplace operator on the torus. +11 + +Integrating by parts and using the regularization indicated above, we obtain +c0 = 1 +4π ln ε + O(1) , +Z ∼ 1 + 1 +2α′� +ln ε + O(1) +� +R + O(α′2) . +To calculate the coefficients of the R2, R2 +µν, and R2 +µαβν terms we note that +aR2 +bR2 +µν +cR2 +µαβν = (a+ 1 +2b+c)∂µ∂νhµν∂α∂βhαβ +(a+ 1 +4b)∂2hµ +µ∂2hα +α +... (3.16) +On the other hand, the coefficient a is in fact known (it is related to the coefficient of +the R term), since R and R2 effectively arise from the expansion of the exponential eR. +Thus +a = 1 +8 ln2 ε + O(ln ε) . +(3.17) +We note that like the finite part of the c0 in (3.15), the coefficients of ln ε terms in a +and b are not unique, i.e. depend on a regularization scheme.3 Finding the coefficient +λ1 = a + 1 +2b + c and λ2 = a + 1 +4b in (3.16) and using (3.17), we can calculate b and c. +Expanding (3.14) to order ∂4h, we obtain +λ1 = 32 +� +d2zd2z′∂zD∂¯zD∂z′D∂¯z′D , +λ2 = 4 +� +d2zd2z′∂z∂¯zD +�� +z=z′∂z′∂¯z′D +�� +z=z′ +� +D2(z, z) + D2(z, z′)) . +(3.18) +From this it follows that +λ1 = 1 +4 ln ε + O(1) , +λ2 = 1 +16 ln2 ε + O(ln ε) , +b = −1 +4 ln2 ε + O(ln ε) , +c = 1 +4 ln ε + O(1) . +(3.19) +Thus, the expression for Z has the form +Z = Z1 +� +[dτ] +� +dDy +√ +G +� +1 + 1 +2α′ ln εR ++ 1 +8α′2 ln2 εR2 − 1 +4α′2 ln2 εR2 +µν + 1 +4α′2 ln εR2 +µαβν + . . . +� +. (3.20) +The coefficient of R2 +µαβν is consistent with the renormalizability of Z (cf. (2.44) with +χ = 0 and (2.42)), as it is easily seen from the well known expression [12] for the +two-loop βG-function +0 = +dZ +d ln ε = +∂Z +∂ ln ε − βG +µν +∂Z +∂Gµν +, +βµν +G = α′Rµν + 1 +2α′2RµαβγRν +αβγ + O(α′3) . +Note that, in fact, we have effectively calculated the local βG-function of the σ-model +on a torus. It coincides with that on a sphere, as expected. The direct calculation +of βG on a torus was also performed in [16]. Compared to [16] where cumbersome +expressions arose and cutoff regularization of the sums was applied, our calculation +using Z is rather simple. We should stress that the possibility of deriving βG from Z +3Note that the ambiguity of the ln ε terms in a and b does not affect the coefficient c as a+ 1 +2b ∼ O(1). +12 + +is a distinctive feature of the torus geometry: there is an R2 term in the dilaton βφ as +well, but for the torus the e−χφ factor is trivial as χ = 0. +The above method of calculating Z illustrated on the example of the torus which +is based on the use of the trivial measure for xµ, an expansion in hµν = Gµν − δµν, and +a special prescription for subtracting power divergences that ensures the covariance of +Z, is closest in spirit to the usual method of calculating string scattering amplitudes +as correlators of vertex operators. +This approach can be generalized to surfaces of higher genus (where, to ensure +invariance it is necessary to discard δ(2)(z, z) altogether, i.e. to discard the ε−2 diver- +gence and the finite part 1 +6χ term in (2.7)). Integrating by parts in (3.18), one can +prove that the prescription δ(2)(z, z) = 0 is sufficient to verify the universality of the +coefficients of the ln2 ε terms in a and b in (3.16). Note that though the value of the +coefficient of ln ε in λ2 in the general case depends on a choice of regularization, the +value of c(4 − χ) ln ε is the same in all regularizations that preserve the covariance (for +example, in dimensional and in δ(2)(z, z) = 0 regularizations). +4 +Partition function of the N = 1 supersymmetric +σ-model +Let us generalize the results of Sec.2 to the case of the supersymmetric 2d σ-model +related to fermionic (NSR) string in curved background. The important difference from +the bosonic case is the automatic cancellation of power UV divergences. +The action of a fermionic string in flat space is given by (see, e.g., [19]) +I = +1 +2πα′ +� +d4z E D−ˆxµD+ˆxµ , +(4.1) +where d4zE = d2σdθd¯θ sdetEA +M, ˆxµ is a scalar superfield, D− and D+ are superderiva- +tives, and (σ1, σ2, θ, ¯θ) are the coordinates on the supersurface. +For the action of the corresponding supersymmetric σ-model we have +I = +1 +2πα′ +� +d4z E D−ˆxµD+ˆxν Gµν(ˆx) + i +2π +� +d4z E R+− φ(ˆx) . +(4.2) +Here R+− are the components of the two-dimensional curvature tensor, and Gµν and φ +are the graviton and dilaton fields. Note that the Euler characteristic can be written +also as +χ = i +2π +� +d4z E R+− . +(4.3) +The component expansion of ˆxµ is +ˆxµ = xµ + θrψµ +r + iθ¯θF µ . +(4.4) +We shall use the antiperiodic boundary conditions for the field ψ +ψ(ϕ + 2π) = −ψ(ϕ) . +(4.5) +Here ϕ is the polar angle in the complex plane or angle of a cylinder. In this case the +Dirac operator does not have zero modes (but the scalar Laplace operator has). On +13 + +surfaces of higher genera this choice of boundary conditions corresponds to an even +spin structure for ψ +ψ(z + ai) = −ψ(z) , +ψ(z + bi) = −ψ(z) , +(4.6) +where i = 1, . . . , g, and ai and bi are the basis cycles on the Riemann surface. +We shall use the supersymmetric generalization of the heat kernel method used in +Sec.2. The expressions (2.7) and (2.8) become +ˆN′ = +� +d4z E ˆKε − 1 = 1 +2χ − 1 + O(ε2) , +(4.7) +ˆKε = Kε(σ, σ)δ2(θ, θ) = i +4πR+− + O(ε2) . +(4.8) +Note that −1 in (4.7) corresponds to the bosonic zero mode yµ = const. As already +mentioned, in contrast to the bosonic case, here the ε−2 divergence is absent which is a +manifestation of the two-dimensional supersymmetry which also forbids the standard +tachyon term in the σ-model action (cf. (2.2)). +To calculate Z we separate in ˆx the zero mode, ˆx = y + ˆη, Dˆx = dDy Dˆη. The +terms in the action (4.2) that contribute in the one-loop approximation have the form +I = +1 +2πα′ +� +d4z E D−ˆηµD+ˆηνGµν(y) + i +2π +� +d4z E R+−φ(y) . +(4.9) +Analogously to (2.9), we obtain +ˆZ0 = ˆ +Z0 exp +�� +1 − 1 +2χ + O(ε2) +� +ln G +� +, +ˆ +Z0 = exp(−1 +2D ln det ′ ˆ∆) . +(4.10) +As in the bosonic case, the factor ( +√ +G)χ can be absorbed into a redefinition of the +dilaton field +φ′ = φ + a ln +� +G(ˆx) . +(4.11) +The value of a is fixed by the condition that ˆZ should be covariant. As a result, a = −1 +2. +To find ˆZ in the two-loop approximation, we choose the integration measure as (cf. +(2.35)) +Dˆx = J dDy Dˆξ , +(4.12) +where J is determined from the normalization condition +� +Dδˆxe−||δˆx||2 = 1 , +||δˆx||2 = +� +d4z E δˆxδˆxν Gµν . +(4.13) +Performing the calculation analogous to the one in the bosonic case and using the +normal coordinates ˆξ, we get +ˆZ = ˆ +Z0 +� +dDy +√ +G e−χφ � +exp +� � +d4z E 1 +3πα′RµανβDγ ˆξµDγ ˆξν ˆξαˆξβ +−πα′�1 +3 +ˆK′ +ε(z, z) + 1 +V +� +Rαβ ˆξαˆξβ + O(α′2) +�� +, +(4.14) +14 + +where ⟨...⟩ is computed with the free gaussian action for the normal coordinate fields +ˆξα. As in the bosonic case, we have redefined ˆξ → (2πα′)1/2ˆξ, taken the one-loop +contribution into account, and have chosen φ=const. As a consequence, +ˆZ = ˆ +Z0 +� +dDy +√ +G e−χφ � +1 − πα′ ˆD(z, z)R + O(α′2) +� +. +(4.15) +Using the regularized expression for ˆD(z, z) (see, e.g., [19]) +ˆD(z, z) = − 1 +2π ln ε + O(1) , +(4.14) becomes +ˆZ = ˆ +Z0 +� +dDy +√ +G e−χφ � +1 + 1 +2α′� +ln ε + O(1) +� +R + O(α′2) +� +, +(4.16) +which (at this leading oder in α′) coincides with the bosonic string expression in (2.20). +For the case of the sphere with a nontrivial dilaton field we get +ˆZ = ˆ +Z0 +� +dDy +√ +G e−2φ � +1 + 1 +2α′� +ln ε + O(1) +�� +R + 2GµνDµDνφ +� ++ O(α′2) +� +. (4.17) +Applying the +∂ +∂ ln ε prescription for “dividing” over the volume of the super-M¨obius +group we obtain +∂ ˆZ +∂ ln ε = 1 +2α′ ˆ +Z0 +� +dDy +√ +G e−2φ � +R + 2D2φ + O(α′) +� +, +(4.18) +that agrees with the expression for the superstring effective action (same as bosonic +action to this order in (3.7)) found using the S-matrix approach. +15 + +References +[1] C. Lovelace, “Strings in Curved Space,” Phys. Lett. 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Tseytlin, “Renormalization of Mobius Infinities and Partition Function Represen- +tation for String Theory Effective Action,” Phys. Lett. B 202 (1988), 81-88. +O. D. Andreev and A. A. Tseytlin, “Generating Functional for Scattering Amplitudes +and Effective Action in the Open Superstring Theory,” Phys. Lett. B 207 (1988), 157- +163 +[6] J. Scherk and J. H. Schwarz, “Dual Models for Nonhadrons,” Nucl. Phys. B 81 (1974), +118-144. +T. Yoneya, “Connection of Dual Models to Electrodynamics and Gravidynamics,” Prog. +Theor. Phys. 51 (1974), 1907-1920 +[7] S. Mandelstam, in Unified String Theories, Proceedings of the Santa Barbara Workshop, +edited by M. Green and D. Gross (World Scientific, Singapore, 1986), p. 526 +P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, “A Simple Expression for the Multiloop +Amplitude in the Bosonic String,” Phys. Lett. B 199 (1987), 49-56; “N String Vertex +and Loop Calculation in the Bosonic String,” Nucl. Phys. B 298 (1988), 527 +[8] V. G. 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Tseytlin, “Partition Function of String σ Model on a Compact Two Space,” Phys. +Lett. B 223 (1989), 165-174 +[12] D. Friedan, “Nonlinear Models in Two Epsilon Dimensions,” Phys. Rev. Lett. 45 (1980), +1057; “Nonlinear Models in Two + Epsilon Dimensions,” Annals Phys. 163 (1985), 318 +[13] B.S. DeWitt, in General Relativity: An Einstein Centenary Survey, edited by S. Hawking +and S. Israel (Cambridge University Press, 1979). +[14] L. Alvarez-Gaume, D. Z. Freedman and S. Mukhi, “The Background Field Method and +the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model,” Annals Phys. +134 (1981), 85 +[15] A. A. Tseytlin, “Conditions of Weyl Invariance of Two-dimensional σ Model From Equa- +tions of Stationarity of ’Central Charge’ Action,” Phys. Lett. B 194 (1987), 63; Phys. +Lett. B 178 (1986), 34. +H. Osborn, “Renormalization and Composite Operators in Nonlinear σ Models,” Nucl. +Phys. B 294 (1987), 595-620 +[16] I. G. Koh and H. J. Shin, “World sheet topology and target manifold in string theory,” +Phys. Rev. D 36 (1987), 1773 +[17] J. Polchinski, “Evaluation of the One Loop String Path Integral,” Commun. Math. Phys. +104 (1986), 37 +[18] D. Mumford, Tata Lectures on Theta, Vols.1, 2 (Birkh¨auser, Basel, 1983). +[19] E. D’Hoker and D. H. Phong, “The Geometry of String Perturbation Theory,” Rev. +Mod. Phys. 60 (1988), 917 +17 + diff --git a/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/load_file.txt b/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..853e1edd5070d2678ed5186dc2c327b37daf9d31 --- /dev/null +++ b/-tE1T4oBgHgl3EQfCwIW/content/tmp_files/load_file.txt @@ -0,0 +1,611 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf,len=610 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='02867v1 [hep-th] 7 Jan 2023 Covariant calculation of the partition function of the two-dimensional sigma model on compact two-surfaces O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Andreev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Metsaev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Tseytlin Department of Theoretical Physics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Lebedev Physical Institute, Leninski prospect 53, Moscow 117924, USSR (submitted 17 July 1989) Abstract Motivated by string theory connection, a covariant procedure for perturbative calculation of the partition function Z of the two-dimensional generalized σ- model is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The importance of a consistent regularization of the measure in the path integral is emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The partition function Z is computed for a number of specific 2-manifolds: sphere, disk and torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Published in: Yad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 51 (1990) 564-576 [Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 51 (1990) 359-366] 1 Contents 1 Introduction 2 2 Calculation of partition function of σ-model on compact 2-surfaces 3 3 Partition function on specific 2-surfaces: sphere, disk and torus 9 4 Partition function of the N = 1 supersymmetric σ-model 13 1 Introduction A promising approach to string theory is the so-called σ-model approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It may help elucidate the structure and first principles of string theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' A central role in the σ-model approach is played by the partition function Z of the generalized two-dimensional σ-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Z is closely related to the generation functional for the string S-matrix and to the effective action of the string theory [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The string partition function differs from the usual σ-model partition function by a factor of the volume of the M¨obius group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In the theory of closed strings a possible implementation of the operation of division by the M¨obius group volume is by taking the derivative with respect to the log of the UV cutoff ∂ ∂ ln ε of the regularized partition function ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The reason for this is the presence of a logarithmic divergence [3] in the regularized volume ΩR of the M´obius group [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In the theory of open strings the procedure of “division” by the M¨obius volume reduces to a renormalization of power divergencies as the regularized volume of the M¨obius group SL(2, R) contains only power divergences [5, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The remaining loga- rithmic two-dimensional UV divergences can be interpreted as being due to the mass- less poles in the scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As a result, the renormalized string partition function coincides with the effective action S for the massless modes of the open string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We shall perform the calculation of the partition function of the two-dimensional σ-model on compact surfaces emphasizing the role of the measure in the functional integral in the procedure of calculating the covariant expression for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2 we consider three possible ways of determining the regularized measure that lead to a covariant answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3 we give examples of the calculation of the leading terms in Z for some specific cases of 2-manifolds: the sphere, disk (hemisphere), and the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Taking into account the procedure for dividing by the M¨obius volume, we obtain an alternative to the S-matrix method of [6] for calculating the string effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4 we consider a generalization of this approach to the supersymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us make a comment on the interpretation of infinities that are present in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In addition to the already mentioned M¨obius and other two-dimensional UV infinities, in the case of 2-surfaces of higher genera there exist the so-called modular infinities corresponding to degeneration of the Riemann surfaces [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1 The “modular’ correction to the β-functions corresponds to the infinities associated with the degeneration of trivial cycles [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The partition function Z should be renormalizable with respect 1In the framework of the σ-model approach, in the case of surfaces of higher genera it is necessary to use the Schottky [7] or the branch-point type [8] parameterization for the moduli space in which the on-shell scattering amplitudes have formal SL(2, C) invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 2 to all infinities (modular and local), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' it should be finite after the renormalization corresponding to the complete β-function [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 2 Calculation of partition function of σ-model on compact 2-surfaces We shall consider the bosonic σ-model (µ, ν = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=', D) Z = � [Dx] exp � − I(x) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1) I = 1 4πα′ � d2σ√g �α′ ε2ϕ(x) + ∂axµ∂axνGµν(x) + α′R(2)φ(x) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) defined on a compact closed two-dimensional surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Here Gµν, φ, and ϕ are the bare fields that depend on the two-dimensional cutoff ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The renormalized value of ϕ will be chosen to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The theory is defined by the action I and the measure [Dx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Imposing the requirement of invariance under the general coordinate transformations xµ → x′µ , Gµν → G′ µν = ∂xα ∂x′µ ∂xβ ∂x′ν Gαβ , (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' that upon a transformation of xµ the “coupling constants” Gµν of the theory are also transformed), below we shall consider three ways of calculating the partition function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1) that are consistent with the requirement of this covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us first choose the measure [Dx] to be trivial: [Dx] = � σ dDx(σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' To cancel the power divergences we make use of the bare tachyon field ϕ(x) (with the renormalized value of ϕ set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We separate xµ into a constant and a non-constant parts, xµ = yµ + ηµ, inserting “one” into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' [12]) 1 = � dDy � � dDη δ(D)(x(σ) − y − η) δ(D)(P µ[y, η]) Q[y, η] , Q = det ∂P µ[y − a, η + a] ∂aν ��� a=0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3) where P = 0 is a gauge condition and Q is the ghost determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' One possible choice is P µ = � d2σ√g ηµ , Q = V D , V = � d2σ √g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4) The condition P = 0 implies that η does not contain a zero mode of the Laplace operator (a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We substitute x = y +η into the action and expand it in powers of η: I = 1 4πα′ � d2σ√g �α′ ε2ϕ + ∂aηµ∂aην� Gµν + ∂λGµνηλ + 1 2∂λ∂ρGµνηληρ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � +α′R(2)(φ + 1 2∂µ∂νφηµην + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='5) 3 The leading (one-loop) contribution of the integral over η is Z0 = [det ′(Gµν∆)]−1/2 = exp � − 1 2N′ ln G − 1 2D ln det ′∆ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='6) where N′ is the regularized number of nonzero eigenmodes of the Laplace operator, G = det Gµν and D is the dimensionality of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The number N′ can be expressed in terms of the heat kernel in a familiar way N′ = � d2σ √g Kε − 1 = V 4πε2 + 1 6χ + O(ε2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) Kε = � n fn(σ)fn(σ′) exp(−λnε2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='8) where fn(σ) and λn are, respectively, the eigenfunctions and eigenvalues of the Laplace operator on the two-dimensional surface of Euler number χ = 1 4π � d2σ√gR(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Taking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) into account, we obtain for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='6) Z0 = Z0 exp �� − V 4πε2 − 1 6χ + 1 + O(ε2) � ln G � , Z0 = exp � − 1 2D ln det ′∆ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='9) The dependence of Z on the dilaton field (to order α′2) is easily found from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='5): Z = � dDy Z0 e−χφ� 1 − α′πχ∂µ∂νφ Gµν D(σ, σ) + O(α′2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='10) where D is the regularized Green function of the Laplace operator D(σ, σ′) = � λn̸=0 fn(σ)fn(σ′) λn exp � − λnε2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='11) For ε → 0 it has the form [13] D(σ, σ) = − 1 2π ln ε + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='12) To determine the dependence of Z on the graviton field Gµν it is necessary to consider the two possible one-particle-irreducible two-loop diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Their contribution to Z is found to be Z = � dDy Z0 e−χφ� 1 + c1GµνGλρ∂λ∂ρGµν + c2GµαGνβGρλ∂ρGµν∂λGαβ +c3GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) � , c1 = −1 2πα′D(σ, σ)N′ , c2 = 1 2πα′ � d2σ d2σ′√g � g′ D(σ, σ′) ∂a∂b′D(σ, σ′) ∂a∂b′ D(σ, σ′) , c3 = πα′ � d2σ d2σ′ √g � g′ ∂aD(σ, σ′) ∂a∂b′D(σ, σ′) ∂b′D(σ, σ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='13) 4 We ensure the covariance of Z0 by means of the special choice of the bare fields φ′ = φ + a ln � G(x) , ϕ′ = ϕ + b ln � G(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='14) Substituting x = y +η and expanding ln � G(x) in powers of η we obtain the following correction to the action in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) ∆I = 1 4πα′ � d2σ√g α′ � b ε2 + aR(2)�� ln √ G + 1 4Gµν∂λ∂ρGµνηληρ − 1 4GµβGνα∂ρGµν∂λGαβηρηλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='15) The values of a and b are calculated from the condition that Z0 has a required covariant form (Z0 ∼ √ G) a = −1 6 , b = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16) We now find the correction to Z from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='15) taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16) and the final expression for Z0 ∆Z = Z0 � dDy √ G e−χφ� 1 + 1 2πα′�1 6χ + V 4πε2 � D(σ, σ) × � GµνGλρ∂λ∂ρGµν − GµαGνβGρλ∂ρGµν∂λGαβ + O(α′2) �� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='17) As a result, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='13), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='17), we obtain Z = Z0 � dDy √ Ge−χφ� 1 − πα′χD(σ, σ)∂µ∂νφGµν + ˜c1GµνGλρ∂λ∂ρGµν + ˜c2GµαGνβGρλ∂ρGµν∂λGαβ + ˜c3GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) � , ˜c1 = c1 + 1 2πα′(1 6χ + V 4πε2)D(σ, σ) , ˜c2 = c2 − 1 2πα′(1 6χ + V 4πε2)D(σ, σ) , ˜c3 = c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='18) After the ci’s have been calculated using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='12), the power divergences cancel and the dependence of Z on ε takes the form Z = Z0 � dDy √ G e−χφ � 1 + 1 2α′χ(ln ε + O(1))∂µ∂νφ Gµν − 1 4α′(ln ε + O(1))GµνGλρ∂λ∂ρGµν + 1 8α′(ln ε + O(1))GµαGνβGρλ∂ρGµν∂λGαβ + 1 4α′(ln ε + O(1))GµλGνβGρα∂ρGµν∂λGαβ + O(α′2) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='19) Using in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='19) the expression for the target space scalar curvature R in terms of Gµν and integrating by parts we observe that we can rewrite Z in the manifestly covariant form Z = Z0 � dDy √ G e−χφ � 1 + 1 2α′� ln ε + O(1) �� R + χD2φ � + O(α′2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20) 5 where Dµ in D2 is the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Next, let us consider the manifestly covariant method of calculating Z based on the expansion for the action and the measure in normal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us define the measure [Dx] by the formal product [Dx] = � σ dDx(σ) � G(x(σ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='21) To preserve the general covariant invariance in the regularized theory it is necessary to regularize the measure and the action in a consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We choose the regularized expression for the measure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='21) in the form [Dx] = � σ dDx(σ) eM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='22) M = 1 2 � d2σ√g ln G(x)Kε(σ, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='23) Now let us set xµ = yµ + ηµ(y, ξ) where ξµ is the tangent vector to the geodesic joining the points yµ and yµ + ηµ ηµ = ξµ − 1 2Γµ αβξαξβ − 1 6 � ∂γΓµ αβ − 2Γλ γαΓµ λβ � ξαξβξγ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='24) The expansions of the action and measure in powers of ξ have the form [14] I = 1 4πα′ � d2σ√g � ∂aξµ∂aξν� Gµν + 1 3Rµλρνξλξρ + 2 45Rλµρ γRανβγξλξρξαξβ + O(ξ5) � + α′R(2)� φ + Dµφ ξµ + 1 2DµDνφ ξµξν + O(ξ3) �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='25) M = 1 2 � d2σ√g Kε(σ, σ) � ln G − 1 3Rµνξµξν + O(ξ3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='26) Since the kinetic term is invariant under a constant shift ξ → ξ + a and ξ may contain a constant part under the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4), it is desirable to fix the symmetry y → y − a, η → η + a by means of another gauge condition [12] P µ = � d2σ√g ξµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='27) In this case the ghost determinant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3) is Q = det � � d2σ√g λµ ν � , λµ ν = ∂ξµ(y, η) ∂ην − ∂ξν(y, η) ∂ηµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='28) Its covariant expression takes the form Q = V D exp � − 1 3V � d2σ√g Rµνξµξν + O(ξ3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='29) To determine the measure in the y integral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � dDy √ G, it is necessary to take into account not only the one-loop contribution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='6) but also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Using that the regularized number of eigenvalues is N = � d2σ√g Kε(σ, σ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='30) 6 and also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7), we arrive at the expression for the covariant measure √ G in the integral over y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In fact, √ G is the contribution of the only (constant) zero mode of the Laplace operator on the compact surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The partition function Z then takes the form Z = � dDy √ G e−χφ F(R, DR, Dφ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='31) It is not difficult to calculate the first terms of the expansion of F in powers of α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='25), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='26), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='29), we obtain Z = Z0 � dDy √ G e−χφ� 1 + α′(a1 + a2 + a3)R + α′b1D2φ + O(α′2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='32) The coefficients a1 and b1 correspond to contributions from the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='25), a2 arises from the measure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='26), and a3 from the ghost determinant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The expressions for these coefficients in terms of the Green functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='11) have the following appearance a1 = π 3 � d2σ√g ∂a∂′aD(σ, σ′)|σ=σ′D(σ, σ) − ∂aD(σ, σ′)|σ=σ′∂′aD(σ, σ′)|σ=σ′ , a2 = −π 3 � d2σ√gKε(σ, σ)D(σ, σ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='33) a3 = − 2π 3V � d2σ√gD(σ, σ) , b1 = −1 4 � d2σ√gR(2)D(σ, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Explicit calculations give a1 = −1 6N′ ln ε + ¯a1 , a2 = 1 6N ln ε + ¯a2 , a3 = 1 3 ln ε + ¯a3 , b1 = 1 2χ ln ε + ¯b1 , a0 = a1 + a2 + a3 = 1 2 ln ε + ¯a0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='34) where the ¯ai and ¯bi are finite constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It is easy to see that the power infinities cancel, and the resulting expression for Z in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='32) coincides with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us now consider one more method of calculating Z, which is explicitly covariant and turns out to be simpler in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Here we define the measure [Dx] as follows [Dx] = J dDy [Dξ] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='35) where the factor J is fixed from the normalization condition � [Dδx] e−||δx||2 = � dDδy � [Dδξ] J e−||δx||2 = 1 , ||δx||2 = 1 4πα′ � d2σ√g δxµδxν Gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='36) The expression for ||δx||2 expanded in normal coordinates has the form ||δx||2 = 1 4πα′ � d2σ√g � Gµν + 1 3Rµλ1ρ1νξλ1ξρ1 − 2 45Rµλ1ρ1 γ1Rα1νβ1γ1ξλ1ξρ1ξα1ξβ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' �� δyµ + δξµ + 1 3Rµ λ2ρ2κξλ2ξρ2δyκ − 1 45Rµ λ2ρ2γ2Rγ2α2β2κξλ2ξρ2ξα2ξβ2δyκ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' �� δyν + δξν + 1 3Rν λ3ρ3σξλ3ξρ3δyσ − 1 45Rν λ3ρ3γ3Rγ3α3β3σξλ3ξρ3ξα3ξβ3δyσ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 7 Integrating successively over δy and δξ, we find J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Taking into account the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='35) for the action, we have Z = Z0 � dDy √ Ge−χφ� exp � � d2σ√g �πα′ 3 Rµλνρ∂aξµ∂aξνξλξρ − πα′ 3 K′ ε(σ, σ)Rµνξµξν − πα′ V Rµνξµξν − 4π2α′2 45 Rλµρ γRανβγ∂aξµ∂aξνξλξρξαξβ + π2α′2� 4 45K′ ε(σ, σ) + 2 3V � Rµλρ γRµ αβγξλξρξαξβ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � − � d2σd2σ′√g � g′π2α′2�1 9K′ ε 2(σ, σ′) + 2K′ ε(σ, σ′) 9V + 1 V 2 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='37) × RµρλνRµ αβ νξλ(σ)ξρ(σ)ξα(σ′)ξβ(σ′) + O(α′3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We have redefined y → (2πα′)1/2y and ξ → (2πα′)1/2ξ, set φ to be constant for simplic- ity, and took the one-loop contribution into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Starting from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='37), we easily find the expression for the order α′ terms in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It is given by the first three terms in the exponent in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The contribution of the second term cancels that of the first one so that the coefficient of R turns out to be proportional to D(σ, σ), so that as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20) we get Z = Z0 � dDy √ G e−χφ � 1 + 1 2α′(ln ε + const)R + O(α′2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='38) The divergent parts of the coefficients of the R2 and R2 µν terms are calculated in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' One gets for the R2 term Z = Z0 � dDy √ Ge−χφ� 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' + 1 2π2α′2D2(σ, σ)R2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='39) The divergent contribution to the coefficient of the R2 µν term comes effectively only from the vertex −π2α′2 V R2ξξξξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Z = Z0 � dDy √ Ge−χφ� 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' − π2α′2D2(σ, σ)RµνRµν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='40) The methods of computing Z considered above admit a natural generalization to the case of 2d surfaces with boundaries (with free open string or Neumann boundary conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Then the Green function D is replaced by the Neumann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' There are new (linear) power divergencies which can be canceled by a redefinition of the values of the boundary analogs of the tachyon and dilaton couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The σ-model action in this case has the form I = 1 4πα′ � d2σ√g �α′ϕ ε2 + ∂axµ∂axνGµν + α′R(2)φ � + 1 2π � ds �ϕ′ ε + Kφ′� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='41) where K is the extrinsic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It is necessary to set φ = φ′ to ensure that the constant part of the dilaton couples to the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It should be emphasized that the above calculation of Z was done for surfaces of any genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' However, we did not integrate over the moduli space of the Riemann surfaces and, therefore, the logarithmic divergences found are only the ordinary local ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 8 The expression Z is renormalizable with respect to these local infinities on a surface of an arbitrary genus (ψi = (G, φ)) dZ d ln ε = ∂Z ∂ ln ε − βi ∂Z ∂ψi = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='42) where βi = − d d ln εψi are the local β-functions of the σ-model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20)) βG µν = α′Rµν + O(α′2) , βφ = 1 6D − 1 2α′D2φ + O(α′2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='43) Assuming that Z is renormalizable also at the next order and using the known expressions for the α′2 terms in the β-functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='43) [12, 15], we find the following expression for the logarithmically divergent term in Z to order α′2 Z = λ � dDy √ G e−χφ � 1 + 1 2 ln ε � α′R + 1 8(4 − χ) α′2RµαβνRµαβν� + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='44) We shall also confirm the coefficient of the RµαβνRµαβν term directly in the case of the torus (χ = 0) in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us note also that the (ln ε)2 coefficients of R2 and R2 µν that we found in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='39) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='40) are consistent with the renormalizability of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 3 Partition function on specific 2-surfaces: sphere, disk and torus Let us now consider the calculation of Z for some simplest surfaces: the sphere, disk, and torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In these cases the coefficients of the leading terms in the α′ expansion of Z can be found explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Let us start with the 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In spherical coordinates the eigenfunctions and eigenvalues of the Laplace operator have the form fn,m = Yn,m(θ, φ) , λn,m = n(n + 1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1) where the Yn,m are the orthonormal spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The regularized expression for the Green’s function has the form D(σ, σ′) = � n̸=0 n � m=−n 1 n(n + 1)e−n(n+1)ε2 Y ∗ n,m(θ, ϕ)Yn,m(θ′, ϕ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) At coincident points, it becomes D(σ, σ) = 1 4π � n̸=0 2n + 1 n(n + 1)e−n(n+1)ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3) The leading terms in expansion in ε → 0 are easily calculated using the Euler-Maclaurin resummation formula D(σ, σ) = − 1 2π ln ε + γ − 1 4π + ε2 6π + O(ε4) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4) 9 where γ is the Euler constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We note that the ln ε and ε2 terms can be calculated from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) using integration over ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Taking (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4) into account, we can write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20) as (here χ = 2) Z = Z0 � dDy √ Ge−2φ� 1 + α′� R + 2D2φ) �1 2ln ε + a + O(ε2) � + O(α′2) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='5) where a = 1 4(γ − 1) is a scheme-dependent constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It is easy to see that Z is renormalizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' making the replacement Gµν = G(R) µν −ln ε βG µν and φ = φ(R)−ln ε βφ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='43)) we get rid of the logarithmic divergences and thus find Z = Z0 � dDy √ Ge−2φ� 1 + aα′� R + 2D2φ) + O(α′2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='6) Note that this expression is not the same as the closed string effective action obtained using the S-matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The reason is that the generating functional for the string tree-level S-matrix is given by Ω−1Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e Z divided by the volume of the group SL(2, C) of M¨obius transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The presence of a logarithmic singularity in the regularized volume of SL(2, C) suggest that one can think of ∂ ∂ ln ε as a possible realization of the operation of division by Ω in the case of closed strings [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Indeed, as follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='5), ∂Z ∂ ln ε = 1 2α′Z0 � dDy √ G e−2φ � R + 2D2φ + O(α′) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) which agrees with the effective action found from the tree-level closed string S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The calculation of Z for disk topology (with a metric of half-sphere) almost analogous to the case of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' A new feature is that in view of the presence of the boundary, we impose the Neumann boundary condition at the boundary of half-sphere ∂θxµ�� θ= π 2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='8) The expansion of the fluctuation field η in eigenfunctions of the Laplace operator on the disk has the form η(θ, φ) = � n,m an,mYn,m , n + m = 2k , n ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='9) and the expression for the regularized Neumann function at coincident points is (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3)) D(σ, σ) = ∞ � n=1 1 2πn e−n(n+1)ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='10) Using the Euler-Maclaurin formula, we obtain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4)) D(σ, σ) = − 1 2π ln ε + γ 4π + 1 4ε − 5 12πε2 + O(ε3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='11) The expression for Z is the same as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='18), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20) with D(σ, σ) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The power divergences ε−2 and ε−1 in Z on the disk are canceled by renormalizing the tachyon fields ϕ and ϕ′, respectively (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='41)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In the case of the 2-torus we shall depart from the scheme used above, which was based on the heat kernel regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' This is due to the technical difficulties of calculating the sums with the spectral e−λnε2 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2 We shall consider the τ-parametrization, in which the torus is represented as a (τ, 1) parallelogram on the complex z-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The string σ-model partition function on the torus has the form [17] Z = � F d2τ 4πτ 2 2 e4πτ2 (2πτ2)12|f(e2πiτ)|−48 � Dx e−I , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='12) � Dx e−I0 = 1 , I = I0 + Iint , f(e2πiτ) = ∞ � n=1 (1 − e2πinτ) , where the fundamental region F is specified by the conditions −1 2 < τ1 ≤ 1 2, |τ| > 1 , τ = τ1 + iτ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We shall consider only the dependence of Z on the metric Gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' By studying the dependence of Z on G|muν = δµν + hµν using the expansion in powers of hµν, we will then restore the coefficients of the R, R2, R2 µν and R2 µαβν terms (assuming that the scheme used for the regularization and renormalization preserves the covariance of Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Since the metric on the parallelogram is flat, it is possible to use the following regularization prescription D(z, z) = − 1 2π ln ε , δ(2)(z, z) = 0 , corresponding to discarding of power divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' This prescription ensures the co- variance of Z without need for a nontrivial measure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The Green function on the torus has the form [17] D(z, z′) = − 1 4π ln |θ(z, z′)|2 |θ′(0)|2 + 1 2τ2 � Im(z − z′) �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='13) where θ(z, z′) is the theta function ϑ11(z, z′) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We redefine x → (2πα′)1/2x and expand the σ-model action I in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) in powers of η = x − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Then Z = ⟨Z⟩ , Z = � Dη exp � − 1 2 � d2σ√g∂aηµ∂aην� δµν + hµν + (2πα′)1/2∂λhµνηλ +πα′∂ρ∂λhµνηρηλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' �� , ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='⟩ = � dDy � [dτ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='14) The coefficient of R is found from the hµν□hµν term (R = 1 4hµν□hµν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As a result, Z ∼ 1 + 2πα′c0R + O(α′2) , c0 = 4 � d2zd2z′� ∂z∂z′D∂¯zD∂¯z′D + ∂z∂¯z′D∂¯zD∂z′D +∂¯z∂¯z′D∂zD∂z′D + ∂¯z∂z′D∂zD∂¯z′D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='15) 2Note that in [16] the authors used a regularization based on a cutoff on the upper limits of the sums over eigenmodes of the Laplace operator on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 11 Integrating by parts and using the regularization indicated above, we obtain c0 = 1 4π ln ε + O(1) , Z ∼ 1 + 1 2α′� ln ε + O(1) � R + O(α′2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' To calculate the coefficients of the R2, R2 µν, and R2 µαβν terms we note that aR2 +bR2 µν +cR2 µαβν = (a+ 1 2b+c)∂µ∂νhµν∂α∂βhαβ +(a+ 1 4b)∂2hµ µ∂2hα α +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16) On the other hand, the coefficient a is in fact known (it is related to the coefficient of the R term), since R and R2 effectively arise from the expansion of the exponential eR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Thus a = 1 8 ln2 ε + O(ln ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='17) We note that like the finite part of the c0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='15), the coefficients of ln ε terms in a and b are not unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' depend on a regularization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3 Finding the coefficient λ1 = a + 1 2b + c and λ2 = a + 1 4b in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='17), we can calculate b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Expanding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='14) to order ∂4h, we obtain λ1 = 32 � d2zd2z′∂zD∂¯zD∂z′D∂¯z′D , λ2 = 4 � d2zd2z′∂z∂¯zD �� z=z′∂z′∂¯z′D �� z=z′ � D2(z, z) + D2(z, z′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='18) From this it follows that λ1 = 1 4 ln ε + O(1) , λ2 = 1 16 ln2 ε + O(ln ε) , b = −1 4 ln2 ε + O(ln ε) , c = 1 4 ln ε + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='19) Thus, the expression for Z has the form Z = Z1 � [dτ] � dDy √ G � 1 + 1 2α′ ln εR + 1 8α′2 ln2 εR2 − 1 4α′2 ln2 εR2 µν + 1 4α′2 ln εR2 µαβν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20) The coefficient of R2 µαβν is consistent with the renormalizability of Z (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='44) with χ = 0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='42)), as it is easily seen from the well known expression [12] for the two-loop βG-function 0 = dZ d ln ε = ∂Z ∂ ln ε − βG µν ∂Z ∂Gµν , βµν G = α′Rµν + 1 2α′2RµαβγRν αβγ + O(α′3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Note that, in fact, we have effectively calculated the local βG-function of the σ-model on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' It coincides with that on a sphere, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The direct calculation of βG on a torus was also performed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Compared to [16] where cumbersome expressions arose and cutoff regularization of the sums was applied, our calculation using Z is rather simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We should stress that the possibility of deriving βG from Z 3Note that the ambiguity of the ln ε terms in a and b does not affect the coefficient c as a+ 1 2b ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 12 is a distinctive feature of the torus geometry: there is an R2 term in the dilaton βφ as well, but for the torus the e−χφ factor is trivial as χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The above method of calculating Z illustrated on the example of the torus which is based on the use of the trivial measure for xµ, an expansion in hµν = Gµν − δµν, and a special prescription for subtracting power divergences that ensures the covariance of Z, is closest in spirit to the usual method of calculating string scattering amplitudes as correlators of vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' This approach can be generalized to surfaces of higher genus (where, to ensure invariance it is necessary to discard δ(2)(z, z) altogether, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' to discard the ε−2 diver- gence and the finite part 1 6χ term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Integrating by parts in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='18), one can prove that the prescription δ(2)(z, z) = 0 is sufficient to verify the universality of the coefficients of the ln2 ε terms in a and b in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Note that though the value of the coefficient of ln ε in λ2 in the general case depends on a choice of regularization, the value of c(4 − χ) ln ε is the same in all regularizations that preserve the covariance (for example, in dimensional and in δ(2)(z, z) = 0 regularizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 4 Partition function of the N = 1 supersymmetric σ-model Let us generalize the results of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2 to the case of the supersymmetric 2d σ-model related to fermionic (NSR) string in curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The important difference from the bosonic case is the automatic cancellation of power UV divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The action of a fermionic string in flat space is given by (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=', [19]) I = 1 2πα′ � d4z E D−ˆxµD+ˆxµ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='1) where d4zE = d2σdθd¯θ sdetEA M, ˆxµ is a scalar superfield, D− and D+ are superderiva- tives, and (σ1, σ2, θ, ¯θ) are the coordinates on the supersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' For the action of the corresponding supersymmetric σ-model we have I = 1 2πα′ � d4z E D−ˆxµD+ˆxν Gµν(ˆx) + i 2π � d4z E R+− φ(ˆx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) Here R+− are the components of the two-dimensional curvature tensor, and Gµν and φ are the graviton and dilaton fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Note that the Euler characteristic can be written also as χ = i 2π � d4z E R+− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='3) The component expansion of ˆxµ is ˆxµ = xµ + θrψµ r + iθ¯θF µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='4) We shall use the antiperiodic boundary conditions for the field ψ ψ(ϕ + 2π) = −ψ(ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='5) Here ϕ is the polar angle in the complex plane or angle of a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' In this case the Dirac operator does not have zero modes (but the scalar Laplace operator has).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' On 13 surfaces of higher genera this choice of boundary conditions corresponds to an even spin structure for ψ ψ(z + ai) = −ψ(z) , ψ(z + bi) = −ψ(z) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='6) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' , g, and ai and bi are the basis cycles on the Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' We shall use the supersymmetric generalization of the heat kernel method used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='8) become ˆN′ = � d4z E ˆKε − 1 = 1 2χ − 1 + O(ε2) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) ˆKε = Kε(σ, σ)δ2(θ, θ) = i 4πR+− + O(ε2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='8) Note that −1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7) corresponds to the bosonic zero mode yµ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As already mentioned, in contrast to the bosonic case, here the ε−2 divergence is absent which is a manifestation of the two-dimensional supersymmetry which also forbids the standard tachyon term in the σ-model action (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' To calculate Z we separate in ˆx the zero mode, ˆx = y + ˆη, Dˆx = dDy Dˆη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' The terms in the action (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='2) that contribute in the one-loop approximation have the form I = 1 2πα′ � d4z E D−ˆηµD+ˆηνGµν(y) + i 2π � d4z E R+−φ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='9) Analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='9), we obtain ˆZ0 = ˆ Z0 exp �� 1 − 1 2χ + O(ε2) � ln G � , ˆ Z0 = exp(−1 2D ln det ′ ˆ∆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='10) As in the bosonic case, the factor ( √ G)χ can be absorbed into a redefinition of the dilaton field φ′ = φ + a ln � G(ˆx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='11) The value of a is fixed by the condition that ˆZ should be covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As a result, a = −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' To find ˆZ in the two-loop approximation, we choose the integration measure as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='35)) Dˆx = J dDy Dˆξ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='12) where J is determined from the normalization condition � Dδˆxe−||δˆx||2 = 1 , ||δˆx||2 = � d4z E δˆxδˆxν Gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='13) Performing the calculation analogous to the one in the bosonic case and using the normal coordinates ˆξ, we get ˆZ = ˆ Z0 � dDy √ G e−χφ � exp � � d4z E 1 3πα′RµανβDγ ˆξµDγ ˆξν ˆξαˆξβ −πα′�1 3 ˆK′ ε(z, z) + 1 V � Rαβ ˆξαˆξβ + O(α′2) �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='14) 14 where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='⟩ is computed with the free gaussian action for the normal coordinate fields ˆξα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As in the bosonic case, we have redefined ˆξ → (2πα′)1/2ˆξ, taken the one-loop contribution into account, and have chosen φ=const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' As a consequence, ˆZ = ˆ Z0 � dDy √ G e−χφ � 1 − πα′ ˆD(z, z)R + O(α′2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='15) Using the regularized expression for ˆD(z, z) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=', [19]) ˆD(z, z) = − 1 2π ln ε + O(1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='14) becomes ˆZ = ˆ Z0 � dDy √ G e−χφ � 1 + 1 2α′� ln ε + O(1) � R + O(α′2) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='16) which (at this leading oder in α′) coincides with the bosonic string expression in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' For the case of the sphere with a nontrivial dilaton field we get ˆZ = ˆ Z0 � dDy √ G e−2φ � 1 + 1 2α′� ln ε + O(1) �� R + 2GµνDµDνφ � + O(α′2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='17) Applying the ∂ ∂ ln ε prescription for “dividing” over the volume of the super-M¨obius group we obtain ∂ ˆZ ∂ ln ε = 1 2α′ ˆ Z0 � dDy √ G e−2φ � R + 2D2φ + O(α′) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='18) that agrees with the expression for the superstring effective action (same as bosonic action to this order in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content='7)) found using the S-matrix approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 15 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Lovelace, “Strings in Curved Space,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' B 135 (1984), 75-77;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' “Stability of String Vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' A New Picture of the Renormalization Group,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' B 273 (1986), 413-467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} +page_content=' E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfCwIW/content/2301.02867v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 04992aad28550da825dde293907a5c9cc5481f59..42ac6de76ee62c452ca2065687dc073813a58e9d 100644 --- a/.gitattributes +++ b/.gitattributes @@ -258,3 +258,5 @@ KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf filter=lfs diff=lfs merge=lfs -tex k9FIT4oBgHgl3EQfriuF/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text hNE4T4oBgHgl3EQfrQ2w/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf filter=lfs diff=lfs merge=lfs -text +ptAzT4oBgHgl3EQf5v4s/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf b/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..41efbc99f5932fe743a478c44b2fa708810e107e --- /dev/null +++ b/1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:35e941e24a3835698670a18e4793195f914cfb58826b28e6a5732faf46610fac +size 1353780 diff --git a/2NFAT4oBgHgl3EQfkh06/content/tmp_files/2301.08611v1.pdf.txt b/2NFAT4oBgHgl3EQfkh06/content/tmp_files/2301.08611v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9bc88583b4824bf2c4d91a9c62ccd0a1631198d --- /dev/null +++ b/2NFAT4oBgHgl3EQfkh06/content/tmp_files/2301.08611v1.pdf.txt @@ -0,0 +1,265 @@ +Verification of crossbar-based lattice through +modeling technique +Rajesh Kumar Datta +Dept. of Electrical and Computer Engineering +University of Texas at Dallas +Dallas,Texas, USA +rajesh.datta@utdallas.edu +Abstract—The use of Nano crossbar-based switching lattice +implementation of Boolean functions has been proposed as an +alternative to traditional CMOS-based implementations [1] in +digital circuits. As Moore’s law is expected to come to an end soon +[2], the use of crossbar-based switching lattice implementation +may be a solution to meet the demands of future electronic +designs. In recent years, various methods and tools have been +proposed for implementing boolean functions with crossbar struc- +tures. In this work, a method for verifying crossbar-based lattice +has been proposed and implemented. This kind of verification +will be necessary for any design that utilizes this crossbar-based +implementation of boolean logic. +Index Terms—Cross-bar-based lattice, lattice modeling. +I. INTRODUCTION +Four-terminal switching networks, also known as crossbar- +based circuits, are a potential alternative to traditional CMOS- +based circuits in the semiconductor industry as the latter +face limits in terms of further miniaturization. Four-terminal +switching networks have four terminals per switch, rather than +the two in traditional CMOS switches, and can be used to +implement the same Boolean functions using fewer switches. +This technology has the potential to address the end of Moore’s +Law, which refers to the trend of decreasing CMOS transistor +dimensions [1] [3]. Two-terminal switches can be used to +implement Boolean functions through switching networks that +are arranged in series or parallel configurations. Each switch +is controlled by a Boolean literal, with a value of ‘1’ turning +the switch ON and ‘0’ turning it OFF. The computation of the +Boolean function is performed by taking the product of the +literals in each valid path through the switching network. In +contrast, four-terminal switches have four terminals arranged +in a rectangular lattice and can be either mutually ON or +mutually OFF. These switches can also be controlled by +Boolean literals, with a value of ‘1’ turning the switch ON +and ‘0’ turning it OFF. Four-terminal switches offer more +flexibility in terms of connections, as they can be made at the +four sides of the rectangular switch. These networks, known as +switching lattices, can be used to implement the same Boolean +functions as two-terminal switches. Different methods ( [4] [5] +[6] ) show the implementation of the boolean function with +four terminal switching lattices. In this work, a method has +been presented for verifying the accuracy of crossbar-based +(four-terminal) implementations of a given function. This is +ON +a +c +b +d +TOP +BOTTOM +OFF +ON +OFF +Function= ab + cd +a +c +b +d +TOP +BOTTOM +Switching Lattice +Cross-bar +based switch +Two-terminal +switch +Fig. 1. Two-terminal switches have two connection points and can be either +on or off, depending on whether the terminals are closed or open, respectively. +In contrast, crossbar switches have four terminals and can be used to form a +switching lattice, which is a network of these switches. The valid connections +from the top to bottom plates of the switching lattice can be used to implement +the PRODUCT terms of any function. +accomplished by modeling the characteristics of the lattice +and comparing the output with the expected result. By doing +so, it is possible to determine whether the implementation is +correct or not. +II. FOUR TERMINAL-BASED NETWORK +The idea of using two-dimensional arrays of four-terminal +switches is not new. Akers introduced the four-terminal switch +model in a seminal paper in 1972 [7]. In recent years, the +four-terminal switch model has gained renewed attention due +to advances in technology [8] [9]. Boolean functions can be +implemented using crossbar-type switches [10], [11]. Figure 1 +and 2 summarize the basic concept. +III. VERIFICATION PROCESS THROUGH MODELING OF +LATTICE NETWORK +Given a lattice network, the position of the Boolean liter- +als can be identified. However, as the size of the network +increases, it becomes more difficult to verify the function +implemented by the network due to the rapid increase in +the number of Sum-of-Products (SOP) terms. To simplify the +process of verifying the Boolean function implemented by a +given lattice network, we can create a model of the network +and find the SOPs implemented by it. This can make the +verification process easier. +Formation of a Lattice +arXiv:2301.08611v1 [cs.ET] 20 Jan 2023 + +d +b +c +f +e +a +b +f +a +b +c +a +d +b +e +TOP +c +f +BOTTOM +3 by 3 network +Implemented function: +g h i + g h e f+ g h e b c +d e f ++d e b c+d e h i +a b c + a b e f ++a b e h i +3 by 2 network +a +d +b +e +TOP +f +BOTTOM +c +g +h +i +Fig. 2. +(Left side) The implementation of the function X = abc+ abef+ +debc+ def can be performed using either crossbar-based or two-terminal-based +circuits. The crossbar-based implementation requires 6 switches, while the +two-terminal-based implementation requires 11 switches [12]. Lines from Top +to Bottom shows the path which implements product terms of the function. +In larger functions, the size of the network can be significantly reduced by +using crossbar-based circuits. (Right side) A 3 by 3 switching lattice is also +shown in the figure. +In a lattice network with 3 rows and 3 columns (as shown +in Figure 2), there are a total of 9 four-terminal switches. +The top part of the lattice is referred to as ‘TOP’ and the +bottom part is referred to as ‘BOTTOM’. Every path that +connects the TOP to the BOTTOM is considered a Product +term of the Boolean function being implemented by the lattice +network. The lattice network can be represented as a graph, +with each switch being a node. To find the paths through the +lattice network, a Depth-first search (DFS) algorithm is used. +However, not all of the paths found by the DFS algorithm are +valid. Nonvalid paths, called ‘superset paths’, must be removed +from the design in order to accurately represent the Boolean +function being implemented. +Path formation +To generate paths in a 3 by 3 lattice, we begin by checking +the adjacent nodes (or ‘children’) of the source four-terminal +switch. Each four-terminal switch can be connected to a +maximum of four other switches, so each switch can have +up to four children. In the 3 by 3 lattice shown in Figure +2, the source node has three children: ‘a’, ‘d’, and ‘g’. We +then choose one of these children, such as ‘a’, and check its +children. ‘a’ has one child, ‘b’, which has two children: ‘e’ +and ‘c’. If we choose ‘c’, the path reaches the bottom of the +lattice and the product term becomes ‘a b c’. If we choose ‘e’ +instead, it has two children: ‘h’ and ‘f’. If we choose ‘f’, the +path reaches the bottom and the product term becomes ‘a b +e f’. This process is repeated recursively until all nodes and +their children have been examined. Nodes that have already +been marked as used in a previous path are not checked again. +Valid path selection To determine if a generated path is a +superset of another path, we have two options: we can either +generate all the paths first and then check for supersets, or we +can check the path as it is being generated. The latter option +is faster, especially for large functions with many paths. When +adding a new node to the path, we can check if it is a child of +any previous node in the path, except for the one that brought +us to the new node. This avoids the need to generate all the +paths and then remove the supersets. +Repetition of literals +To generate the maximum number of paths in a lattice +structure with repeated literals, we can build a basic model +by treating all the literals as distinct and then replacing the +basic literals with the original, repeated literals. We can then +remove any superset paths created by the repetition of literals +to get the final set of paths. To design a model that behaves +like a cross-bar or four-terminal lattice, we need to consider +all these relevant features. Once the model is designed, we +can obtain the boolean function it implements and verify +that it is the same as the target function that the lattice +structure was intended to implement. This will ensure that +the model accurately represents the intended behavior of the +lattice structure. +IV. IMPLEMENTATION +To implement the design, the approach described in earlier +sections was followed. The input for this design was the +positions of the variables (literals) in the lattice network. A +model of the lattice was then created and used to retrieve +the implemented function, following the design rules. As an +example, a 3 by-3 lattice network and its resulting function +were provided in figure 2. By following the steps outlined in +previous sections, it is possible to confirm the accuracy of the +implemented function. It was implemented in ‘C’ language +and the process was very accurate and faster to verify any +kind of cross-bar-based implementation of any function. +V. CONCLUSION +In this study, we presented a method for verifying any +switching lattice network by modeling it. This approach can +be used to verify the implementation of any function with a +cross-bar lattice network. +REFERENCES +[1] Mustafa Altun and Marc D Riedel. Logic synthesis for switching lattices. +IEEE Transactions on Computers, 61(11):1588–1600, 2012. +[2] Manek Dubash. Moore’s law is dead, says gordon moore. Techworld. +com, 13, 2005. +[3] Serzat Safaltin, Oguz Gencer, M Ceylan Morgul, Levent Aksoy, Seba- +hattin Gurmen, Csaba Andras Moritz, and Mustafa Altun. Realization +of four-terminal switching lattices: Technology development and circuit +modeling. In 2019 Design, Automation & Test in Europe Conference & +Exhibition (DATE), pages 504–509. IEEE, 2019. +[4] Levent Aksoy and Mustafa Altun. Novel methods for efficient realization +of logic functions using switching lattices. +IEEE Transactions on +Computers, 69(3):427–440, 2020. +[5] M Ceylan Morg¨ul and Mustafa Altun. Optimal and heuristic algorithms +to synthesize lattices of four-terminal switches. Integration, 64:60–70, +2019. +[6] Muhammed Ceylan Morgul and Mustafa Altun. +Synthesis and opti- +mization of switching nanoarrays. +In 2015 IEEE 18th International +Symposium on Design and Diagnostics of Electronic Circuits & Systems, +pages 161–164, 2015. +[7] Sheldon B Akers. A rectangular logic array. In 12th Annual Symposium +on Switching and Automata Theory (swat 1971), pages 79–90. IEEE, +1971. +[8] Malgorzata Chrzanowska-Jeske, Yang Xu, and Marek Perkowski. Logic +synthesis for a regular layout. VLSI Design, 10(1):35–55, 1999. +[9] Malgorzata Chrzanowska-Jeske and Alan Mishchenko. +Synthesis for +regularity using decision diagrams [logic ic synthesis and layout]. In +2005 IEEE International Symposium on Circuits and Systems, pages +4721–4724. IEEE, 2005. + +[10] Matthew M Ziegler and Mircea R Stan. +Cmos/nano co-design for +crossbar-based molecular electronic systems. +IEEE Transactions on +Nanotechnology, 2(4):217–230, 2003. +[11] Mary M Eshaghian-Wilner, Amar H Flood, Alex Khitun, J Fraser +Stoddart, and Kang Wang. +Molecular and nanoscale computing and +technology. In Handbook of nature-inspired and innovative computing, +pages 477–509. Springer, 2006. +[12] Rajesh Kumar Datta. Implementing boolean functions with switching +lattice networks, 2022. + diff --git a/2NFAT4oBgHgl3EQfkh06/content/tmp_files/load_file.txt b/2NFAT4oBgHgl3EQfkh06/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3d0efeb27efe97d1dd3a3ad7c27a35c5e511035 --- /dev/null +++ b/2NFAT4oBgHgl3EQfkh06/content/tmp_files/load_file.txt @@ -0,0 +1,126 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf,len=125 +page_content='Verification of crossbar-based lattice through modeling technique Rajesh Kumar Datta Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' of Electrical and Computer Engineering University of Texas at Dallas Dallas,Texas, USA rajesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content='datta@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content='edu Abstract—The use of Nano crossbar-based switching lattice implementation of Boolean functions has been proposed as an alternative to traditional CMOS-based implementations [1] in digital circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' As Moore’s law is expected to come to an end soon [2], the use of crossbar-based switching lattice implementation may be a solution to meet the demands of future electronic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In recent years, various methods and tools have been proposed for implementing boolean functions with crossbar struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In this work, a method for verifying crossbar-based lattice has been proposed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This kind of verification will be necessary for any design that utilizes this crossbar-based implementation of boolean logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Index Terms—Cross-bar-based lattice, lattice modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' INTRODUCTION Four-terminal switching networks, also known as crossbar- based circuits, are a potential alternative to traditional CMOS- based circuits in the semiconductor industry as the latter face limits in terms of further miniaturization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Four-terminal switching networks have four terminals per switch, rather than the two in traditional CMOS switches, and can be used to implement the same Boolean functions using fewer switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This technology has the potential to address the end of Moore’s Law, which refers to the trend of decreasing CMOS transistor dimensions [1] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Two-terminal switches can be used to implement Boolean functions through switching networks that are arranged in series or parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Each switch is controlled by a Boolean literal, with a value of ‘1’ turning the switch ON and ‘0’ turning it OFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The computation of the Boolean function is performed by taking the product of the literals in each valid path through the switching network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In contrast, four-terminal switches have four terminals arranged in a rectangular lattice and can be either mutually ON or mutually OFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' These switches can also be controlled by Boolean literals, with a value of ‘1’ turning the switch ON and ‘0’ turning it OFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Four-terminal switches offer more flexibility in terms of connections, as they can be made at the four sides of the rectangular switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' These networks, known as switching lattices, can be used to implement the same Boolean functions as two-terminal switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Different methods ( [4] [5] [6] ) show the implementation of the boolean function with four terminal switching lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In this work, a method has been presented for verifying the accuracy of crossbar-based (four-terminal) implementations of a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This is ON a c b d TOP BOTTOM OFF ON OFF Function= ab + cd a c b d TOP BOTTOM Switching Lattice Cross-bar based switch Two-terminal switch Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Two-terminal switches have two connection points and can be either on or off, depending on whether the terminals are closed or open, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In contrast, crossbar switches have four terminals and can be used to form a switching lattice, which is a network of these switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The valid connections from the top to bottom plates of the switching lattice can be used to implement the PRODUCT terms of any function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' accomplished by modeling the characteristics of the lattice and comparing the output with the expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' By doing so, it is possible to determine whether the implementation is correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' FOUR TERMINAL-BASED NETWORK The idea of using two-dimensional arrays of four-terminal switches is not new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Akers introduced the four-terminal switch model in a seminal paper in 1972 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In recent years, the four-terminal switch model has gained renewed attention due to advances in technology [8] [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Boolean functions can be implemented using crossbar-type switches [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Figure 1 and 2 summarize the basic concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' VERIFICATION PROCESS THROUGH MODELING OF LATTICE NETWORK Given a lattice network, the position of the Boolean liter- als can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' However, as the size of the network increases, it becomes more difficult to verify the function implemented by the network due to the rapid increase in the number of Sum-of-Products (SOP) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' To simplify the process of verifying the Boolean function implemented by a given lattice network, we can create a model of the network and find the SOPs implemented by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This can make the verification process easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Formation of a Lattice arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content='08611v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content='ET] 20 Jan 2023 d b c f e a b f a b c a d b e TOP c f BOTTOM 3 by 3 network Implemented function: g h i + g h e f+ g h e b c +d e f +d e b c+d e h i +a b c + a b e f +a b e h i 3 by 2 network a d b e TOP f BOTTOM c g h i Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' (Left side) The implementation of the function X = abc+ abef+ debc+ def can be performed using either crossbar-based or two-terminal-based circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The crossbar-based implementation requires 6 switches, while the two-terminal-based implementation requires 11 switches [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Lines from Top to Bottom shows the path which implements product terms of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In larger functions, the size of the network can be significantly reduced by using crossbar-based circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' (Right side) A 3 by 3 switching lattice is also shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In a lattice network with 3 rows and 3 columns (as shown in Figure 2), there are a total of 9 four-terminal switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The top part of the lattice is referred to as ‘TOP’ and the bottom part is referred to as ‘BOTTOM’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Every path that connects the TOP to the BOTTOM is considered a Product term of the Boolean function being implemented by the lattice network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The lattice network can be represented as a graph, with each switch being a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' To find the paths through the lattice network, a Depth-first search (DFS) algorithm is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' However, not all of the paths found by the DFS algorithm are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Nonvalid paths, called ‘superset paths’, must be removed from the design in order to accurately represent the Boolean function being implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Path formation To generate paths in a 3 by 3 lattice, we begin by checking the adjacent nodes (or ‘children’) of the source four-terminal switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Each four-terminal switch can be connected to a maximum of four other switches, so each switch can have up to four children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In the 3 by 3 lattice shown in Figure 2, the source node has three children: ‘a’, ‘d’, and ‘g’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' We then choose one of these children, such as ‘a’, and check its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' ‘a’ has one child, ‘b’, which has two children: ‘e’ and ‘c’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' If we choose ‘c’, the path reaches the bottom of the lattice and the product term becomes ‘a b c’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' If we choose ‘e’ instead, it has two children: ‘h’ and ‘f’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' If we choose ‘f’, the path reaches the bottom and the product term becomes ‘a b e f’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This process is repeated recursively until all nodes and their children have been examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Nodes that have already been marked as used in a previous path are not checked again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Valid path selection To determine if a generated path is a superset of another path, we have two options: we can either generate all the paths first and then check for supersets, or we can check the path as it is being generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The latter option is faster, especially for large functions with many paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' When adding a new node to the path, we can check if it is a child of any previous node in the path, except for the one that brought us to the new node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This avoids the need to generate all the paths and then remove the supersets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Repetition of literals To generate the maximum number of paths in a lattice structure with repeated literals, we can build a basic model by treating all the literals as distinct and then replacing the basic literals with the original, repeated literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' We can then remove any superset paths created by the repetition of literals to get the final set of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' To design a model that behaves like a cross-bar or four-terminal lattice, we need to consider all these relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Once the model is designed, we can obtain the boolean function it implements and verify that it is the same as the target function that the lattice structure was intended to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This will ensure that the model accurately represents the intended behavior of the lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IMPLEMENTATION To implement the design, the approach described in earlier sections was followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' The input for this design was the positions of the variables (literals) in the lattice network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' A model of the lattice was then created and used to retrieve the implemented function, following the design rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' As an example, a 3 by-3 lattice network and its resulting function were provided in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' By following the steps outlined in previous sections, it is possible to confirm the accuracy of the implemented function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' It was implemented in ‘C’ language and the process was very accurate and faster to verify any kind of cross-bar-based implementation of any function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' CONCLUSION In this study, we presented a method for verifying any switching lattice network by modeling it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' This approach can be used to verify the implementation of any function with a cross-bar lattice network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' REFERENCES [1] Mustafa Altun and Marc D Riedel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Logic synthesis for switching lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE Transactions on Computers, 61(11):1588–1600, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [2] Manek Dubash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Moore’s law is dead, says gordon moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Techworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' com, 13, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [3] Serzat Safaltin, Oguz Gencer, M Ceylan Morgul, Levent Aksoy, Seba- hattin Gurmen, Csaba Andras Moritz, and Mustafa Altun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Realization of four-terminal switching lattices: Technology development and circuit modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In 2019 Design, Automation & Test in Europe Conference & Exhibition (DATE), pages 504–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [4] Levent Aksoy and Mustafa Altun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Novel methods for efficient realization of logic functions using switching lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE Transactions on Computers, 69(3):427–440, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [5] M Ceylan Morg¨ul and Mustafa Altun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Optimal and heuristic algorithms to synthesize lattices of four-terminal switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Integration, 64:60–70, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [6] Muhammed Ceylan Morgul and Mustafa Altun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Synthesis and opti- mization of switching nanoarrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In 2015 IEEE 18th International Symposium on Design and Diagnostics of Electronic Circuits & Systems, pages 161–164, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [7] Sheldon B Akers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' A rectangular logic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In 12th Annual Symposium on Switching and Automata Theory (swat 1971), pages 79–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [8] Malgorzata Chrzanowska-Jeske, Yang Xu, and Marek Perkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Logic synthesis for a regular layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' VLSI Design, 10(1):35–55, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [9] Malgorzata Chrzanowska-Jeske and Alan Mishchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Synthesis for regularity using decision diagrams [logic ic synthesis and layout].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In 2005 IEEE International Symposium on Circuits and Systems, pages 4721–4724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [10] Matthew M Ziegler and Mircea R Stan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Cmos/nano co-design for crossbar-based molecular electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' IEEE Transactions on Nanotechnology, 2(4):217–230, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [11] Mary M Eshaghian-Wilner, Amar H Flood, Alex Khitun, J Fraser Stoddart, and Kang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Molecular and nanoscale computing and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' In Handbook of nature-inspired and innovative computing, pages 477–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' [12] Rajesh Kumar Datta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} +page_content=' Implementing boolean functions with switching lattice networks, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFAT4oBgHgl3EQfkh06/content/2301.08611v1.pdf'} diff --git a/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/2301.03063v1.pdf.txt b/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/2301.03063v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..61e25a3943cc44bcde6cef883380c6dc96f6cf85 --- /dev/null +++ b/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/2301.03063v1.pdf.txt @@ -0,0 +1,2611 @@ + +1 + +INFLATION AND VALUE CREATION: AN ECONOMIC AND PHILOSOPHIC +INVESTIGATION + +Gennady Shkliarevsky + + +Abstract: The subject of this study is inflation—a problem that has plagued America +and the world over the last several decades. Despite a rich trove of scholarly studies and +a wide range of tools developed to deal with inflation, we are nowhere near a solution of +this problem. We are now in the middle of the inflation that threatens to become a +stagflation or even a full recession; and we have no idea what to prevent this outcome. +This investigation explores the real source of inflation. Tracing the problem of inflation +to production, it finds that inflation is not a phenomenon intrinsic to economy; rather, it is +a result of inefficiencies and waste in our economy. The investigation leads to a +conclusion that the solution of the problem of inflation is in achieving full efficiency in +production. Our economic production is a result of the evolution that is propelled by the +process of creation. In order to end economic inefficiencies, we should model our +economic practice on the process that preceded production and has led to its emergence. +In addition, the study will outline ways in which our economic theory and practice must +be changed to achieve full efficiency of our production. Finally, the study provides a +critical overview of the current theories of inflation and remedies that are proposed to +deal with it. + + +Key words: inflation, business cycle, recession, value creation, Keynesianism, neo- +classical economics, evolution, and the process of creation. + +DOI: 10.13140/RG.2.2.30512.23046 + +Introduction + + +Inflation is a very familiar word in economics. It is also a highly popular topic +these days. Not a day passes without hearing this word from politicians, economists, +mainstream media sources, or even in casual conversations. The reason for this +popularity is the fact that the world economy is in the grips of inflation, and not just an +ordinary one. For one thing, the inflation is high. Even in the U.S. where the level of +inflation is now the lowest in the world, it is close to 9%. This inflation does not affect +one particular or even group of countries; it is a truly worldwide phenomenon. Even in +developed G-7 countries it reaches over 10% and is even much higher in less developed +and underdeveloped countries.1 It is a global phenomenon. Fabio Vighi offers this +characterization: + +We have entered a global cycle of secular inflation that is unique in +history. The cynical attempt to preserve a system based on the ontological + + +2 +assumption of permanent monetary injections now entails the controlled +demolition of the real economy and the world it supports.2 + + +So far, all efforts to curb this inflation by raising interests rates have failed. Many +observers point to an imminent danger of this inflation becoming a run-away inflation or +hyperinflation. Even worse, it can turn into a stagflation, similar to the one that erupted +in the mid-1970s that had very severe economic and political ramifications from which +our economy has not recovered to this day. + +The consequences of the current inflation can be even worse. For example, if the +inflation of the 1970s led to the demise of the Democratic Party for over a decade, this +inflation may very well wipe it out completely. The current inflation has already led to +the wave of political discontent. This wave has already led to the election of Donald +Trump and to the establishment of Republican control over the House of Representatives. +There is no doubt that this inflation will play a very important role in the 2024 elections. +Many researchers fear that this inflation will bring upon us increasing “ideological +manipulation and authoritarian violence.”3 + +Inflation has always been one of the macroeconomic phenomena that attracted +much attention. Inflation and its counterpart, deflation, have always provoked heated +debates.4 Decades of research have expanded our knowledge of various factors that drive +inflations. Although we have come to understand a lot about inflation, we still do not +have a clear idea as to the root cause of inflation. We are not even sure whether inflation +is necessarily a bad thing.5 In fact, many economic planners currently believe that we +need inflation and one or two per cent inflation is actually good for our economy. We +have also learned that deflation--the opposite of inflation--is not much better than its +counterpart. Japan is a sad example of its pernicious impact.6 The result of the confusion +and controversies surrounding inflation is the lack of coherent and clear policies that +would curb the current inflation. Despite the fact that the Fed is currently increasing +interest rates to fight the inflation, there are many prominent economists who criticize the +Fed’s approach. Joseph Stiglitz sees in it nothing but pain and no benefits. He writes: +“As a new Roosevelt Institute report that I co-authored shows, any benefits from the extra +interest-rate-driven reduction in inflation will be minimal, compared to what would have +happened anyway.”7 We seem to be permanently deadlocked as we are trying, on one +hand, to avoid high unemployment and, on the other, to prevent run-away price increases. + +The purpose of this article is to find a way out of the current conundrum related to +inflation. As part of the search for a solution, the article will provide a critical survey of +the current views on inflation and popular approaches in dealing with it. Much of the +current research on inflation focuses on various factors that drive inflations. No doubt, +this knowledge is important but it is not sufficient. The factors that drive inflation are +relevant but they seem to be phenomena that attend inflation, rather than cause it. For +example, we certainly accept the notion that growing money supply drives price +increases, but this notion tells us nothing about the reason why our policy planners decide +to boost the supply of money, even though we know perfectly well the consequences of +such policy. In many cases, economic planners resort to this policy in order to contain +unemployment, which suggests that our economy is always in a double jeopardy, +oscillating between inflation and unemployment. There is a vital need to determine what + + +3 +and why sets price stability and employment on a collision course. This article will +address all these questions and issues related to them. + + +Current Views on Inflation and Its Causes + + +Definitions + + +The term “inflation” in relation to economic processes came into use in the +second half of the 19th century.8 Back then the term “inflation” referred primarily to the +expansion of the currency. This view is still popular today, particularly among neo- +classical economists. According to Milton Friedman, for example, inflation “is always +and everywhere a monetary phenomenon.”9 Only later, economists began to define +inflation in terms of price increases that might result from monetary expansion but could +also have other causes.10 + +Today, most economists think about inflation as price increases for goods and +services, or “how much more expensive the relevant set of goods and/or services has +become over a certain period, most commonly a year.”11 There are certainly wordier +definitions. The Encyclopedia Britannica offers the following expanded version: + +Inflation, in economics, is collective increases in the supply of money, in +money incomes, or in prices. Inflation is generally thought of as an +inordinate rise in the general level of prices.12 + +In his recent contribution, Adam Tooze offers an even lengthier formulation: + +In discussing inflation, economics abstracts from idiosyncratic shocks. +Inflation is defined as a general upward pressure on all prices, independent +of idiosyncratic supply shocks. Inflation, in this sense, is a +macroeconomic, aggregate concept . . . Inflation can thus be defined as a +shift in the terms of trade between (1) money and (2) goods, as +experienced (3) by a particular group of people and (4) captured by a +particular statistical apparatus.13 + +No matter how long or how sophisticated current definitions are, none of them really +ventures beyond the mere description of inflation and factors attending to it. + + +Perceived Causes of Inflation + + +There is no shortage of theories that try to explain what causes inflation.14 They +are different and diverse. Yet all they do is create confusion. They all focus on some +specific factor or factors attending inflation and try to reduce their explanation what they +see as the cause. The factors they stress are often valid, but together they create a +contradictory picture that defies understanding. + + +4 + +Perhaps the most common explanation of what causes inflation is a monetarist +one. It is also by far the oldest. David Hume recognized the link between the expansion +of money supply and price increases in his essay “Of Money” that appeared in 1752. +This explanation remains popular to this day.15 According to this school of thought, there +is only one cause of inflation and it is an over-issue of inconvertible money.16 Milton +Friedman is the best-known representative of the monetarist school. In his concise, if +restrictive, formulation, inflation is a monetary phenomenon. As he put it, inflation “is +and can be produced only by a more rapid increase in the quantity of money than in +output.”17 + +The reason for the continued popularity of the monetarist explanation is the fact +that it enjoys more empirical support than any other economic theory.18 However, +despite its long history and the substantial factual evidence, the predicted association +between money supply and inflation remains disputed.19 For one thing, this association +works only over long periods of time. Also, there are some examples of variations in +money supply that do not necessarily negatively correlate with price levels. For example, +decreases in the supply of money do not necessarily lead to price increases. During the +Great Depression decreases in money relative to real incomes were associated with +decreases, not increases, in price levels.20 + +Also, the monetarist explanation of what causes inflation is not unambiguous. +The link between money supply and the availability of goods/values goes both ways. If +the increase in money supply can cause price increases, so can the decline in the +production of goods. The connection between inflation and production is a common +preoccupation among economists. David McMillan, for example, explains the high rate +of inflation in Britain by the sluggish growth of productivity and low rate of investment +that lags behind Britain’s main economic competitors—the United States, Germany, and +France.21 The results of a study that has tested the link between inflation and economic +growth in Sri Lanka show that there is “a long run negative and significant relationship +between the economic growth and inflation.”22 The study shows a similar but short-term +impact for China.23 A study by Hrushikesh Mallick also reveals the negative correlation +between inflation and growth in India.24 + +Government spending is yet another popular explanation for inflation. +Conservatives have expertly used this explanation in the past as a way of defeating their +opponents who favor the expansion of government programs. Ronald Reagan effectively +employed this cudgel by arguing that extensive and expensive government programs had +a devastating effect on American economy and were the principal cause of the stagflation +that occurred in the mid-1970s. One can hear very similar criticisms of the government +these days. In his contribution “Inflation and Growth: Some Theory and Evidence” Max +Gillman challenges liberal economists and politicians who explain the current high rate of +inflation by the war in Ukraine, “corporate greed,” or short-term supply issues. The +cause, in his view, is “bad public policy.”25 Nathan Benefield makes a similar charge in +his editorial. “Washington politicians,” he opines, “pretend they don’t know what’s +behind runaway inflation, but voters know.” Pointing to what he sees is the real issue, +Benefield quotes James Carville, a Democratic strategist, who quipped on one occasion: +“It’s the spending, stupid.”26 + +This list of explanations is far from complete. The number of factors that +researchers cite as causing, contributing to, or driving inflation is much larger. It is + + +5 +beyond the scope of this study to examine them all. They include supply bottlenecks, +unequal distribution of money, sluggish price adjustment, too “few” unemployed, and +many others.27 There are even sociological perspectives on inflation that focus on the +role, interests, and expectations of social actors as a cause of inflationary tendencies.28 In +reading various authors, one gets an impression that researchers base their explanations +on their theoretical or political commitments; and that what they present as causes of +inflation are merely their subjective preferences among factors that attend inflations. +Gregory Mankiw appears to say as much when he writes about economists “dirty little +secret” since they often speak “not just as economic scientists, but also as political +philosophers.”29 Alexander Barta observes: + + . . . [I]t is elementary to recognise that the very measurement of +inflation, i.e. the determination of the basket of goods and services whose +prices are taken into account as much as the very production of the +statistic itself, is a selective, theory-driven and, indeed, political choice +already--there is no such thing as “objective inflation” . . . Data are +business. Data are political. And that is particularly pertinent in the case +of inflation, because inflations are contentious. They generate winners and +losers.30 + + +There are no known factors cited as a cause of inflation that would not be put in +doubt by empirical evidence. Even the most commonly cited cause—money growth—is +not without counter examples. As has already been mentions, during the Great +Depression of the 1930s money relative to income and the price levels were both in +decline.31 + +The examination of the current explanations of what causes inflation shows that +researchers have different views on this subject. The causes they cite—supply of money, +production rates, structure of the economy, distribution of wealth and money, savings and +shopping habits, and many others—are merely co-dependent economic variables +attending inflations, rather than causing it. Barta correctly points out that inflation cannot +follow from any of these co-dependent variables because variations in any of them +simply leads to “recalibration of the system of relative prices by the Walrasian +Auctioneer, but not to a rise in the general price level.”32 On close analysis, what is often +perceived as a cause of inflation seems to be a symptom—a variable attending to +inflation—rather than its cause. + + +Inflation Remedies + + +There are two principal approaches in dealing with inflation. One emphasizes the +role of the government in managing inflation. The other recommends relying on market +forces, rather than government fiat, to achieve price stability. + + + +1. Inflation Management + + + +6 + +The roots of the first approach rest on Keynesian economic theory. Keynes +challenged neoclassical economics and its reliance on market forces to regulate the +economy. The pro-active approach advocated by Keynes proposed that the government +was to use a variety of levers available to it to manage economy and inflation. + +According to this approach, inflation is neither bad, nor good; or rather, it is bad +and good at the same time. Good inflation is one that stays at a recommended level of +1% to 2% a year or 10% to 12% for developing countries.33 The proponents of this +approach maintain that while high inflation can be harmful, too little inflation or even +negative inflation, or deflation, can also weaken the economy. Japan, for example, has +experienced a long period of practically no economic growth primarily because of +deflation. Following the financial crisis of 2007, the Fed and other central banks around +the world promoted low interest rates and other monetary incentives to make sure that +liquidity stays sufficient; they euphemistically called it “quantitative easing.”34 + +The main thrust of the Keynesian approach and its variations is to provide a +framework in which the government can manage economic activity by varying its +expenditures and receipts or by influencing the level of private investment through +interest rates and money supply.35 The followers of Keynes do not seek to eliminate +inflation. Rather they try to ameliorate its most harmful effects and keep it at a +manageable level. + +The Federal Reserve, or the Fed, is the main institution that regulates the +economy. The Fed has two jobs: one is to maintain price stability and the other is to +maximize employment. When the economy is struggling and inflation is too low, the Fed +will lower interest rates or buy assets to increase the amount of available cash. When the +economy is expanding too quickly and inflation rises, the Fed will typically raise interest +rates or sell assets to reduce the volume of cash in circulation. Borrowing money +becomes expensive, which slows economic growth and brings down the level of +inflation.36 The theory that guides the Fed and most central banks aims at mild inflation +at a level of one or two per cent.37 However, this theory is little more than just a theory. + +The policy actions of the Fed and other central banks in the face of inflation may +include raising interest rates at which central banks provide reserves to financial +intermediaries, buying and selling government securities, changing reserve requirements, +and others.38 This approach is very eclectic. Its proponents offer a variety of policies and +policy combinations that are supposed to help curb inflation. They propose to lower +health costs, reform the tax code to raise more revenue, limit discretionary spending, +reduce consumption-oriented spending, cut aid to states, reduce costs of energy, trade, +and procurement, and much else.39 Even a more restrained fiscal policy is not off-limits. +As President Biden has remarked numerous times in the current inflation period, +“bringing down the deficit is one way to ease inflationary pressures.” This could include +avoiding further deficit-boosting measures; lowering health care costs; raising tax +revenue; reducing consumption-oriented spending; promoting work, savings, and +investment; and/or lowering energy, trade, and procurement costs.40 + +Even when the American government abandoned the policy of low interest rates +in response to inflationary pressures, many Keynesians criticized this move by insisting +that giving up this policy was wrong. Joseph Stiglitz, for example, has argued in his +article that in a recent issue of Project Syndicate that high interest rates bring nothing but +pain to the economy and have no benefits.41 + + +7 + +One should not get an impression that this approach to inflation is limited to +Democrats. President Donald Trump throughout his presidency publicly pressured the +Federal Reserve to keep interest rates low and to keep pursuing bond-buying monetary +expansions. In one of his tweets, Trump wrote in April 2019: + +We have the potential to go up like a rocket if we did some lowering of +rates, like one point, and some quantitative easing . . . Yes, we are doing +very well at 3.2% GDP, but with our wonderfully low inflation, we could +be setting major records &, at the same time, make our National Debt start +to look small!”42 + + +As the current inflation continues to grow, even the current American government +that is dominated by Keynesians has launched a policy of high interests in expectation +that this measure will reduce inflation even if at the cost of employment. One can +surmise that the intention now is to keep on alternating between the policy of high +interest rates and the policy of high employment. There is not much theory behind this +constant alternation; it seems to be more of a trial-and-error approach driven by wishful +hope with little empirical support that somehow the economy will grow out of the current +predicament. + + + +2. Price control + + +Perhaps the most radical policy that exemplifies the approach that favors +government intervention is price control. There are few economists and politicians who +advocate this policy, but it is on the menu of possibilities. One can never trust politicians +who may be opposed to price control one day and then change their position the +following day. + +Power brokers tried this policy in the 1970s when inflation was devastating family +earnings and when the buying power of common Americans was severely reduced.43 +President Nixon preached to Americans in 1965 that “the lesson that government price +fixing doesn’t work is never learned.” When campaigning for president, Nixon pledged +that he would “not take this nation down the road of wage and price controls.” However, +in 1971, when inflation reached six per cent, Nixon, against his own previous judgment, +began to pressure the business community, with little or no success, to hold down prices +and wages. Prices continued to grow.44 Trying to deflect criticism, President Carter +blamed rising prices and recession on OPEC, also with little success. + +President Biden’s initial response to rising prices was dismissive. He simply +declared them temporary. When prices continued to grow, Biden blamed rising prices on +the pandemic. When the war in Ukraine erupted, Biden quickly put the blame on Putin. +In his view and that of other Democrats, the rising prices justified emergency action. +Elizabeth Warren and the host of other left-learning Democrats rushed to vilify “greedy +corporations” for exploiting the pandemic. Later in 2021 The New York Times ran a trial +balloon. The paper wrote approvingly that as rising inflation was threatening Biden’s +presidency, he was turning to “the federal government’s antitrust authorities to try to +tame red-hot price increases.”45 Politicians certainly took the cue. Representative Jamaal + + +8 +Bowman (D-N.Y.) obliged the president by introducing the Emergency Price +Stabilization Act, calling on the government to “build the capacity to establish limits on +the growth of certain prices, and to otherwise strategically regulate such prices, in order +to stabilize the cost of essential goods and services.”46 Targeting price stability as a way +to achieve higher rates of economic growth is also popular in countries with emerging +markets. 47 + + + +3. The Neo-Classical Approach + + +As has already been mentioned, the other approach toward inflation has originated +in neo-classical economics. The advocates of this approach stress that the availability of +resources to produce goods, services, and particularly technological change is a major +factor affecting growth.48 The theory that underlies this pro-growth perspective +maintains that producing more goods and services in a shorter time would cut costs per +unit, raise supply, and thus put downward pressure on prices.49 The secret to curing +inflation, in their view, is pro-growth policies that create incentives for more goods, more +employment, less government spending and sound money. When production grows, +prices go down.50 However, in contrast to interventionists, the pro-market proponents +insist on reducing the role of the government. + +The policies pursued in the market-oriented approach stress the need to reduce +government interventions into the economy. Such reduction takes several forms. First of +all, it involves severe restrictions to the government’s capacity to increase the supply of +money—either by limiting the issue of paper money or by preventing the government +from manipulating interest rates. The idea is that if the government stops playing with +money supply, investors can get a realistic picture of their opportunities, better calculate +risks, and have sober expectations about profits. + +Also, the proponents of the market-oriented approach advocate the reduction of +the federal budget by eliminating high-cost government programs. Cutting government +programs was a signature policy introduced by Ronald Reagan and continued under Bill +Clinton. Lawrence Summers, who served as Bill Clinton’s Treasury secretary, rocked the +Democratic establishment last year by predicting that his party’s excessive spending +would cause inflation.51 For Nathan Benefield, Senior Vice President of the +Commonwealth Foundation--Pennsylvania’s free-market think tank—government +spending is the main factor that generates inflation. He succinctly summarizes the reason +for the current inflation by paraphrasing the Democratic strategist James Carville: “It’s +the spending, stupid.” In his view, if policy makers are serious about fighting inflation, +they should start with fiscal restraint.52 + +In accordance with the pro-market approach, reduction of government spending +should go hand-in-hand with cutting taxes. Tax cuts will give more money to both +business owners and regular working families; and more money for them will invigorate +the market, ensure growth, and cut inflation. There is no shortage of calls from both the +right and the left on lawmakers demanding to enact a comprehensive tax reform that +would ease burdens on working families and enable small businesses to hire more +workers and raise wages. In their article “Economic Growth, Not Austerity, Is the + + +9 +Answer to Inflation” Arthur Laffer and Stephen Moore cite historical examples that prove +their point. They write: + +History proves growth doesn’t cause inflation. In the 1920s, when the +highest tax rate was cut from 73% to 25%, real GDP soared and the price +level fell. In the 1960s, tax cuts and pro-growth policies led to an +economic expansion, stable prices and budget surpluses.53 + + +Lastly, many economists and politicians also see the need to reform or even +eliminate various excessive regulations imposed by the government that constrain +economic activity and negatively impact production. Those who support reducing +regulations argue that such reforms would help producers to control costs and unleash +energy production to lower electricity and fuel prices.54 + + +Critique of the Proposed Remedies + + +As the above shows, recipes for dealing with inflation are different and diverse. +The proponents of each approach mostly focus on the advantages of what they propose +and prefer to ignore disadvantages. However, an objective assessment needs to take into +account all sides of what is proposed—both positive and negative. Criticisms in this +section will focus on shortcomings of these recipes, both those specific to each one and +those common to all. + + +1. Managing Inflation + + +Perhaps the most serious disadvantage of the managing inflation approach is the +fact that it has proven to be completely ineffective in the current inflation period. The +proponents of this approach have claimed that maintaining a tolerable level of inflation at +one or two percent a year is relatively harmless and will have no serious ramifications for +the economy. In their view, this approach should allow making financially sound +decisions on saving, investing and borrowing money.55 + +However, trying to manage inflation at a desired level has proven to be very +difficult. Since 1921 secular inflation in the United States and throughout the world has +been a permanent presence in the economy. There have been rises and declines. The +current secular inflation has grown steadily in the last decade or so and reaches today the +level of close to 9% in America. Among all developed countries, the inflation is the +highest in Great Britain reaching the ominous 10%. All through this period, the +Democrats defended and continue to defend the policy of high government spending and +expensive government programs. The evidence from both industrialized and developing +countries supports the view that inflation causes lower real growth. As Woo S. Jung and +Peyton Marshall conclude in their article on inflation and economic growth: “The use of +inflationary finance as a means to force additional savings and to increase capital +formation appears to be an unwise strategy for economic development.”56 Manipulating +different variables to maintain the recommended level of 1% to 2% inflation is like + + +10 +balancing economy on a pinhead: even if you achieve this balance, it will be extremely +precarious, affected by various destabilizing economic and non-economic factors. + +In extreme cases inflation can become a run-away inflation, or even +hyperinflation.57 One of the best-known cases of hyperinflation is Weimar Germany in +1923 when one American dollar was worth 4 trillion German marks. Money lost all +value and people shifted to barter, using goods with stable value as currency. For +example, pianos became currency during the German inflation.58 The post-war inflation +in Hungary reached 41.9 quadrillion percent.59 In 2008, Zimbabwe experienced one of +the worst cases of hyperinflation ever. The estimated annual inflation level at one point +was 500 billion percent. 60 + +High inflation can cause a decline of production. When high inflation and +production decline occur at the same time, the result is what is called stagflation. +Stagflations have particularly devastating effects. One may recall the stagflation in +America during the mid-1970s when high interest rates, rising prices, and economic +stagnation created an extremely explosive political situation that led to the collapse of the +Democratic Party and brought Ronald Reagan to power and almost a decade of +Republican rule. + +The current inflation also has a very strong potential to turn into a stagflation.61 +Top economists and bankers have already cautioned about this possibility. Those who +have sounded alarm include Allianz and Gramercy’s chief economic adviser Mohamed +El-Erian and Goldman Sachs CEO David Solomon. Both have pointed out that the +current inflation is entrenched and widespread throughout the world. The World Bank +has also issued multiple warnings to the effect that if economy remains sluggish, inflation +may very well end in a stagflation in many countries, including in the United States.62 +Even Paul Krugman, of The New York Times, who is usually optimistic regarding current +inflation, has expressed concerns over the persistently “hot job market.”63 One economist +writes about the current mood among economists that “they also know that this hardly +original escamotage can only lead to runaway inflation, and then hyperinflation. What +takes place today as a matter of monetary normality used to characterise wartime +economies, namely direct financing via the money presses.”64 + +Many leading economists fear that today’s high levels of inflation, and the Fed’s +commitment to containing it, could trigger a recession as early as 2023.65 For example, +the vast majority of economists at 23 large financial institutions have already confirmed +this gloomy prediction only a few days into 2023.66 In his article for Project Syndicate, +Nouriel Roubini predicts an “unavoidable crash” of the economy in the near future.67 +The Fed is currently trying to contain inflation with high interest rates that should +eventually reach 5.1%. At this rate, the unemployment may reach the level as high as +4.6% and possibly even higher. Yet despite these efforts the inflation remains resilient. +Even economists who try to moderate fears of recession, such as Jeffrey Frankel, still +acknowledge that in the next two years a worldwide recession is entirely within the range +of possibilities.68 + + +2. Price control + + +Price control is a very radical policy that should be used with great caution. It is +certainly a legitimate policy that has often been used in dire circumstance of war when + + +11 +goods are in short supply. Governments successfully used this policy in combination +with the policy of distribution of resources. However, using this policy to fight inflation +is totally useless and even damaging. + +The government controlled prices in the Soviet Union through much of the +country’s history. It worked reasonably well in the time of WWII when many goods, +particularly food products, were very scarce. However, continuing this policy in +peacetime, particularly in the late 1970s and early 1980s, resulted in horrible economic +distortions and inefficiencies. Indeed, the inflation was low on Soviet planners’ books, +but it was merely pushed underground and out of sight. Money continued to lose their +value but prices were kept stable by government fiat. Many citizens began to run away +from cheap money into goods with stable and low prices. The most notorious example of +this flight was the fact that many farmers used cheap bread to feed their livestock. The +result was enormous hoarding and chaotic disappearance of goods. By not addressing the +source of inflation and pretending that it did not exist, just because prices did not change, +Soviet economic planners simply made inflation uncontrollable. + +There are many stories about that period in the late 1970s and 1980a in the Soviet +Union. Many who observed the erstwhile Soviet economy documented empty shelves +and long lines to buy even the most essential items (toilet paper and vodka were two +items particularly in high demand). Researchers habitually attributed these shortages to +production inefficiencies. Indeed, production in the Soviet Union was inefficient, but no +more than usually. The real story behind these shortages was that farms and industries +continued to produce goods at their normal rate, but these goods disappeared into the +thriving black market even before they left enterprises. Crafty speculators hoarded goods +and sold them for high profit. Many goods, particularly perishable food products, rotted +in underground warehouses, while people waited for days in huge lines to buy even the +semi-spoiled leftovers. + +Such are the lessons of price control. No wonder that political leaders rarely +resort to this policy, despite occasional calls from academics.69 President Nixon, for +example, who at one point talked about price control, still relied on appeals to the +business community to limit wages and prices on a voluntary basis. + + + + +3. Reducing the Role of the Government in the Economy + + +The approach that sees the solution of the problem of inflation in reducing the role +of government in the economy and in relying on market forces to restore economic +stability has several major shortcomings. For one thing, the reduction of money supply +as a cure for inflation will inevitably slow down production, which will also inadvertently +lead to unemployment. The proponents of this approach argue that high level of +unemployment is simply unavoidable. It is, in their view, a necessary evil that we simply +have to endure. However, many people who end up on the receiving end of this policy +and have to pay a heavy price for it are not particularly receptive to the idea that their +suffering paves the road to future happiness. Unemployment, declining living standards, +growing gap between the rich and the poor are sure to lead to political tensions, ravaging +instability, and social cataclysms. + + +12 + +Many neo-classical economists advocate this perspective. Ronald Reagan was +perhaps its best-known practitioner. However, there have also been quite a few +Keynesians among those who recommended this approach. In 1980, for example, when +inflation in America stood at double-digit, Paul Samuelson, who was the first American +to win the Nobel Prize in economic sciences, wrote that “five to ten years of austerity, in +which the unemployment rate rises to an eight or nine percent average and real output +inches upward at barely one or two percent per year, might accomplish a gradual taming +of U.S. inflation.”70 Closer to our time Lawrence Summers, who served as Bill Clinton’s +Treasury secretary, has also argued that American economy needs several years of +unemployment above 5% or 10% unemployment for one year to contain inflation.71 This +proposal, if implemented, was sure to leave millions of Americans without jobs. + +Cutting government programs is another prong of the austerity approach. Both +Ronald Reagan and Bill Clinton pursued the reduction of government spending by cutting +federal programs, including welfare programs. This policy has certainly had an adverse +effect, causing pain and suffering among the most vulnerable members of society. +Summers rocked the Democratic establishment in 2021 by predicting that his party’s +excessive spending would cause inflation, insisting that government spending should be +cut.72 + +The most obvious negative side to this approach is that it leads to much suffering +for a great number of people who would have to endure long periods of unemployment +and reduced government assistance. Indeed, they would face enormous difficulties in +trying to cope on their own with the deteriorating conditions of their life. Predictably, +this policy will inevitably lead to the growing gap between the rich and the poor. In fact, +this growing gap has been persistent in America and the world for the last several +decades. The erosion of the standard of living for vast number of people is sure to +generate social tensions and conflicts that will disrupt social peace and stability. +Although many proponents predict that this period of increased suffering will be limited, +nobody can really tell how long it may last. Few doubt, though, that the longer it lasts, +the more unpredictable and dangerous will be the social and political consequences. + +However, these adverse consequences do not exhaust all the negative effects of +the austerity approach. It may also profoundly distort the structure of the economy that +will become more oriented toward the wealthy consumers, rather than the middle class. +Production will cater to those with money. As a result, the market will offer more luxury +real estate, more private jets and extravagant yachts, and more conspicuous consumption +for the rich, rather than benefits for the middle and low ranks of society. One should +remember that mass production oriented toward the average citizen was what made +American economy a success story. An increased production of high-end goods is +unlikely to replicate this success. + +As many argue, inflation, even relatively small inflation, is dangerous for the +economy. Hrushikesh Mallick finds that “inflation rates have a significant adverse +impact on economic growth.”73 A number of researchers point out that the relationship +between inflation and economic growth goes both ways. While low productivity can +result in inflationary tendencies, inflation also makes borrowing money more expensive, +which slows economic growth.74 We must address inflation. Yet on close examination, +all proposed remedies augur ill for our economic future. Many economists and economic +planners have publicly voiced their pessimism. They do not see that any of these + + +13 +remedies promise relief in the current conditions. Using combinations of these +remedies—as, for example, the Fed has been doing in the current inflation, by pursuing +both policies of low and high interest rates--looks also extremely problematic. As Fabio +Vighi aptly summarized, “For most of us, then, the future seems to offer a choice +between structural stagflation (stagnant economy with high inflation) and an abrupt +deflationary depression--like a choice between bleeding to death and suffering a heart +attack.”75 + + +Understanding Inflation + + +Inflation and Business Cycle + + +As has already been pointed out earlier, there is no unanimity among economists +and economic planners in their views of inflation.76 The only general conclusion that one +can draw from their discussions is that inflation is a result of some complex imbalance in +the economy; and that inflation is only one of the symptoms of this imbalance. +Researchers have failed to explain why economy becomes periodically unstable. They +merely recognize the existence of such periodic phenomena that they call “business +cycle.” + +Business cycle is the pattern of economic booms and busts that are experienced +by all developed economies.77 Columbia Electronic Encyclopedia attributes the first +formulation of the theory of business cycles to French physician Clement Juglar who was +the first to recognize, in 1862, that economic fluctuations associated with the boom-and- +bust were a characteristic feature of all economic systems.78 The Great Depression that +struck the United States and the world in 1929 was an important catalyst that stimulated +much interest in business cycles—these periods of rapid economic expansions followed +by economic slowdowns and contractions.79 + +Theories explaining business cycles are numerous and diverse. They focus on +different factors that supposedly trigger economic contractions in business cycles, but +that hardly amounts to an explanation as to why business cycles are there in the first +place. Despite the fact that business cycles have been the subject of intense scrutiny for a +very long time, the universal recognition today is that we simply do not know the why of +business cycles. The best that researchers do is merely point out the existence of +economic fluctuations that they explain with such vagaries as “chaotic market processes.” +Matthew Shapiro concludes that most theories “take the answer to this question to be +axiomatic”—that is, the cycles are merely assumed to be part of economic reality.80 +James Mirrlees echoes a similar view that “recessions are in some degree inevitable” and +“are bound to happen.81 In his colorful metaphorical description Mirrlees compares +attempts to steer economy away from downturns with efforts “to sail a straight line in a +boat with wind direction constantly shifting, and sometimes blowing a gale.”82 + +Neo-classical economists are very vocal and consistent in pointing to monetary +interventions as perhaps the most significant factor that contributes to economic +contractions. Murray Rothbard is one of many who confidently claims that the "’boom- +bust’ cycle is generated by monetary intervention in the market, specifically bank credit + + +14 +expansion to business.”83 However, even he recognizes that the problem of business +cycles is “one of general boom and depression.”84 He dismisses economic fluctuations as +the source of depressions. Rothbard writes: + +We may, therefore, expect specific business fluctuations all the time. +There is no need for any special "cycle theory" to account for them. There +is nothing here to account for a general business depression—a +phenomenon of the true "business cycle” . . . The explanation of +depressions, then, will not be found by referring to specific or even +general business fluctuations per se.85 + +Rothbard finds that there is something unexplainable in the behavior of many +experienced business people who are “misled” by the availability of cheap credit. One +can see Rothbard’s sense of profound amazement at something incomprehensible when +he writes: “In short, how did all the country's astute businessmen come to make such +errors together, and why were they all suddenly revealed at this particular time? This is +the great problem of cycle theory.”86 + +In his reflection on the Great Depression, Milton Friedman, an acknowledged +doyenne of neo-classical economics, also makes a claim that "monetary developments [in +the early 1930s] were the major explanation for the depth and the length of the +contraction.” He further explains: “As I've said over and over again, I'm not saying that +that [monetary developments] caused the initial recession . . . And I don't doubt for a +moment that the collapse of the stock market in 1929 played a role in the initial +recession.”87 + +Thus, most neo-classical economists advise to take business cycles and +depressions as a given and even embrace them. Arguing that business cycles, including +depressions, are an essential part of economy, Rothbard, for example, suggests that rather +than fight depressions, we should change our perspective on them from pessimistic to +optimistic. We should view them as actually serving a useful purpose. Downturns, in his +view, are the way that economy “adjusts to the wastes and errors of the boom, and +reestablishes efficient service of consumer desires.”88 “The depression,” Rothbard +rhapsodizes, “far from being an evil scourge, is the necessary and beneficial return of the +economy to normal after the distortions imposed by the boom. The boom, then, requires +a ‘bust.’"89 Rothbard is not alone in proposing a change in attitude toward economic ups +and downs. According to James Mirrlees, “The general conclusion [among economists +today] is that we should encourage people not to worry too much about asset price +fluctuations. Then they will be happier; and they will not be so likely to reduce their +consumption spending when markets crash.”90 + +Not everyone agrees with neo-classical economists in viewing business cycles as +an intrinsic part of economy. Such 20th-century theorists as John Maurice Clark and +Joseph Schumpeter have attempted to find cures for economic instability, rather than +describe it, in the manner of many 19th century theorists, simply as a natural +phenomenon.91 Tejvan Pettinger, among many, considers this view to be controversial. +Critics disagree with the view that economic downturns have a beneficial role to play +because they “shake up” economy, weed out “inefficient” firms, and create incentives for +cutting costs and operating efficiently. They argue that in a recession, even “’good + + +15 +efficient’ firms can go out of business leading to a permanent loss of productive +efficiency.”92 Rendig Fels is another critic who rejects the notion of the inherent nature +of business cycles. He sees “little evidence of a built-in tendency of the American +economy to generate cycles.”93 Yet even detractors offer no insight as to why these +cycles exist. + +Indeed, there is a great deal of truth in the argument of neo-classical economists +that monetary interventions are a major contributor to economic downturns in business +cycles. However, one should recognize that the introduction of monetary interventions +by the government was not a whim. It was largely a response to the existence of business +cycles. The current inflation did not start merely because some policy makers decided to +institute low interest rates. For example, one important reason for lowering interest rates +to almost zero in the current inflation period was to prevent deflation and production +decline caused by the global financial crisis that started in 2007. After the outbreak of +the crisis the US Federal Reserve and other central banks around the world kept interest +rates low for a prolonged period of time and have instituted other monetary policies to +ensure that financial systems have plenty of liquidity.94 + +When John Maynard Keynes published The General Theory of Employment, +Interest, and Money in 1936, the world economy was in ruins. The book was an +important watershed in macroeconomics. Keynes’s theory did not explain business +cycles but it argued that monetary interventions by the government are the way to +mediate their most adverse consequences.95 + +Initially, the response to Keynes’s theory was limited to scholarly circles. It was +only after WWII that the theory swept away the influence of the classical orthodoxy and +became the main tool for guiding economies. The “Keynesian Revolution” got under +way.96 The most important factor in this new development was the rising wave of +recessions in the 1950s and 1960s. During that period many economic planners and +policy makers came to believe that there was a direct trade-off between unemployment +and inflation. They came on the side of inflation to keep unemployment down.97 Paul +Samuelson and Robert Slow, two Nobel laureates in economics, forcefully argued in +support of maintaining the price index at 4 to 5 per cent a year as “the necessary cost” of +keeping employment around 3%.98 When Nixon was blamed for the on-going recession +in 1971, he reportedly quipped: “We’ll take inflation if necessary, but we can’t take +unemployment.”99 + + +The Nature of Production + + +The connection often made between inflation and business cycle indicates that +both these phenomena are interrelated. Inflation is one of the possible ways in which +economic instability manifests itself; and so does downturn of production. In other +words, inflation is associated with economic instability that is part of the business cycle. +We do not know what causes this instability, nor do we understand why business cycles +are there. What we do know, however, is that these phenomena are intimately related to +production. Therefore, in order to understand inflation and business cycles we must look +closer at the process of production. + + +16 + +We commonly view production in economic terms, as the action of making or +manufacturing from components or raw materials, or the process of being so +manufactured. According to one definition, + +Production is the process of making or manufacturing goods and products +from raw materials or components. In other words, production takes inputs +and uses them to create an output which is fit for consumption—a good or +product which has value to an end-user or customer.100 + +In a contribution to Economic Discussions Sanket Suman defines production as “the +organised activity of transforming resources into finished products in the form of goods +and services; the objective of production is to satisfy the demand for such transformed +resources.”101 Finally, Wikipedia offers the following extensive formulation: + +Production is the process of combining various inputs, both material (such +as metal, wood, glass, or plastics) and immaterial (such as plans, or +knowledge) in order to create output. Ideally this output will be a good or +service which has value and contributes to the utility of individuals.[1] +The area of economics that focuses on production is called production +theory, and it is closely related to the consumption (or consumer) theory of +economics.102 + + +There is, however, another and broader view of production as a totality of human +interactions with reality. German idealists tended to have such comprehensive view of +production. In this thinking, cognition, or knowledge acquisition, is also a form of +production. As Hegel put it, reality in this sense is “process and result rolled into one.”103 +Karl Marx also viewed production in terms of the broad interrelationship between +humans and nature. Consider the following passage from his Economic and Philosophic +Manuscripts of 1844 in which Marx writes: “Each of his human relations to the world – +seeing, hearing, smelling, tasting, feeling, thinking, observing, experiencing, wanting, +acting, loving – in short, all the organs of his individual being . . . are . . . in their +orientation to the object the appropriation of human reality.”104 This view of production +does not contradict the purely economic perspective; it simply represents a broader +approach. It is in this broader approach that I propose to explore production and its place +in human life. + +Production is a form of interaction between humans and reality. It is a sensuous, +or physical form of interaction. Humans are a product of the evolution. Therefore, +human interactions with reality are also a product of the evolution. The evolution is a +universal process that sustains our universe. Consequently, production is a result of the +process that plays the most essential role in the existence of our universe. + +The evolution originates in conservation that is ubiquitous throughout the +universe. The roots of conservation are in the very unique nature of our universe. The +universe is all that is there. Nothing can come into it from outside because there is no +outside; nothing can disappear from it because there is nowhere to disappear. +Consequently, everything must be conserved. + + +17 + +Conservation requires resources; and resources are always limited. Therefore, +new resources are vital for conserving our universe. These necessary resources cannot +come into our universe from outside. They have to be created inside the universe. +Therefore, conservation requires creation of new and increasingly more powerful levels +of organization that offer new possibilities that have not existed prior to their creation. +These new possibilities offer access to new resources; they represent such new resources. +Thus, the universe is impossible without the creation of new and increasingly more +powerful levels of organization;105 and the creation of such levels of organization, or their +production, is what the evolution is all about. + +Since humans are products of the evolution, their interactions with nature have +inherited the main features of the process of creation. We also live under the constraint +of limited resources. Therefore, in order to sustain our life we have to create new and +increasingly more powerful levels of organization that provide access to new resources. +This is the essence of our production, including economic production. + + +Value Creation and Inflation + + +Newly created and increasingly more powerful levels of organization offer new +possibilities that are the main resource for sustaining our existence. Therefore, these new +and increasingly more powerful levels of organization represent an enormous value for us +since they sustain our existence. They are the source of value. Value creation is the main +function of our interactions with nature; it is also the main function of our economic +activity, or production. + +What does the creation of new and increasingly more powerful levels of +organization involve? A level of organization consists of interconnected functional +operations. In order for these operations to persist, they must be conserved. Action is +what conserves operations: the more operation is active, the better it is conserved. + +In order to bring an operation into an active state, or to activate it, the operation +needs something that will trigger its activity. The more often the operation is triggered +into action, the more active it is, and the better it is conserved. Consequently, the more +triggers an operation has, the better it is conserved. + +If an operation combines with another operation to create a new whole, it will +double the number of triggers that bring it into action—one that triggers one operation +and another that triggers the other operation. Since interactions among functional +operations conserve them, such interactions are favored by the evolution. By combining +with each other, operations acquire new possibilities and, therefore, become more +powerful in comparison with when they were on their own. Each new level of +organization that we create, or produce, increases our power and, thus, represents value. +In order to conserve this value, we must create another and even more powerful level of +organization, or a new value. Constant value creation is the main function of production. + +Thus, interactions among functional operations create new levels of organization +that offer more possibilities and, consequently, are more powerful than those from which +they have emerged. This added power represents value. Production is about the creation +of value, which means that production involves the creation of new and increasingly +more powerful levels of organization. The only way to conserve value is to conserve the + + +18 +level of organization that represents this value; and the only way to conserve this level of +organization is by creating a new level that is more powerful, and consequently has more +value, than the one from which it has emerged. + +The conclusion that follows from the above is that the primary function of +production is to conserve functional operations sustained by a particular level of +organization. Production is ultimately about conservation; and conservation can only +result from creating new values. Failure to create new and more powerful levels of +organization means that existing possibilities have not been realized. Without realizing +these possibilities, they cannot be conserved; in other words, value is not conserved. +Non-conservation of value is a sure sign that production is not efficient and that available +resource are either underutilized or simply wasted. Inefficient production does not create +value and, consequently, sustains losses. Production losses are at the root of economic +instabilities and disruptions. + +All economic values have monetary equivalents. Money plays the most essential +role of realizing the possibilities that each value offers. We use money to attract +resources and labor to production. Money does not have intrinsic value, after all money +is a mere convention, a piece of paper with something printed on it--and now even not +that; today money is merely an electronic signal transmitted from one source to another. +The value of money is the equivalent of the real value. When real value is not conserved, +when all possibilities associated with this value are not realized, the real value these +possibilities represent diminishes, or depreciates; and when value depreciates, the money +equivalent of this real value that is supposed to be used for realizing these possibilities +also loses its worth. Money becomes cheap and the cheapening of money results in price +increases, which is the essence of inflation. + + +The New Economic Practice + + +The Model for the New Economic Practice + + +The philosophic exploration of production shows the connection between +production as an evolutionary phenomenon and the universal process of creation. This +exploration leads to several important conclusions. First of all, it makes clear that +inflation is an aberration that is in no way intrinsic to production. There is nothing in the +process of creation that warrants this aberration. The growth in the supply of money is +not the ultimate source of inflation. It is merely an effect of the depreciation of money +that results from inefficient production that leads to the depreciation of real value and the +monetary equivalent of this real value. The depreciation of real value indicates that our +production fails to conserve this value. + +Production must be efficient. There is hardly a single economist, businessman, +economic policy maker, and even a lay individual who would dispute this maxim. +Efficiency results only from the full utilization of all available resources and possibilities. +Wasted resources and squandered possibilities make economy inefficient. Efficiency is +what makes economy possible; it is the pillar on which our economies rest. All negative +economic phenomena—instabilities, crises, inflation, unemployment, and others—are + + +19 +results of inefficient production. In order to understand our economic problems, +including inflation, we must understand why we fail to use resources efficiently, create +wastage; in other words, why we do not conserve value. + +The philosophic investigation that traces production to the universal process of +creation shows that production is an evolutionary phenomenon. It emerges in the course +of the evolution driven by the universal process of creation and, therefore, production +inherits all the main features of this process. The distinct feature of the process of +creation is the full utilization of all available possibilities. In fact, this process can only +work on the basis of such full utilization. Anything less than full utilization makes the +creation of new and increasingly more powerful levels of organization impossible. Thus, +full efficiency is the essential property of the process of creation. Therefore, the main +requirement of efficient production is the full utilization of all available resources. + +There are other important features of the process of creation that efficient +production must replicate. One important feature of the process of creation is universal +inclusion. The full utilization of all available possibilities is the key to the creation of a +new and more powerful level of organization. Only such utilization conserves all +available possibilities, which is the only way to create a new level of organization that is +more powerful than the one from which it has emerged. Only the combination of all +available possibilities makes the new emerging level greater, or more powerful, than the +sum total of its parts. If any possibility is excluded, it will not be conserved; in other +words, it will be wasted, which is how we define inefficiency. As a result of exclusion, +the emergent level of organization will not exceed the power the level from which it has +emerged. + +The process of creation is a dynamic process. It constantly evolves. The feature +that makes such dynamism possible is the balance between equilibrium and +disequilibrium that one can observe in the process of creation. The equilibration of +specific operations (i.e., possibilities) increases equilibrium. However, equilibration also +creates a new and more powerful level of organization. Since this new level has greater +power, it represents disequilibrium. Thus, as equilibrium grows, so does disequilibrium. +The two are always in balance and complement each other. + +The creation of combinations conserves operations (possibilities). They do not +disappear. No operation consumes another operation. Both are conserved. No operation +that belongs to a given level of organization is superior to another operation sustained by +the same level. Therefore, their interactions are interactions among equals, which means +that these interactions are non-hierarchical. However, in the course of interactions, +operations create a new and more powerful level of organization. Therefore, they create +a hierarchy. Consequently, the balance between hierarchical and non-hierarchical +interactions is another important feature of the process of creation. Non-hierarchical +interactions create new levels of organization and hierarchical interactions conserve and +optimize these new levels. + +The above features are those that have already been recognized as having the +essential role in the process of creation; they make it possible. In order to create an +economic practice that would be able to replicate the process of creation in its efficiency, +all these features must become part of the new economic practice. There may be some +other features that we still do not know about and that we will recognize as we continue + + +20 +to learn more about the process of creation in the course of the realization of the new +practice. + + +Changes Required for the New Economic Practice + + +In order to make our economy efficient, avoid wastage, and prevent instabilities, +the new economic practice should use the process of creation as its main organizing +principle. In this case, the new economic practice will include all the main features of the +process of creation. As a result, the new economic practice will be different from the +current practice in some critical respects. This new practice will also lead to rethinking +of some key concepts that are relevant to economic practice. + +In contrast to the existing practice, the new practice will involve a much greater +emphasis on creativity. Investing in the production of new ideas is clearly an emerging +trend in the current practice. In the new practice, investments in the production of new +ideas will constitute the increasingly growing portion of total investments. Moreover, +much of these new investments will be focused on radical innovation that is associated +with the emergence of new and increasingly more powerful levels of organization. + + + +1. The New Conception of Management and Leadership + + +The emphasis on creativity will inevitably affect our conception of management +and leadership. In contrast to the current economic practice that is dominated by +hierarchical interactions, there will be a much greater emphasis on the role of non- +hierarchical interactions. Non-hierarchical interactions play the main role in creating new +combinations. They are the source of creativity. Hierarchical interactions conserve and +optimize what non-hierarchical interactions create. As has been explained, the process of +creation requires a balance between the two types of interactions. Maintaining this +balance will be one of the main preoccupations of the new economic practice. + +The top-down approach dominates our current economic (and not only economic) +practice. Indeed, there are some experimental trends that try to moderate hierarchical +domination. But even these new trends do not go nearly far enough in balancing +hierarchical and non-hierarchical interactions. For one thing, these innovative trends +limit the scope of balancing only to top economic and managerial elites and excludes a +large number of people involved in the process of production and exchange, including but +not limited to workers, employees and even small and medium-size businesses. In the +United States, for example, small and medium size businesses do not qualify for a +generous support of the kind that goes to corporate giants, such as GM, Ford, or major +mega-banks. While the market certainly has a non-hierarchical structure, our managerial +culture remains by and large hierarchical.106 The top economic and managerial elites +essentially adhere to principles of hierarchical control, rather than to the non-hierarchical +mode associated with the market. The dominant neo-liberal approach in economic +management has been pursuing and to a significant degree has achieved the merger of the +government and economic elites. Rather than combine the hierarchical principle of +government bureaucracy with the non-hierarchical principle of the market, this approach + + +21 +has strengthened the concentration of power in the elites and enhanced the hierarchical +principle in our society. It has not balanced hierarchical and non-hierarchical +interactions. + +There is a growing number of researchers who recognize the need for a genuine +combination of hierarchical and non-hierarchical principles. One popular trend is the so- +called hybrid solutions, that is, solutions that still see hierarchical and nonhierarchical +interactions as ontologically separate but seek some format in which their coexistence +and limited cooperation can become possible. These solutions are largely eclectic and do +not achieve a true integration.107 John Kotter, the chief innovation officer at Kotter +International and a professor emeritus of the Harvard Business School, typifies this +approach. In his view, hierarchies and networks are two separate structures that excel at +what they do best. Kotter recognizes that hierarchies are very good at optimizing and are +capable of effecting small and medium-size changes but cannot perform large-scale +innovative transformations. He explains: + +But I am referring to something far bigger: large-scale organizational +change, such as a company redesigning its entire business model, or +accomplishing its most important strategic objectives of the decade, or +changing its portfolio of product offerings. And there is no evidence to +suggest that the Hierarchy allows for such changes, let alone that it +effectively facilitates them.108 + +In Kotter’s view, the future lies in the coexistence of the two structures in one business +organization. In his own words: + +All of this has led me to believe that the successful organization of the +future will have two organizational structures: a Hierarchy, and a more +teaming, egalitarian, and adaptive Network. Both are designed and +purposive. While the Hierarchy is as important as it has always been for +optimizing work, the Network is where big change happens. It allows a +company to more easily spot big opportunities and then change itself to +grab them.109 + + +Coexistence is a far cry from a genuine integration. It presupposes that the +competition between co-existing entities will continue and will be merely moderated. +Ultimately, they will not cooperate but they will try to avoid interfering with each other. +This solution is certainly not enough. The two types of interactions should not merely +co-exist, which is a very unstable arrangement based on constant competition, but truly +cooperate and complement each other. They should be part of the same decision-making +process, not merely the two sides in a compromise decision. Compromise solutions +involve emphasizing commonalities and suppressing differences. Yet, differences, not +commonalities, are the main source of radical innovation. + +Hybrid solutions offer a rich plethora of interesting ideas regarding possible +mechanisms of interactions between hierarchies and networks. However, as all eclectic +solutions, they do not have a solid theoretically grounding and tend to have internal +contradictions. Nothing illustrates this shortcoming better than the discussion of such + + +22 +critical subject as the relationship between leaders/managers and networks/employees. +Opinions on this score vary widely: from a more activist role of leaders/ managers as +enablers110 to a weaker role of regulators and filterers of external information,111 to an +even weaker role as facilitators of critical discourse and enhancers of local interactions +among network agents.112 Some even believe that the desired goal can be achieved +without structural changes by merely modifying the rationale for the role of hierarchies +and by educating managers in the values and merits of organizational democracy. Martin +Clarke and David Butcher, for example, see education and the principle of voluntarism +they borrow from political philosophy as vehicles for reconciling hierarchies and +networks in organizational structures.113 + +There is no doubt that the literature on hybrid solutions certainly deserves serious +attention. It addresses many aspects of what is obviously a very complex and seemingly +intractable problem. Many of ideas articulated in hybrid solutions are undoubtedly very +useful. But even all together, they hardly measure up to the magnitude of the task, which +leaves quite a few researchers dissatisfied and vying for a comprehensive solution. In +their essay “Simplistic vs. Complex Organization: Markets, Hierarchies, and Networks +in an Organizational Triangle,” Wolfram Elsner, Gero Hocker and Henning Schwardt +make an argument for just such a comprehensive solution. In their view, “… pure market +and hierarchy, including their potential formal hybrids, are an empirically void set.” +Rather, “coordination forms” in the real world, they argue, “have to be conceptualized in +a fundamentally different way. A relevant organizational space must reflect the +dimensions of a complex world.”114 + +In making their appeal to complexity of the real world, Elsner, Hocker and +Schwardt suggest that the division between hierarchical and non-hierarchical interactions +is not real, it is merely conceptual;115 that in reality, the two types of interactions are +closely entangled with each other, although they fail to explain the nature of this +entanglement. Numerous other researchers support the approach that centers on the +entanglement of hierarchical and non-hierarchical interactions and the complexity of their +relationship. Antoine Danchin points to the ubiquity of networks and hierarchies in +nature and their complementary relationship.116 Joan Roelofs challenges the simplistic +view of networks as spontaneously resistant to hierarchies and naturally prone to +democracy. As she maintains, + +…some participants in network governance are vastly more powerful than +others. As for “civil society” organizations, support from corporate or +private foundations is essential to almost all civil rights, social justice or +environmental organizations that wish to be viable and visible; the funders +exert control in many ways.117 + + +While the above perspectives serve as a valuable source of insights, they +ultimately do not resolve the problem of the relationship between networks and +hierarchies. Despite their astute and nuanced observations on the nature of this +relationship, they still see hierarchies and networks as ontologically separate. In their +view, tensions between networks118 and hierarchies can only be ameliorated, but they will +ultimately always remain and be a potential source of conflict.119 + + + +23 + +The perception that networks and hierarchies are polar opposites, perennially in +tension and conflict with each other, contradicts what we know about systems in nature. +As has been explained earlier, systems conserve themselves by forming bonds, or what +Humberto Maturana and FranciscoVarela called structural coupling,120 with other +systems in their environment as part of the process known as self-organization, thus +creating new organized totalities.121 The process of creating a new organized totality +gives rise to the operation that regulates the functioning of this totality. And that is what +a system is: an organized totality of coordinated operations with a common regulatory +mechanism. + +Since the regulation of a system is a product of combining the capabilities of its +constituent parts, the level of organization that supports regulation is more powerful than +the level of organization of each subsystem or their sum total. The emergence of this +more powerful level of organization creates a hierarchy. As one can see, regulation is a +product of interactions among subsystems. It supervenes on local interactions and vitally +depends on them for its own existence. + +The regulatory function also needs to be conserved. For this reason, it has to form +strong bonds that would activate it; and first and foremost, it should have strong bonds +with the subsystems it regulates. The process of forming bonds between the global level +of regulation and the level of local interactions results in the integration of the two levels +of the system and the adaptation of local interactions to the global operations. Regulation +can facilitate such adaptation. When the weaker operations adapt to the more powerful +level that sustains regulation, they change and gain in power; the re-equilibration of these +enhanced operations increases the power of the regulatory level. Thus, the entire system +evolves. + +This observation regarding systems in general suggests that in social systems the +role of leaders and hierarchies, which also operate at a more powerful level of +organization, must be very similar. By virtue of their position, leaders can enormously +facilitate the integration of systems because they have access and can observe both the +global and the local level of interactions. In order to integrate the system they regulate, +leaders must resort to reflective coding—the procedure that Gödel used in his famous +proof of consistency and completeness.122 It is a creative task because it creates a level of +organization that can incorporate both global functions and local interactions as its +particular cases. + +This role leaders and hierarchies has nothing to do with command and control, +that is, transmitting decisions from those above to those below and overseeing their +implementation. Leaders must appreciate the enormous creative power of local +interactions and be closely attuned to their variations and modifications. Since they rely, +or supervene, so much in what they do on interactions among network agents, or +subsystems of the system, they should promote, regulate, and facilitate these interactions, +not dominate them and impose on them their will. It is a sensitive, delicate, and highly +creative role that involves both cooperation and two-way adaptation. Those who operate +at the global level and those involved in local interactions are, in a way, equal +participants in a common creative enterprise of ensuring the conservation and evolution +of the system that they constitute. + +Because of their location in the liminal space between the system and its +environment, hierarchies and leaders are in a position to reflect critically (that is, + + +24 +observing at the same time the system and also themselves in performing their +function)123 on all interactions among all the local agents and subsystems of the system. +The latter, by virtue of their position, can reflect only on local interactions. For this +reason, the position of leaders makes possible for them to see new and more powerful +possibilities emerging in interactions within the system, as well as recognize, promote, +and facilitate the utilization of these possibilities. + +The creation of new and increasingly more powerful levels of organization that +propels the system’s evolution is incompatible with the relationship of exclusion and +domination. It requires cooperation and close interaction in common creative work. +Such cooperation can only be effective if there is a balance between hierarchical and non- +hierarchical interactions, between hierarchies and networks--managers and leaders, on +one hand, and employees, on the other.124 Leaders should not see their role as that of +ultimate arbiters whose word is decisive and final—far from it. The notion of a leader as +the ultimate arbiter without whom there will be chaos and instability is a result of a +profoundly flawed view that excludes the process of creation from its frame of vision +and, as a consequence, leads to a failure in understanding how systems function and +evolve. This view makes impossible to have clear and rational validation criteria that can +help choose the most powerful level of organization. As has been argued elsewhere, the +current approach largely relies on subjective choices of those at the top of the +hierarchy.125 The lack of such objective and rational criteria of validation is the main +reason why we now tend to defer decisions to top managers. In the absence of such +criteria, all decisions are subjective and all are equal. Recognizing all decisions as equal +is likely to lead to chaos and instability and nobody wants to argue for disorder. As a +result, the common current default is to defer to the decision of those who are at the +highest level in the hierarchy because even a bad decision that preserves order is deemed +better than chaos and instability. How many businesses have paid a heavy price for +limitations of those at the top of their hierarchies? + + + +2. Assessment of Economic Performance + + +The current economic practice uses value as the most important criterion for +assessing results of production. However, we greatly differ in our ways of understanding +value. Some use market share in determining value, others profitability, still others prices +of company’s shares on the stock market; and there is more. The choice depends on how +one defines value. + +There are many definitions of value currently in use. IGI Global, for example, +identifies eight different definitions of value and its list is not the most extensive.126 With +this number of definitions, their differences and diversity do not come as a surprise. +Perhaps the most common formulation defines value as “a measure of the benefit +provided by a good or service to an economic agent.” Currency is the most common +measure of value; consequently, money is the most common way to quantify value. +However, it is not the only measure of value. In another formulation value is “the +maximum amount of money a specific actor is willing and able to pay for the good or +service."127 Caroline Benton, for example, defines value in terms of preferences of + + +25 +individuals who “determine the economic value of a good or service and the trade-offs +that they will be willing to make to obtain it.”128 + +Although in the above formulations money usually objectifies value, the influence +of subjectivity in valuation is unmistakable: value is based on preferences. Preferences +do not have to be monetary either. Yusuke Kuwayama, Justine Huetteman, and Bethany +Mabee, for example, do not consider money as the only and even the most important way +of assessing value. “Simply put,” they argue, “things that have ‘value’ are useful to you, +improve your situation, or simply make you happy or more secure.”129 All these are +more or less subjective factors. + +There are some formulations that try to see value as something calculable. The +emphasis on calculability is supposed to give value the appearance of something +objective. Yet even such formulations cannot escape subjectivity. Market Business News +defines value as “a calculation of the profits an asset has either produced or may produce +in the future.” Value is “a measure of the benefit a product or service provides an +economic agent (person or company).” Ultimately, the willingness of the agent to pay for +something, his or her preferences, still appear in the estimation of value. Finally, all +formulations stress that economic value is not the same as market price that is regarded as +more objective than individual preferences.130 + +Finally, there is this gem of a definition that makes the determination of value a +total exercise in futility. A team from the Corporate Finance Institute offers the +following formulation: + +The economic value of a business is the business’s contribution to the +global gross domestic product (GDP). The most common method of +estimating economic value is the counter-factual method. The counter- +factual method states that the economic value of a business is the +difference between the current global GDP and the hypothetical global +GDP if the business did not exist.131 + + +Even a brief overview of some of the examples of the current definitions +of value shows their inadequacy. They are not uniform. They often contradict +each other. Also, and probably even more importantly, these definitions are to +greater or lesser extent subjective. Basing our assessment of economic efficiency +on such subjective understanding of value cannot result in an objective valuation. +We need a uniform and objective understanding of what constitutes value. The +economic practice that uses the process of creation as its main organizing +principle offers such understanding. + +As the earlier discussion of the process of creation makes clear, efficiency +depends on the use of resources and possibilities. A system—any system, +including economic ones—is efficient if and only if it fully utilizes all resources +and possibilities available to it. Only by combining all these resources and +possibilities, a system can create a new level of organization that will make it +more powerful. Such system will conserve itself and will make an evolutionary +advance. The new level of organization will offer access to new resources and +new possibilities. + + +26 + +In order to be efficient, a system must be inclusive. In order to survive, +business organizations must know their resources, both physical and mental. The +latter are particularly important in our time because of the growing emphasis on +creativity and the production of ideas. If resources are known, one can easily +determine whether production uses all of them or just some. Also, production +involves the creation of combinations of all available operations. In other words, +each operation should establish connections with all other operations. If a system +that has n-number of operations functions efficiently, it will produce the number +of operations that will be equal to n2. In other words, an efficiently functioning +business organization will have an exponential growth.132 Conversely, if a system +demonstrates an exponential growth, it must be functioning with full efficiency. +Profits will necessarily reflect this efficiency. + +Inclusivity and exponential growth are not subjective criteria. They do not +depend on individual preferences. These criteria are derivatives from the process +of creation that is not our subjective construct. This process is not our creation. +On the contrary, this process has created the human race. This process is the +source of value that is determined in objective terms of inclusivity and +exponential growth. + + + +3. The new conception of production and consumption + + +The preceding discussion of the new economic practice focused primarily on +production. However, there is another dimension that is also relevant for determining +economic efficiency. This dimension is consumption. Efficient economy is one in which +all products are consumed and nothing goes to waste. Economic efficiency includes both +production and consumption. + +Despite the fact that efficient economy involves a close interrelationship between +production and consumption, there is a strong tendency in our economic science to +separate the two, even though it recognizes, at least to some extent, their close +interrelationship. Our economists and business people accept the notion that ultimately +we should strive for a full utilization of what we produce. Otherwise, our production will +be inefficient and sustain losses and waste. + +However, despite this recognition, our economic thinking also sees production +and consumption as two ontological opposites. This thinking generally associates +production with appreciation, or value creation, and consumption with depreciation or the +erosion of value. We know very well, for example, that when we buy a new car, the car +will lose a significant part of what we paid for it as soon as it leaves the parking lot of the +dealership. + +The analysis of the process of creation shows that production and consumption +are two closely interrelated aspects of one integral whole. They are merely analytical +categories, not ontological entities that exist separately from each other. When +subsystems form bonds with each other, they assimilate each other. In other words, they +include each other in their own functional operations, which is essentially a form of +consumption. However, the inclusion of a different entity into the subsystem’s functional +operations modifies these operations, thus producing something new that did not exist + + +27 +prior to inclusion. Moreover, even to be open to a possibility of inclusion, a subsystem +must create a construct that allows it to perceive another subsystem in its environment. +Such perception does not come automatically. A system must have a capacity to perceive +difference. Regulation has this capacity to perceive something different form the entity it +regulates. + +Our perception of reality offers a good illustration of this point. In order to +perceive, our mind has to create a mental construct that makes this perception possible. +The act of perceiving assimilates the perceived object and thus stabilizes and conserves +the mental construct that makes the perception of this particular object possible. Thus, +the production of perceived reality goes hand in hand with consumption of this reality. +We cannot perceive what we have not first constructed. Infant’s experience of reality is +defined by inherent sensory-motor operations. Mental constructs that make possible for +the infant to perceive permanent objects are a result of the combination of sensory-motor +operations that create permanent mental images.133 + +Thus, production and consumption go hand-in-hand together. They are both +aspects of the process that creates new properties and new possibilities. Production and +consumption are analytical, rather than ontological categories. As the analysis of the +process of creation shows, the conservation of a given level of organization creates a new +and more powerful level of organization. The emerging level of organization supervenes +on the level from which it emerges; in other words, the level of organization that gives +rise the new level of organization is a resource that is consumed in the process that +creates the new level. One can also represent this relationship between consumption and +production as a balance between equilibration and the production of disequilibrium. In +this conception, disequilibrium is a resource for equilibration; and equilibration results in +a new disequilibrium. + +What practical consequences will the recognition of the unity of consumption and +production? How will it affect our economic practice? + +Organizing our economic activities around the process of creation will end the +tendency to dissociate production from consumption; it will make the interrelationship +between production and consumption effective and efficient. By complementing each +other, they both will be able to grow exponentially and will make possible an exponential +growth of our entire economy. One can see the contours of this new economic +organization in the comment of Alan Webber who concludes: “In the end, the location of +the new economy is not in the technology, be it the microchip or the global +telecommunications network. It is in the human mind.”134 + +In contrast to ordinary goods, knowledge does not depreciate when used. On the +contrary, it appreciates. In other words, its value grows. Our assimilation of ideas +creates new and increasingly more powerful levels of organization that give rise to even +more innovative ideas, approaches, and decisions. By producing new and increasingly +more powerful levels of organization we generate new knowledge and ideas that lead to +economic growth. Knowledge, for example, is one important product that does not +depreciate. It only appreciates when consumed, as its consumption leads to new and +increasingly more powerful levels of organization that give rise to new knowledge and +ideas. As Thomas Davenport and Lawrence Prusak have noted, “ideas breed new ideas, +and shared knowledge stays with the giver while it enriches the receiver.”135 + + +28 + +The realization that production and consumption are intimately related will +change our patterns of investment. The economic practice that uses the process of +creation as its main organizing principle will focus investments on the creation of new +and increasingly more powerful levels of organization, not merely on putting new items +on store shelves. Such new pattern of investment will be becoming increasingly +important with the on-going shift of the emphasis in our economy from production of +things to production of knowledge and ideas. + +The change in the pattern of investment will enhance our production and +economic efficiency. In today’s economy many goods that reach the market often face +no demand. They are either drastically discounted or completely trashed. With the new +pattern of investment and the emphasis on the production of knowledge and ideas we will +avoid this waste. As our production grows exponentially, so will our consumption; yet +we will not be facing a situation that what we are not able to consume what we have +produced. Just like our production, our consumption can also grow infinitely; and this +growth will not be posing any threats to our planet or the universe. On the contrary, they +will help us in solving numerous problems that we face. Moreover, they will also help us +anticipate future problems and find their solutions even before these problems emerge. +Finally, the new patterns of production and consumption will help us live better and have +a more satisfying and happier life. + + + +4. Education + + +The importance of knowledge for economic production is hard to overestimate. +Knowledge has always been a major contributor to economic progress. Its contribution is +particularly important in this day and age when our economy puts premium on +knowledge production and creativity; our economy is increasingly about knowledge +production.136 + +Education is one important sphere of our civilization that directly relates to +knowledge. Traditionally, storage and dissemination of the existing knowledge has been +the main preoccupation of education. By transmitting knowledge to new generations of +young men and women, the system of education prepared them for becoming productive +members of society. Modernity self-consciously used education to set our civilization +onto a new course. Since the very beginning of the industrialization, the connection +between education and economic progress acquired particular importance. Educational +institutions became the main breeding ground where students were trained to become part +of the labor force that propelled the industrial development. They became the workers, +technicians, managers, business organizers who built and operated the growing number +of factories and plants. The connection between education and economic progress has +become particularly important in this day and age when the production of innovative +ideas, approaches, and decisions has become the most important production factor. + +Due to the close association between education and economic progress, economic +needs have always shaped our educational system. There is no doubt that the transition to +the new economic practice will necessitate changes in our education and teaching +methodology. + + +29 + +As has already been pointed out, educational institutions were traditionally +preoccupied mostly with the dissemination of knowledge. Our educational system +exposed students to the vast body of information and skills that were accumulated in the +course of the evolution of human civilization. However, it rarely, if at all, taught students +the habits and skills required for producing new knowledge. Indeed, many talented +young men and women engaged in knowledge production and became the source of +radical innovations. However, their success owed very little to the educational system. +The manifestation of their genius occurred contrary to this system, rather than because of +it. + +Our educational system does pitifully little to enhance and develop the creative +capacity of young people. Generally, our educational institutions relegate creative type +of activities to art and literature courses.137 There is very little room for creativity in +mainstream courses in sciences, including social sciences, and math. As a result, more +often than not, education stifles students’ creative impulses and suppresses their capacity +to create. This situation is hardly an accident. Our knowledge of what makes creation +possible is meager. The sad fact has been and remains that our understanding of the +process of creation remains very rudimentary. As a result, our ability to control this +process is extremely limited. Margaret Boden, one of the pre-eminent researchers in the +field, draws the following conclusion in her influential book on creativity: + +Our ignorance of our own creativity is very great. We are not aware of all +the structural constraints involved in particular domains, still less of the +ways in which they can be creatively transformed. We use creative +heuristics, but know very little about what they are or how they work. If +we do have any sense of these matters, it is very likely tacit rather than +explicit: many people can be surprised by a novel harmony, but relatively +few can explicitly predict even a plagal cadence.138 + +This situation must change with the onset of the new economic practice. Some radical +innovations must take place in our theory of education and teaching methodology. + +It is beyond the scope of this study to provide a detailed description of the kind of +changes that are in order. The transformation of our education is a project that is only +beginning. Many of the required changes will become clear only in the course of this +project’s implementation. However, the description of some essential features is possible +today. + +Whatever new forms our education will take, the process of creation should be +their main model and inspiration. Therefore, the recognition of the importance of this +process, further study of this process, and a comprehensive understanding of it are +essential. Our educational system should use the process of creation as its main +organizing principle. + +The new teaching methodology must include important features that are known +and characterize the process of creation. Universal inclusion and empowerment must +constitute the basis of the new methodology. Teaching should also observe important +balances that sustain the process of creation: the balance between equilibration and the +production of disequilibrium, or the balance between equilibrium and disequilibrium, and +the balance between hierarchical and non-hierarchical interactions. One can find a more +detailed description of the proposed innovations elsewhere.139 + + +30 + + +Conclusion + + +Economic production is, in more ways than one, a very important building block +of our civilization. It provides means for our sustenance, helps us control our +environment, and makes our life more comfortable, enjoyable, and satisfying. But its +importance transcends these utilitarian purposes. Economic production strengthens and +develops our capacity to create, reinforcing our connection to the universal process of +creation and, thus, to our entire universe. + +This article argues that our economic problems, of which inflation is only one, are +due to the fact that our economic production is inefficient. Its inefficiency is a result of +the inadequate use of resources and possibilities available to us, most importantly human +capacity to create an infinite number of new and increasingly more powerful levels of +organization. This inadequacy is not fortuitous; it is not a result of our mistakes or lack +of industriousness and diligence. It is a result of our lack of understanding of the roots of +our own existence—the universal process that led to the rise of humanity and has been +driving the evolution of human civilization. To this day, we have not grasped the +important role this process plays in our relationship with reality.140 + +The article shows the connection between economic production and the process of +creation. It also shows that the reason why our economy is inefficient is the fact that we +do not know what factors makes the process of creation so efficient; consequently, we +cannot replicate these factors in our economic activities. As the article shows, these +factors include universal inclusion, the balance between hierarchical and non-hierarchical +interactions, as well as the balance between equilibrium and disequilibrium. Only full +utilization of all available resources and possibilities can create new and increasingly +more powerful levels of organization that can give rise to new ideas, decisions, and +approaches. Such efficient use of resources will lead to constant, stable, and exponential +economic growth without inflations, business cycles, and economic contractions and +crises. + +Understanding the process of creation is only the first step in addressing the +problem of our economic inefficiency. As this article has argued, we have to acquire +knowledge of the process of creation and use this knowledge in establishing the new +economic practice that will use the process of creation as its main organizing principle. +Such new practice will involve changes in investment patterns, conception of +management and leadership, as well as re-conceptualization of consumption and the +relationship between production and consumption. + +Indeed, some may make an argument that even though we do not understand the +process of creation and its role in our relationship with reality, our civilization has been +able to make a remarkable progress in the course of its history. This fact only reinforces +the main point made in this article. It shows the enormous power of the process of +creation that has helped our civilization to evolve to this point despite our disregard of +this process and our failure to utilize fully its infinite possibilities. In a way, we have +been able to make a remarkable progress despite ourselves. + +However, our civilization can progress only so far if it continues to ignore the +process of creation. The way we have pursued progress so far has limitations.141 In order + + +31 +to continue the progress of our civilization into the future, we need to embrace, +understand, and use this process efficiently to transcend our self-imposed limit. Our +production has reached a limit of what it can achieve without availing ourselves fully of +the enormous resources and possibilities that the process of creation offers. Our +economic production has evolved to the point when our capacity to create is rapidly +becoming the most important factor without which our production simply cannot evolve +any further and help us in solving the problems we now face. + +Our civilization badly needs new ideas,142 including new ideas in the way we run +our economies. So far, there have been two main approaches in organizing our +production. Liberalism promotes one approach and socialism another. Both versions +ignore the process of creation; and both have reached their limit. The many disruptions +of the world economy are an eloquent indication of the inadequacy of the current +approaches. + +Capitalism is the term that has been much abused and maligned; the meaning of +capitalism has been perverted and ideologically misconstrued. Capitalism is essentially +about the growth of capital. Capitalists are people who focus their skills, energy, and +resources on increasing capital, that is, on creating new value. The rise of capitalism +represented a realization that making capital grow and creating value is so important that +we should concentrate our will and resources on promoting constant growth. The growth +of capital has become a self-conscious goal. + +In this sense, Western liberalism and Marxist socialism are not very different +from each other, despite their assertions to the contrary. Marx’s teaching that has +probably done more than anything else to discredit capitalism is in this sense very +capitalist in its spirit and orientation. It is also about capital growth; it is also about value +creation. In fact, Marx’s main argument against capitalism is that capitalism has flaws +that do not allow growing economy indefinitely. Socialism, according to Marx, was +supposed to remove these fetters and to ensure unlimited growth. His argument for the +capacity of socialism to accomplish this task has been a dismal failure. However, the +passion and conviction that ring in this argument are genuine; they reflect his aspiration +for infinite growth that is essentially “capitalist” in its nature in the sense in which +capitalism has been defined above. The similarity in the pursuit of economic growth +between liberalism and socialism is perhaps one reason why Western liberalism has +proven to be receptive to socialist ideas. Many even mainstream progressive liberals in +the United States today see solutions to the current problems in socialist policies, such as +redistribution of wealth and the involvement of the state in managing the economy. + +The rise of capitalism was an important evolutionary development. The idea that +the survival of our civilization vitally depends on perennial creation of value and capital +growth is our articulation of what we have inherited in the course of the evolution from +the process of creation. Our anthropocentric bias has prevented us from grasping the +cosmic significance of this process. We have failed in moving beyond our vague +intuitions about it. The anthropocentric attribution of the emergence of capitalism +exclusively to humans and their decisions has overshadowed the intuition of early +capitalists who believed that their ability to increase capital was divine in its origin. +These “secular monks” created much value and acquired enormous wealth, yet they did +not use this wealth for the sake of hedonistic enjoyment; they relished with the abandon + + +32 +of religious ecstasy in their very capacity to create value that they felt brought them +closer to God the Creator. + + +33 +ENDNOTES + + +1 David McMillan, “Inflation: There’s a Vital Way to Reduce It That Everyone +Overlooks—Raise Productivity,” World Economic Forum, June 9, 2022, +https://www.weforum.org/agenda/2022/06/inflation-there-s-a-vital-way-to-reduce-it-that- +everyone-overlooks-raise-productivity/. + +2 Fabio Vighi, “A System on Life Support,” The Philosophical Salon (blog), September +5, 2022. https://thephilosophicalsalon.com/a-system-on-life-support/. + +3 Vighi, “A System on Life Support.” + +4 Alexander Barta, “Inflation in Economic Theory | Exploring Economics,” Exploring +Economics, 2021, https://www.exploring-economics.org/en/discover/inflation/. + +5 Kat Tretina, “Is Inflation Good Or Bad?” Forbes, September 14, 2022. +https://www.forbes.com/advisor/investing/is-inflation-good-or-bad/. + +6 Ceyda Oner, “Inflation: Prices on the Rise,” Finance and Development IMF, July 15, +2022, 30–31, https://www.imf.org/en/Publications/fandd/issues/Series/Back-to- +Basics/Inflation. + +7 Joseph E. Stiglitz, “All Pain and No Gain from Higher Interest Rates,” Project +Syndicate, December 8, 2022. https://www.project-syndicate.org/commentary/fed- +interest-rate-increases-counterproductive-all-pain-no-gain-by-joseph-e-stiglitz-2022-12. + +8 Spang, Rebecca L. “The Rise of Inflation | Rebecca L. Spang.” Cabinet. Accessed +November 27, 2022. https://cabinetmagazine.org/issues/50/spang.php. + +9 Friedman, Milton. The Counter-Revolution in Monetary Theory: First Wincott +Memorial Lecture, Delivered at the Senate House, University of London, 16 September, +1970. 1970. London: The Institute of Economic Affairs, 1970. + +10 Barta, “Inflation in Economic Theory | Exploring Economics.” + +11 Oner, “Inflation: Prices on the Rise,” p. 30. + +12 “Inflation—The ‘Cost-Push’ Theory,” Britannica,” +https://www.britannica.com/topic/inflation-economics. + +13 Adam Tooze, “Chartbook Newsletter #22: How do you count inflation? Tracking +Weimar's hyperinflation.,” 2021, https://adamtooze.substack.com/p/chartbook- +newsletter-22. + + + +34 + +14 “What Causes Inflation?” Forbes, May 28, 2022, +https://www.forbes.com/advisor/investing/what-causes-inflation/; “What Causes +Inflation?” Investopedia, https://www.investopedia.com/ask/answers/111314/what- +causes-inflation-and-does-anyone-gain-it.asp; “Understanding Inflation,” Pacific +Investment Management Company LLC., https://europe.pimco.com/en- +eu/resources/education/understanding-inflation; “What Causes Inflation?” Stanford News +(blog), September 6, 2022, https://news.stanford.edu/2022/09/06/what-causes-inflation/. + +15 Gerald P. Dwyer Jr. and R. W. Hafer, “Are Money Growth and Inflation Still +Related?” Economic Review (07321813), vol. 84, no. 2 (Quarter 1999), p. 32; Peter +Ireland, “The Classical Theory of Inflation and Its Uses Today,” Presented at the Shadow +Open Market Committee Meeting, November 3, 2014, p. 1. + +16 “Monetarist Theory of Inflation: Meaning & Examples,” StudySmarter US, +https://www.studysmarter.us/explanations/macroeconomics/economic- +performance/monetarist-theory-of-inflation/. + +17 Milton Friedman, Counter-Revolution in Monetary Theory (London: The Institute of +Economic Affairs, 1970. + +18 Ireland, “The Classical Theory of Inflation and Its Uses Today,” p. 1. + + +19 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?” p. 32. + +20 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?” p, 37. + +21 David McMillan, “Inflation: There’s a Vital Way to Reduce It That Everyone +Overlooks--Raise Productivity,” World Economic Forum, June 9, 2022, +https://www.weforum.org/agenda/2022/06/inflation-there-s-a-vital-way-to-reduce-it-that- +everyone-overlooks-raise-productivity/. + +22 Pradana M. 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Maturana and Francisco J. Varela, The Tree of Knowledge: The +Biological Roots of Human Understanding (Boston & London: Shambhala, 1998); +Humberto Maturana, “Autopoiesis, Structural Coupling and Cognition: A History of +These and Other Notions in the Biology of Cognition,” Cybernetics & Human Knowing, +vol. 9, no. 3–4 (2002), pp. 5–34. + + +121 Ilya Prigogine and Isabelle Stengers, Order out of Chaos: Man’s New Dialogue with +Nature (Toronto ; New York, N.Y: Bantam Books, 1984); Stuart A. Kauffman, The +Origins of Order: Self-Organization and Selection in Evolution, 1 edition (New York: +Oxford University Press, 1993); Niklas Luhmann, Social Systems (Stanford: Stanford +University Press, 1995); John A. Buck and Gerard Endenburg, “The Creative Forces of +Self-Organization,” Sociocratic Center, Rotterdam, Netherlands, 2016, +https://sociocracyconsulting.com/wp-content/uploads/2016/04/CreativeForces- +updated2012.pdf. + + +122 Ernest Nagel and James R. 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1993. https://hbr.org/1993/01/whats-so-new-about-the-new-economy. + +Weinstock, Lida R. “Introduction to U.S. Economy: The Business Cycle and Growth.” +Congressional Research Service, January 13, 2022, +https://sgp.fas.org/crs/misc/IF10411.pdf; + + diff --git a/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/load_file.txt b/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed46a1375bcc3bc04606d6d406f8367d87e66b9c --- /dev/null +++ b/3NE1T4oBgHgl3EQfSQNJ/content/tmp_files/load_file.txt @@ -0,0 +1,1937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf,len=1936 +page_content='1 INFLATION AND VALUE CREATION: AN ECONOMIC AND PHILOSOPHIC INVESTIGATION Gennady Shkliarevsky Abstract: The subject of this study is inflation—a problem that has plagued America and the world over the last several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Despite a rich trove of scholarly studies and a wide range of tools developed to deal with inflation, we are nowhere near a solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We are now in the middle of the inflation that threatens to become a stagflation or even a full recession;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and we have no idea what to prevent this outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This investigation explores the real source of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Tracing the problem of inflation to production, it finds that inflation is not a phenomenon intrinsic to economy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' rather, it is a result of inefficiencies and waste in our economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The investigation leads to a conclusion that the solution of the problem of inflation is in achieving full efficiency in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our economic production is a result of the evolution that is propelled by the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to end economic inefficiencies, we should model our economic practice on the process that preceded production and has led to its emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In addition, the study will outline ways in which our economic theory and practice must be changed to achieve full efficiency of our production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Finally, the study provides a critical overview of the current theories of inflation and remedies that are proposed to deal with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Key words: inflation, business cycle, recession, value creation, Keynesianism, neo- classical economics, evolution, and the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='13140/RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='30512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='23046 Introduction Inflation is a very familiar word in economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is also a highly popular topic these days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Not a day passes without hearing this word from politicians, economists, mainstream media sources, or even in casual conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The reason for this popularity is the fact that the world economy is in the grips of inflation, and not just an ordinary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For one thing, the inflation is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Even in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' where the level of inflation is now the lowest in the world, it is close to 9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This inflation does not affect one particular or even group of countries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it is a truly worldwide phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Even in developed G-7 countries it reaches over 10% and is even much higher in less developed and underdeveloped countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1 It is a global phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Fabio Vighi offers this characterization: We have entered a global cycle of secular inflation that is unique in history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The cynical attempt to preserve a system based on the ontological 2 assumption of permanent monetary injections now entails the controlled demolition of the real economy and the world it supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2 So far, all efforts to curb this inflation by raising interests rates have failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many observers point to an imminent danger of this inflation becoming a run-away inflation or hyperinflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Even worse, it can turn into a stagflation, similar to the one that erupted in the mid-1970s that had very severe economic and political ramifications from which our economy has not recovered to this day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The consequences of the current inflation can be even worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For example, if the inflation of the 1970s led to the demise of the Democratic Party for over a decade, this inflation may very well wipe it out completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The current inflation has already led to the wave of political discontent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This wave has already led to the election of Donald Trump and to the establishment of Republican control over the House of Representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is no doubt that this inflation will play a very important role in the 2024 elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many researchers fear that this inflation will bring upon us increasing “ideological manipulation and authoritarian violence.”3 Inflation has always been one of the macroeconomic phenomena that attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation and its counterpart, deflation, have always provoked heated debates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='4 Decades of research have expanded our knowledge of various factors that drive inflations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Although we have come to understand a lot about inflation, we still do not have a clear idea as to the root cause of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We are not even sure whether inflation is necessarily a bad thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='5 In fact, many economic planners currently believe that we need inflation and one or two per cent inflation is actually good for our economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We have also learned that deflation--the opposite of inflation--is not much better than its counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Japan is a sad example of its pernicious impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='6 The result of the confusion and controversies surrounding inflation is the lack of coherent and clear policies that would curb the current inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Despite the fact that the Fed is currently increasing interest rates to fight the inflation, there are many prominent economists who criticize the Fed’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Joseph Stiglitz sees in it nothing but pain and no benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' He writes: “As a new Roosevelt Institute report that I co-authored shows, any benefits from the extra interest-rate-driven reduction in inflation will be minimal, compared to what would have happened anyway.”7 We seem to be permanently deadlocked as we are trying, on one hand, to avoid high unemployment and, on the other, to prevent run-away price increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The purpose of this article is to find a way out of the current conundrum related to inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As part of the search for a solution, the article will provide a critical survey of the current views on inflation and popular approaches in dealing with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Much of the current research on inflation focuses on various factors that drive inflations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' No doubt, this knowledge is important but it is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The factors that drive inflation are relevant but they seem to be phenomena that attend inflation, rather than cause it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For example, we certainly accept the notion that growing money supply drives price increases, but this notion tells us nothing about the reason why our policy planners decide to boost the supply of money, even though we know perfectly well the consequences of such policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In many cases, economic planners resort to this policy in order to contain unemployment, which suggests that our economy is always in a double jeopardy, oscillating between inflation and unemployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is a vital need to determine what 3 and why sets price stability and employment on a collision course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This article will address all these questions and issues related to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Current Views on Inflation and Its Causes Definitions The term “inflation” in relation to economic processes came into use in the second half of the 19th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='8 Back then the term “inflation” referred primarily to the expansion of the currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This view is still popular today, particularly among neo- classical economists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' According to Milton Friedman, for example, inflation “is always and everywhere a monetary phenomenon.”9 Only later, economists began to define inflation in terms of price increases that might result from monetary expansion but could also have other causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='10 Today, most economists think about inflation as price increases for goods and services, or “how much more expensive the relevant set of goods and/or services has become over a certain period, most commonly a year.”11 There are certainly wordier definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The Encyclopedia Britannica offers the following expanded version: Inflation, in economics, is collective increases in the supply of money, in money incomes, or in prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation is generally thought of as an inordinate rise in the general level of prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='12 In his recent contribution, Adam Tooze offers an even lengthier formulation: In discussing inflation, economics abstracts from idiosyncratic shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation is defined as a general upward pressure on all prices, independent of idiosyncratic supply shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation, in this sense, is a macroeconomic, aggregate concept .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation can thus be defined as a shift in the terms of trade between (1) money and (2) goods, as experienced (3) by a particular group of people and (4) captured by a particular statistical apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='13 No matter how long or how sophisticated current definitions are, none of them really ventures beyond the mere description of inflation and factors attending to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Perceived Causes of Inflation There is no shortage of theories that try to explain what causes inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='14 They are different and diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yet all they do is create confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They all focus on some specific factor or factors attending inflation and try to reduce their explanation what they see as the cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The factors they stress are often valid, but together they create a contradictory picture that defies understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 4 Perhaps the most common explanation of what causes inflation is a monetarist one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is also by far the oldest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' David Hume recognized the link between the expansion of money supply and price increases in his essay “Of Money” that appeared in 1752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This explanation remains popular to this day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='15 According to this school of thought, there is only one cause of inflation and it is an over-issue of inconvertible money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='16 Milton Friedman is the best-known representative of the monetarist school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In his concise, if restrictive, formulation, inflation is a monetary phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As he put it, inflation “is and can be produced only by a more rapid increase in the quantity of money than in output.”17 The reason for the continued popularity of the monetarist explanation is the fact that it enjoys more empirical support than any other economic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='18 However, despite its long history and the substantial factual evidence, the predicted association between money supply and inflation remains disputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='19 For one thing, this association works only over long periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Also, there are some examples of variations in money supply that do not necessarily negatively correlate with price levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For example, decreases in the supply of money do not necessarily lead to price increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' During the Great Depression decreases in money relative to real incomes were associated with decreases, not increases, in price levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='20 Also, the monetarist explanation of what causes inflation is not unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The link between money supply and the availability of goods/values goes both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If the increase in money supply can cause price increases, so can the decline in the production of goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The connection between inflation and production is a common preoccupation among economists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' David McMillan, for example, explains the high rate of inflation in Britain by the sluggish growth of productivity and low rate of investment that lags behind Britain’s main economic competitors—the United States, Germany, and France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='21 The results of a study that has tested the link between inflation and economic growth in Sri Lanka show that there is “a long run negative and significant relationship between the economic growth and inflation.”22 The study shows a similar but short-term impact for China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='23 A study by Hrushikesh Mallick also reveals the negative correlation between inflation and growth in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='24 Government spending is yet another popular explanation for inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Conservatives have expertly used this explanation in the past as a way of defeating their opponents who favor the expansion of government programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Ronald Reagan effectively employed this cudgel by arguing that extensive and expensive government programs had a devastating effect on American economy and were the principal cause of the stagflation that occurred in the mid-1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can hear very similar criticisms of the government these days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In his contribution “Inflation and Growth: Some Theory and Evidence” Max Gillman challenges liberal economists and politicians who explain the current high rate of inflation by the war in Ukraine, “corporate greed,” or short-term supply issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The cause, in his view, is “bad public policy.”25 Nathan Benefield makes a similar charge in his editorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Washington politicians,” he opines, “pretend they don’t know what’s behind runaway inflation, but voters know.” Pointing to what he sees is the real issue, Benefield quotes James Carville, a Democratic strategist, who quipped on one occasion: “It’s the spending, stupid.”26 This list of explanations is far from complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The number of factors that researchers cite as causing, contributing to, or driving inflation is much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is 5 beyond the scope of this study to examine them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They include supply bottlenecks, unequal distribution of money, sluggish price adjustment, too “few” unemployed, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='27 There are even sociological perspectives on inflation that focus on the role, interests, and expectations of social actors as a cause of inflationary tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='28 In reading various authors, one gets an impression that researchers base their explanations on their theoretical or political commitments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and that what they present as causes of inflation are merely their subjective preferences among factors that attend inflations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Gregory Mankiw appears to say as much when he writes about economists “dirty little secret” since they often speak “not just as economic scientists, but also as political philosophers.”29 Alexander Barta observes: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' [I]t is elementary to recognise that the very measurement of inflation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' the determination of the basket of goods and services whose prices are taken into account as much as the very production of the statistic itself, is a selective, theory-driven and, indeed, political choice already--there is no such thing as “objective inflation” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Data are business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Data are political.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' And that is particularly pertinent in the case of inflation, because inflations are contentious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They generate winners and losers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='30 There are no known factors cited as a cause of inflation that would not be put in doubt by empirical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Even the most commonly cited cause—money growth—is not without counter examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As has already been mentions, during the Great Depression of the 1930s money relative to income and the price levels were both in decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='31 The examination of the current explanations of what causes inflation shows that researchers have different views on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The causes they cite—supply of money, production rates, structure of the economy, distribution of wealth and money, savings and shopping habits, and many others—are merely co-dependent economic variables attending inflations, rather than causing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Barta correctly points out that inflation cannot follow from any of these co-dependent variables because variations in any of them simply leads to “recalibration of the system of relative prices by the Walrasian Auctioneer, but not to a rise in the general price level.”32 On close analysis, what is often perceived as a cause of inflation seems to be a symptom—a variable attending to inflation—rather than its cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation Remedies There are two principal approaches in dealing with inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One emphasizes the role of the government in managing inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The other recommends relying on market forces, rather than government fiat, to achieve price stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation Management 6 The roots of the first approach rest on Keynesian economic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Keynes challenged neoclassical economics and its reliance on market forces to regulate the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The pro-active approach advocated by Keynes proposed that the government was to use a variety of levers available to it to manage economy and inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' According to this approach, inflation is neither bad, nor good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' or rather, it is bad and good at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Good inflation is one that stays at a recommended level of 1% to 2% a year or 10% to 12% for developing countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='33 The proponents of this approach maintain that while high inflation can be harmful, too little inflation or even negative inflation, or deflation, can also weaken the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Japan, for example, has experienced a long period of practically no economic growth primarily because of deflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Following the financial crisis of 2007, the Fed and other central banks around the world promoted low interest rates and other monetary incentives to make sure that liquidity stays sufficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' they euphemistically called it “quantitative easing.”34 The main thrust of the Keynesian approach and its variations is to provide a framework in which the government can manage economic activity by varying its expenditures and receipts or by influencing the level of private investment through interest rates and money supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='35 The followers of Keynes do not seek to eliminate inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Rather they try to ameliorate its most harmful effects and keep it at a manageable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The Federal Reserve, or the Fed, is the main institution that regulates the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The Fed has two jobs: one is to maintain price stability and the other is to maximize employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When the economy is struggling and inflation is too low, the Fed will lower interest rates or buy assets to increase the amount of available cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When the economy is expanding too quickly and inflation rises, the Fed will typically raise interest rates or sell assets to reduce the volume of cash in circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Borrowing money becomes expensive, which slows economic growth and brings down the level of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='36 The theory that guides the Fed and most central banks aims at mild inflation at a level of one or two per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='37 However, this theory is little more than just a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The policy actions of the Fed and other central banks in the face of inflation may include raising interest rates at which central banks provide reserves to financial intermediaries, buying and selling government securities, changing reserve requirements, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='38 This approach is very eclectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Its proponents offer a variety of policies and policy combinations that are supposed to help curb inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They propose to lower health costs, reform the tax code to raise more revenue, limit discretionary spending, reduce consumption-oriented spending, cut aid to states, reduce costs of energy, trade, and procurement, and much else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='39 Even a more restrained fiscal policy is not off-limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As President Biden has remarked numerous times in the current inflation period, “bringing down the deficit is one way to ease inflationary pressures.” This could include avoiding further deficit-boosting measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' lowering health care costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' raising tax revenue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' reducing consumption-oriented spending;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' promoting work, savings, and investment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and/or lowering energy, trade, and procurement costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='40 Even when the American government abandoned the policy of low interest rates in response to inflationary pressures, many Keynesians criticized this move by insisting that giving up this policy was wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Joseph Stiglitz, for example, has argued in his article that in a recent issue of Project Syndicate that high interest rates bring nothing but pain to the economy and have no benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='41 7 One should not get an impression that this approach to inflation is limited to Democrats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' President Donald Trump throughout his presidency publicly pressured the Federal Reserve to keep interest rates low and to keep pursuing bond-buying monetary expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In one of his tweets, Trump wrote in April 2019: We have the potential to go up like a rocket if we did some lowering of rates, like one point, and some quantitative easing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yes, we are doing very well at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2% GDP, but with our wonderfully low inflation, we could be setting major records &, at the same time, make our National Debt start to look small!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='42 As the current inflation continues to grow, even the current American government that is dominated by Keynesians has launched a policy of high interests in expectation that this measure will reduce inflation even if at the cost of employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can surmise that the intention now is to keep on alternating between the policy of high interest rates and the policy of high employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is not much theory behind this constant alternation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it seems to be more of a trial-and-error approach driven by wishful hope with little empirical support that somehow the economy will grow out of the current predicament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Price control Perhaps the most radical policy that exemplifies the approach that favors government intervention is price control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There are few economists and politicians who advocate this policy, but it is on the menu of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can never trust politicians who may be opposed to price control one day and then change their position the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Power brokers tried this policy in the 1970s when inflation was devastating family earnings and when the buying power of common Americans was severely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='43 President Nixon preached to Americans in 1965 that “the lesson that government price fixing doesn’t work is never learned.” When campaigning for president, Nixon pledged that he would “not take this nation down the road of wage and price controls.” However, in 1971, when inflation reached six per cent, Nixon, against his own previous judgment, began to pressure the business community, with little or no success, to hold down prices and wages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Prices continued to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='44 Trying to deflect criticism, President Carter blamed rising prices and recession on OPEC, also with little success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' President Biden’s initial response to rising prices was dismissive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' He simply declared them temporary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When prices continued to grow, Biden blamed rising prices on the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When the war in Ukraine erupted, Biden quickly put the blame on Putin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In his view and that of other Democrats, the rising prices justified emergency action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Elizabeth Warren and the host of other left-learning Democrats rushed to vilify “greedy corporations” for exploiting the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Later in 2021 The New York Times ran a trial balloon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The paper wrote approvingly that as rising inflation was threatening Biden’s presidency, he was turning to “the federal government’s antitrust authorities to try to tame red-hot price increases.”45 Politicians certainly took the cue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Representative Jamaal 8 Bowman (D-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=') obliged the president by introducing the Emergency Price Stabilization Act, calling on the government to “build the capacity to establish limits on the growth of certain prices, and to otherwise strategically regulate such prices, in order to stabilize the cost of essential goods and services.”46 Targeting price stability as a way to achieve higher rates of economic growth is also popular in countries with emerging markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The Neo Classical Approach As has already been mentioned, the other approach toward inflation has originated in neo-classical economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The advocates of this approach stress that the availability of resources to produce goods, services, and particularly technological change is a major factor affecting growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='48 The theory that underlies this pro-growth perspective maintains that producing more goods and services in a shorter time would cut costs per unit, raise supply, and thus put downward pressure on prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='49 The secret to curing inflation, in their view, is pro-growth policies that create incentives for more goods, more employment, less government spending and sound money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When production grows, prices go down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='50 However, in contrast to interventionists, the pro-market proponents insist on reducing the role of the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The policies pursued in the market-oriented approach stress the need to reduce government interventions into the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such reduction takes several forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' First of all, it involves severe restrictions to the government’s capacity to increase the supply of money—either by limiting the issue of paper money or by preventing the government from manipulating interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The idea is that if the government stops playing with money supply, investors can get a realistic picture of their opportunities, better calculate risks, and have sober expectations about profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Also, the proponents of the market-oriented approach advocate the reduction of the federal budget by eliminating high-cost government programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Cutting government programs was a signature policy introduced by Ronald Reagan and continued under Bill Clinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Lawrence Summers, who served as Bill Clinton’s Treasury secretary, rocked the Democratic establishment last year by predicting that his party’s excessive spending would cause inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='51 For Nathan Benefield, Senior Vice President of the Commonwealth Foundation--Pennsylvania’s free-market think tank—government spending is the main factor that generates inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' He succinctly summarizes the reason for the current inflation by paraphrasing the Democratic strategist James Carville: “It’s the spending, stupid.” In his view, if policy makers are serious about fighting inflation, they should start with fiscal restraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='52 In accordance with the pro-market approach, reduction of government spending should go hand-in-hand with cutting taxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Tax cuts will give more money to both business owners and regular working families;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and more money for them will invigorate the market, ensure growth, and cut inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is no shortage of calls from both the right and the left on lawmakers demanding to enact a comprehensive tax reform that would ease burdens on working families and enable small businesses to hire more workers and raise wages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In their article “Economic Growth, Not Austerity, Is the 9 Answer to Inflation” Arthur Laffer and Stephen Moore cite historical examples that prove their point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They write: History proves growth doesn’t cause inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In the 1920s, when the highest tax rate was cut from 73% to 25%, real GDP soared and the price level fell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In the 1960s, tax cuts and pro-growth policies led to an economic expansion, stable prices and budget surpluses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='53 Lastly, many economists and politicians also see the need to reform or even eliminate various excessive regulations imposed by the government that constrain economic activity and negatively impact production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Those who support reducing regulations argue that such reforms would help producers to control costs and unleash energy production to lower electricity and fuel prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='54 Critique of the Proposed Remedies As the above shows, recipes for dealing with inflation are different and diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The proponents of each approach mostly focus on the advantages of what they propose and prefer to ignore disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, an objective assessment needs to take into account all sides of what is proposed—both positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Criticisms in this section will focus on shortcomings of these recipes, both those specific to each one and those common to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Managing Inflation Perhaps the most serious disadvantage of the managing inflation approach is the fact that it has proven to be completely ineffective in the current inflation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The proponents of this approach have claimed that maintaining a tolerable level of inflation at one or two percent a year is relatively harmless and will have no serious ramifications for the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In their view, this approach should allow making financially sound decisions on saving, investing and borrowing money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='55 However, trying to manage inflation at a desired level has proven to be very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since 1921 secular inflation in the United States and throughout the world has been a permanent presence in the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There have been rises and declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The current secular inflation has grown steadily in the last decade or so and reaches today the level of close to 9% in America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Among all developed countries, the inflation is the highest in Great Britain reaching the ominous 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' All through this period, the Democrats defended and continue to defend the policy of high government spending and expensive government programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The evidence from both industrialized and developing countries supports the view that inflation causes lower real growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As Woo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Jung and Peyton Marshall conclude in their article on inflation and economic growth: “The use of inflationary finance as a means to force additional savings and to increase capital formation appears to be an unwise strategy for economic development.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='56 Manipulating different variables to maintain the recommended level of 1% to 2% inflation is like 10 balancing economy on a pinhead: even if you achieve this balance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it will be extremely precarious,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' affected by various destabilizing economic and non-economic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In extreme cases inflation can become a run-away inflation, or even hyperinflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='57 One of the best-known cases of hyperinflation is Weimar Germany in 1923 when one American dollar was worth 4 trillion German marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Money lost all value and people shifted to barter, using goods with stable value as currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For example, pianos became currency during the German inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='58 The post-war inflation in Hungary reached 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='9 quadrillion percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='59 In 2008, Zimbabwe experienced one of the worst cases of hyperinflation ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The estimated annual inflation level at one point was 500 billion percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 60 High inflation can cause a decline of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When high inflation and production decline occur at the same time, the result is what is called stagflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Stagflations have particularly devastating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One may recall the stagflation in America during the mid-1970s when high interest rates, rising prices, and economic stagnation created an extremely explosive political situation that led to the collapse of the Democratic Party and brought Ronald Reagan to power and almost a decade of Republican rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The current inflation also has a very strong potential to turn into a stagflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='61 Top economists and bankers have already cautioned about this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Those who have sounded alarm include Allianz and Gramercy’s chief economic adviser Mohamed El-Erian and Goldman Sachs CEO David Solomon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Both have pointed out that the current inflation is entrenched and widespread throughout the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The World Bank has also issued multiple warnings to the effect that if economy remains sluggish, inflation may very well end in a stagflation in many countries, including in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='62 Even Paul Krugman, of The New York Times, who is usually optimistic regarding current inflation, has expressed concerns over the persistently “hot job market.”63 One economist writes about the current mood among economists that “they also know that this hardly original escamotage can only lead to runaway inflation, and then hyperinflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' What takes place today as a matter of monetary normality used to characterise wartime economies, namely direct financing via the money presses.”64 Many leading economists fear that today’s high levels of inflation, and the Fed’s commitment to containing it, could trigger a recession as early as 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='65 For example, the vast majority of economists at 23 large financial institutions have already confirmed this gloomy prediction only a few days into 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='66 In his article for Project Syndicate, Nouriel Roubini predicts an “unavoidable crash” of the economy in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='67 The Fed is currently trying to contain inflation with high interest rates that should eventually reach 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' At this rate, the unemployment may reach the level as high as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='6% and possibly even higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yet despite these efforts the inflation remains resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Even economists who try to moderate fears of recession, such as Jeffrey Frankel, still acknowledge that in the next two years a worldwide recession is entirely within the range of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Price control Price control is a very radical policy that should be used with great caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is certainly a legitimate policy that has often been used in dire circumstance of war when 11 goods are in short supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Governments successfully used this policy in combination with the policy of distribution of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, using this policy to fight inflation is totally useless and even damaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The government controlled prices in the Soviet Union through much of the country’s history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It worked reasonably well in the time of WWII when many goods, particularly food products, were very scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, continuing this policy in peacetime, particularly in the late 1970s and early 1980s, resulted in horrible economic distortions and inefficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, the inflation was low on Soviet planners’ books, but it was merely pushed underground and out of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Money continued to lose their value but prices were kept stable by government fiat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many citizens began to run away from cheap money into goods with stable and low prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The most notorious example of this flight was the fact that many farmers used cheap bread to feed their livestock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The result was enormous hoarding and chaotic disappearance of goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By not addressing the source of inflation and pretending that it did not exist, just because prices did not change, Soviet economic planners simply made inflation uncontrollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There are many stories about that period in the late 1970s and 1980a in the Soviet Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many who observed the erstwhile Soviet economy documented empty shelves and long lines to buy even the most essential items (toilet paper and vodka were two items particularly in high demand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Researchers habitually attributed these shortages to production inefficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, production in the Soviet Union was inefficient, but no more than usually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The real story behind these shortages was that farms and industries continued to produce goods at their normal rate, but these goods disappeared into the thriving black market even before they left enterprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Crafty speculators hoarded goods and sold them for high profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many goods, particularly perishable food products, rotted in underground warehouses, while people waited for days in huge lines to buy even the semi-spoiled leftovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such are the lessons of price control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' No wonder that political leaders rarely resort to this policy, despite occasional calls from academics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='69 President Nixon, for example, who at one point talked about price control, still relied on appeals to the business community to limit wages and prices on a voluntary basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Reducing the Role of the Government in the Economy The approach that sees the solution of the problem of inflation in reducing the role of government in the economy and in relying on market forces to restore economic stability has several major shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For one thing, the reduction of money supply as a cure for inflation will inevitably slow down production, which will also inadvertently lead to unemployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The proponents of this approach argue that high level of unemployment is simply unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is, in their view, a necessary evil that we simply have to endure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, many people who end up on the receiving end of this policy and have to pay a heavy price for it are not particularly receptive to the idea that their suffering paves the road to future happiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Unemployment, declining living standards, growing gap between the rich and the poor are sure to lead to political tensions, ravaging instability, and social cataclysms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 12 Many neo-classical economists advocate this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Ronald Reagan was perhaps its best-known practitioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, there have also been quite a few Keynesians among those who recommended this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In 1980, for example, when inflation in America stood at double-digit, Paul Samuelson, who was the first American to win the Nobel Prize in economic sciences, wrote that “five to ten years of austerity, in which the unemployment rate rises to an eight or nine percent average and real output inches upward at barely one or two percent per year, might accomplish a gradual taming of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' inflation.”70 Closer to our time Lawrence Summers, who served as Bill Clinton’s Treasury secretary, has also argued that American economy needs several years of unemployment above 5% or 10% unemployment for one year to contain inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='71 This proposal, if implemented, was sure to leave millions of Americans without jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Cutting government programs is another prong of the austerity approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Both Ronald Reagan and Bill Clinton pursued the reduction of government spending by cutting federal programs, including welfare programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This policy has certainly had an adverse effect, causing pain and suffering among the most vulnerable members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Summers rocked the Democratic establishment in 2021 by predicting that his party’s excessive spending would cause inflation, insisting that government spending should be cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='72 The most obvious negative side to this approach is that it leads to much suffering for a great number of people who would have to endure long periods of unemployment and reduced government assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, they would face enormous difficulties in trying to cope on their own with the deteriorating conditions of their life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Predictably, this policy will inevitably lead to the growing gap between the rich and the poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In fact, this growing gap has been persistent in America and the world for the last several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The erosion of the standard of living for vast number of people is sure to generate social tensions and conflicts that will disrupt social peace and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Although many proponents predict that this period of increased suffering will be limited, nobody can really tell how long it may last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Few doubt, though, that the longer it lasts, the more unpredictable and dangerous will be the social and political consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, these adverse consequences do not exhaust all the negative effects of the austerity approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It may also profoundly distort the structure of the economy that will become more oriented toward the wealthy consumers, rather than the middle class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production will cater to those with money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result, the market will offer more luxury real estate, more private jets and extravagant yachts, and more conspicuous consumption for the rich, rather than benefits for the middle and low ranks of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One should remember that mass production oriented toward the average citizen was what made American economy a success story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' An increased production of high-end goods is unlikely to replicate this success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As many argue, inflation, even relatively small inflation, is dangerous for the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Hrushikesh Mallick finds that “inflation rates have a significant adverse impact on economic growth.”73 A number of researchers point out that the relationship between inflation and economic growth goes both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' While low productivity can result in inflationary tendencies, inflation also makes borrowing money more expensive, which slows economic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='74 We must address inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yet on close examination, all proposed remedies augur ill for our economic future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many economists and economic planners have publicly voiced their pessimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They do not see that any of these 13 remedies promise relief in the current conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Using combinations of these remedies—as, for example, the Fed has been doing in the current inflation, by pursuing both policies of low and high interest rates--looks also extremely problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As Fabio Vighi aptly summarized, “For most of us, then, the future seems to offer a choice between structural stagflation (stagnant economy with high inflation) and an abrupt deflationary depression--like a choice between bleeding to death and suffering a heart attack.”75 Understanding Inflation Inflation and Business Cycle As has already been pointed out earlier, there is no unanimity among economists and economic planners in their views of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='76 The only general conclusion that one can draw from their discussions is that inflation is a result of some complex imbalance in the economy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and that inflation is only one of the symptoms of this imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Researchers have failed to explain why economy becomes periodically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They merely recognize the existence of such periodic phenomena that they call “business cycle.” Business cycle is the pattern of economic booms and busts that are experienced by all developed economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='77 Columbia Electronic Encyclopedia attributes the first formulation of the theory of business cycles to French physician Clement Juglar who was the first to recognize, in 1862, that economic fluctuations associated with the boom-and- bust were a characteristic feature of all economic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='78 The Great Depression that struck the United States and the world in 1929 was an important catalyst that stimulated much interest in business cycles—these periods of rapid economic expansions followed by economic slowdowns and contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='79 Theories explaining business cycles are numerous and diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They focus on different factors that supposedly trigger economic contractions in business cycles, but that hardly amounts to an explanation as to why business cycles are there in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Despite the fact that business cycles have been the subject of intense scrutiny for a very long time, the universal recognition today is that we simply do not know the why of business cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The best that researchers do is merely point out the existence of economic fluctuations that they explain with such vagaries as “chaotic market processes.” Matthew Shapiro concludes that most theories “take the answer to this question to be axiomatic”—that is, the cycles are merely assumed to be part of economic reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='80 James Mirrlees echoes a similar view that “recessions are in some degree inevitable” and “are bound to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='81 In his colorful metaphorical description Mirrlees compares attempts to steer economy away from downturns with efforts “to sail a straight line in a boat with wind direction constantly shifting, and sometimes blowing a gale.”82 Neo-classical economists are very vocal and consistent in pointing to monetary interventions as perhaps the most significant factor that contributes to economic contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Murray Rothbard is one of many who confidently claims that the "’boom- bust’ cycle is generated by monetary intervention in the market, specifically bank credit 14 expansion to business.”83 However, even he recognizes that the problem of business cycles is “one of general boom and depression.”84 He dismisses economic fluctuations as the source of depressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Rothbard writes: We may, therefore, expect specific business fluctuations all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is no need for any special "cycle theory" to account for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is nothing here to account for a general business depression—a phenomenon of the true "business cycle” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The explanation of depressions, then, will not be found by referring to specific or even general business fluctuations per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='85 Rothbard finds that there is something unexplainable in the behavior of many experienced business people who are “misled” by the availability of cheap credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=" One can see Rothbard’s sense of profound amazement at something incomprehensible when he writes: “In short, how did all the country's astute businessmen come to make such errors together, and why were they all suddenly revealed at this particular time?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This is the great problem of cycle theory.”86 In his reflection on the Great Depression, Milton Friedman, an acknowledged doyenne of neo-classical economics, also makes a claim that "monetary developments [in the early 1930s] were the major explanation for the depth and the length of the contraction.” He further explains: “As I\'ve said over and over again, I\'m not saying that that [monetary developments] caused the initial recession .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=" And I don't doubt for a moment that the collapse of the stock market in 1929 played a role in the initial recession.”87 Thus, most neo-classical economists advise to take business cycles and depressions as a given and even embrace them." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Arguing that business cycles, including depressions, are an essential part of economy, Rothbard, for example, suggests that rather than fight depressions, we should change our perspective on them from pessimistic to optimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We should view them as actually serving a useful purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Downturns, in his view, are the way that economy “adjusts to the wastes and errors of the boom, and reestablishes efficient service of consumer desires.”88 “The depression,” Rothbard rhapsodizes, “far from being an evil scourge, is the necessary and beneficial return of the economy to normal after the distortions imposed by the boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The boom, then, requires a ‘bust.’"89 Rothbard is not alone in proposing a change in attitude toward economic ups and downs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' According to James Mirrlees, “The general conclusion [among economists today] is that we should encourage people not to worry too much about asset price fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Then they will be happier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and they will not be so likely to reduce their consumption spending when markets crash.”90 Not everyone agrees with neo-classical economists in viewing business cycles as an intrinsic part of economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such 20th-century theorists as John Maurice Clark and Joseph Schumpeter have attempted to find cures for economic instability, rather than describe it, in the manner of many 19th century theorists, simply as a natural phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='91 Tejvan Pettinger, among many, considers this view to be controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Critics disagree with the view that economic downturns have a beneficial role to play because they “shake up” economy, weed out “inefficient” firms, and create incentives for cutting costs and operating efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They argue that in a recession, even “’good 15 efficient’ firms can go out of business leading to a permanent loss of productive efficiency.”92 Rendig Fels is another critic who rejects the notion of the inherent nature of business cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' He sees “little evidence of a built-in tendency of the American economy to generate cycles.”93 Yet even detractors offer no insight as to why these cycles exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, there is a great deal of truth in the argument of neo-classical economists that monetary interventions are a major contributor to economic downturns in business cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, one should recognize that the introduction of monetary interventions by the government was not a whim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It was largely a response to the existence of business cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The current inflation did not start merely because some policy makers decided to institute low interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For example, one important reason for lowering interest rates to almost zero in the current inflation period was to prevent deflation and production decline caused by the global financial crisis that started in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' After the outbreak of the crisis the US Federal Reserve and other central banks around the world kept interest rates low for a prolonged period of time and have instituted other monetary policies to ensure that financial systems have plenty of liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='94 When John Maynard Keynes published The General Theory of Employment, Interest, and Money in 1936, the world economy was in ruins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The book was an important watershed in macroeconomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Keynes’s theory did not explain business cycles but it argued that monetary interventions by the government are the way to mediate their most adverse consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='95 Initially, the response to Keynes’s theory was limited to scholarly circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It was only after WWII that the theory swept away the influence of the classical orthodoxy and became the main tool for guiding economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The “Keynesian Revolution” got under way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='96 The most important factor in this new development was the rising wave of recessions in the 1950s and 1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' During that period many economic planners and policy makers came to believe that there was a direct trade-off between unemployment and inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They came on the side of inflation to keep unemployment down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='97 Paul Samuelson and Robert Slow, two Nobel laureates in economics, forcefully argued in support of maintaining the price index at 4 to 5 per cent a year as “the necessary cost” of keeping employment around 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='98 When Nixon was blamed for the on-going recession in 1971, he reportedly quipped: “We’ll take inflation if necessary, but we can’t take unemployment.”99 The Nature of Production The connection often made between inflation and business cycle indicates that both these phenomena are interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inflation is one of the possible ways in which economic instability manifests itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and so does downturn of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, inflation is associated with economic instability that is part of the business cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We do not know what causes this instability, nor do we understand why business cycles are there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' What we do know, however, is that these phenomena are intimately related to production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, in order to understand inflation and business cycles we must look closer at the process of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 16 We commonly view production in economic terms, as the action of making or manufacturing from components or raw materials, or the process of being so manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' According to one definition, Production is the process of making or manufacturing goods and products from raw materials or components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, production takes inputs and uses them to create an output which is fit for consumption—a good or product which has value to an end-user or customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='100 In a contribution to Economic Discussions Sanket Suman defines production as “the organised activity of transforming resources into finished products in the form of goods and services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' the objective of production is to satisfy the demand for such transformed resources.”101 Finally, Wikipedia offers the following extensive formulation: Production is the process of combining various inputs, both material (such as metal, wood, glass, or plastics) and immaterial (such as plans, or knowledge) in order to create output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Ideally this output will be a good or service which has value and contributes to the utility of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' [1] The area of economics that focuses on production is called production theory, and it is closely related to the consumption (or consumer) theory of economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='102 There is, however, another and broader view of production as a totality of human interactions with reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' German idealists tended to have such comprehensive view of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In this thinking, cognition, or knowledge acquisition, is also a form of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As Hegel put it, reality in this sense is “process and result rolled into one.”103 Karl Marx also viewed production in terms of the broad interrelationship between humans and nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Consider the following passage from his Economic and Philosophic Manuscripts of 1844 in which Marx writes: “Each of his human relations to the world – seeing, hearing, smelling, tasting, feeling, thinking, observing, experiencing, wanting, acting, loving – in short, all the organs of his individual being .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' are .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' in their orientation to the object the appropriation of human reality.”104 This view of production does not contradict the purely economic perspective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it simply represents a broader approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is in this broader approach that I propose to explore production and its place in human life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production is a form of interaction between humans and reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is a sensuous, or physical form of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Humans are a product of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, human interactions with reality are also a product of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The evolution is a universal process that sustains our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Consequently, production is a result of the process that plays the most essential role in the existence of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The evolution originates in conservation that is ubiquitous throughout the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The roots of conservation are in the very unique nature of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The universe is all that is there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Nothing can come into it from outside because there is no outside;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' nothing can disappear from it because there is nowhere to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Consequently, everything must be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 17 Conservation requires resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and resources are always limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, new resources are vital for conserving our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' These necessary resources cannot come into our universe from outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They have to be created inside the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, conservation requires creation of new and increasingly more powerful levels of organization that offer new possibilities that have not existed prior to their creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' These new possibilities offer access to new resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' they represent such new resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, the universe is impossible without the creation of new and increasingly more powerful levels of organization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='105 and the creation of such levels of organization, or their production, is what the evolution is all about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since humans are products of the evolution, their interactions with nature have inherited the main features of the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We also live under the constraint of limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, in order to sustain our life we have to create new and increasingly more powerful levels of organization that provide access to new resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This is the essence of our production, including economic production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Value Creation and Inflation Newly created and increasingly more powerful levels of organization offer new possibilities that are the main resource for sustaining our existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, these new and increasingly more powerful levels of organization represent an enormous value for us since they sustain our existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are the source of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Value creation is the main function of our interactions with nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it is also the main function of our economic activity, or production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' What does the creation of new and increasingly more powerful levels of organization involve?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' A level of organization consists of interconnected functional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order for these operations to persist, they must be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Action is what conserves operations: the more operation is active, the better it is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to bring an operation into an active state, or to activate it, the operation needs something that will trigger its activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The more often the operation is triggered into action, the more active it is, and the better it is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Consequently, the more triggers an operation has, the better it is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If an operation combines with another operation to create a new whole, it will double the number of triggers that bring it into action—one that triggers one operation and another that triggers the other operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since interactions among functional operations conserve them, such interactions are favored by the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By combining with each other, operations acquire new possibilities and, therefore, become more powerful in comparison with when they were on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Each new level of organization that we create, or produce, increases our power and, thus, represents value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to conserve this value, we must create another and even more powerful level of organization, or a new value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Constant value creation is the main function of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, interactions among functional operations create new levels of organization that offer more possibilities and, consequently, are more powerful than those from which they have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This added power represents value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production is about the creation of value, which means that production involves the creation of new and increasingly more powerful levels of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The only way to conserve value is to conserve the 18 level of organization that represents this value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and the only way to conserve this level of organization is by creating a new level that is more powerful, and consequently has more value, than the one from which it has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The conclusion that follows from the above is that the primary function of production is to conserve functional operations sustained by a particular level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production is ultimately about conservation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and conservation can only result from creating new values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Failure to create new and more powerful levels of organization means that existing possibilities have not been realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Without realizing these possibilities, they cannot be conserved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' in other words, value is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Non-conservation of value is a sure sign that production is not efficient and that available resource are either underutilized or simply wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inefficient production does not create value and, consequently, sustains losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production losses are at the root of economic instabilities and disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' All economic values have monetary equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Money plays the most essential role of realizing the possibilities that each value offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We use money to attract resources and labor to production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Money does not have intrinsic value, after all money is a mere convention, a piece of paper with something printed on it--and now even not that;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' today money is merely an electronic signal transmitted from one source to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The value of money is the equivalent of the real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When real value is not conserved, when all possibilities associated with this value are not realized, the real value these possibilities represent diminishes, or depreciates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and when value depreciates, the money equivalent of this real value that is supposed to be used for realizing these possibilities also loses its worth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Money becomes cheap and the cheapening of money results in price increases, which is the essence of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The New Economic Practice The Model for the New Economic Practice The philosophic exploration of production shows the connection between production as an evolutionary phenomenon and the universal process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This exploration leads to several important conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' First of all, it makes clear that inflation is an aberration that is in no way intrinsic to production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is nothing in the process of creation that warrants this aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The growth in the supply of money is not the ultimate source of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is merely an effect of the depreciation of money that results from inefficient production that leads to the depreciation of real value and the monetary equivalent of this real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The depreciation of real value indicates that our production fails to conserve this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production must be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is hardly a single economist, businessman, economic policy maker, and even a lay individual who would dispute this maxim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Efficiency results only from the full utilization of all available resources and possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Wasted resources and squandered possibilities make economy inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Efficiency is what makes economy possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it is the pillar on which our economies rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' All negative economic phenomena—instabilities, crises, inflation, unemployment, and others—are 19 results of inefficient production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to understand our economic problems, including inflation, we must understand why we fail to use resources efficiently, create wastage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' in other words, why we do not conserve value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The philosophic investigation that traces production to the universal process of creation shows that production is an evolutionary phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It emerges in the course of the evolution driven by the universal process of creation and, therefore, production inherits all the main features of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The distinct feature of the process of creation is the full utilization of all available possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In fact, this process can only work on the basis of such full utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Anything less than full utilization makes the creation of new and increasingly more powerful levels of organization impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, full efficiency is the essential property of the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, the main requirement of efficient production is the full utilization of all available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There are other important features of the process of creation that efficient production must replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One important feature of the process of creation is universal inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The full utilization of all available possibilities is the key to the creation of a new and more powerful level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Only such utilization conserves all available possibilities, which is the only way to create a new level of organization that is more powerful than the one from which it has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Only the combination of all available possibilities makes the new emerging level greater, or more powerful, than the sum total of its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If any possibility is excluded, it will not be conserved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' in other words, it will be wasted, which is how we define inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result of exclusion, the emergent level of organization will not exceed the power the level from which it has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The process of creation is a dynamic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It constantly evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The feature that makes such dynamism possible is the balance between equilibrium and disequilibrium that one can observe in the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The equilibration of specific operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', possibilities) increases equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, equilibration also creates a new and more powerful level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since this new level has greater power, it represents disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, as equilibrium grows, so does disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The two are always in balance and complement each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The creation of combinations conserves operations (possibilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They do not disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' No operation consumes another operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Both are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' No operation that belongs to a given level of organization is superior to another operation sustained by the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, their interactions are interactions among equals, which means that these interactions are non-hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, in the course of interactions, operations create a new and more powerful level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, they create a hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Consequently, the balance between hierarchical and non-hierarchical interactions is another important feature of the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Non-hierarchical interactions create new levels of organization and hierarchical interactions conserve and optimize these new levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The above features are those that have already been recognized as having the essential role in the process of creation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' they make it possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to create an economic practice that would be able to replicate the process of creation in its efficiency, all these features must become part of the new economic practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There may be some other features that we still do not know about and that we will recognize as we continue 20 to learn more about the process of creation in the course of the realization of the new practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Changes Required for the New Economic Practice In order to make our economy efficient, avoid wastage, and prevent instabilities, the new economic practice should use the process of creation as its main organizing principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In this case, the new economic practice will include all the main features of the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result, the new economic practice will be different from the current practice in some critical respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This new practice will also lead to rethinking of some key concepts that are relevant to economic practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In contrast to the existing practice, the new practice will involve a much greater emphasis on creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Investing in the production of new ideas is clearly an emerging trend in the current practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In the new practice, investments in the production of new ideas will constitute the increasingly growing portion of total investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Moreover, much of these new investments will be focused on radical innovation that is associated with the emergence of new and increasingly more powerful levels of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The New Conception of Management and Leadership The emphasis on creativity will inevitably affect our conception of management and leadership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In contrast to the current economic practice that is dominated by hierarchical interactions, there will be a much greater emphasis on the role of non- hierarchical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Non-hierarchical interactions play the main role in creating new combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are the source of creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Hierarchical interactions conserve and optimize what non-hierarchical interactions create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As has been explained, the process of creation requires a balance between the two types of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Maintaining this balance will be one of the main preoccupations of the new economic practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The top-down approach dominates our current economic (and not only economic) practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, there are some experimental trends that try to moderate hierarchical domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' But even these new trends do not go nearly far enough in balancing hierarchical and non-hierarchical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For one thing, these innovative trends limit the scope of balancing only to top economic and managerial elites and excludes a large number of people involved in the process of production and exchange, including but not limited to workers, employees and even small and medium-size businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In the United States, for example, small and medium size businesses do not qualify for a generous support of the kind that goes to corporate giants, such as GM, Ford, or major mega-banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' While the market certainly has a non-hierarchical structure, our managerial culture remains by and large hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='106 The top economic and managerial elites essentially adhere to principles of hierarchical control, rather than to the non-hierarchical mode associated with the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The dominant neo-liberal approach in economic management has been pursuing and to a significant degree has achieved the merger of the government and economic elites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Rather than combine the hierarchical principle of government bureaucracy with the non-hierarchical principle of the market, this approach 21 has strengthened the concentration of power in the elites and enhanced the hierarchical principle in our society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It has not balanced hierarchical and non-hierarchical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is a growing number of researchers who recognize the need for a genuine combination of hierarchical and non-hierarchical principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One popular trend is the so- called hybrid solutions, that is, solutions that still see hierarchical and nonhierarchical interactions as ontologically separate but seek some format in which their coexistence and limited cooperation can become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' These solutions are largely eclectic and do not achieve a true integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='107 John Kotter, the chief innovation officer at Kotter International and a professor emeritus of the Harvard Business School, typifies this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In his view, hierarchies and networks are two separate structures that excel at what they do best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Kotter recognizes that hierarchies are very good at optimizing and are capable of effecting small and medium-size changes but cannot perform large-scale innovative transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' He explains: But I am referring to something far bigger: large-scale organizational change, such as a company redesigning its entire business model, or accomplishing its most important strategic objectives of the decade, or changing its portfolio of product offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' And there is no evidence to suggest that the Hierarchy allows for such changes, let alone that it effectively facilitates them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='108 In Kotter’s view, the future lies in the coexistence of the two structures in one business organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In his own words: All of this has led me to believe that the successful organization of the future will have two organizational structures: a Hierarchy, and a more teaming, egalitarian, and adaptive Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Both are designed and purposive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' While the Hierarchy is as important as it has always been for optimizing work, the Network is where big change happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It allows a company to more easily spot big opportunities and then change itself to grab them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='109 Coexistence is a far cry from a genuine integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It presupposes that the competition between co-existing entities will continue and will be merely moderated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Ultimately, they will not cooperate but they will try to avoid interfering with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This solution is certainly not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The two types of interactions should not merely co-exist, which is a very unstable arrangement based on constant competition, but truly cooperate and complement each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They should be part of the same decision-making process, not merely the two sides in a compromise decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Compromise solutions involve emphasizing commonalities and suppressing differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yet, differences, not commonalities, are the main source of radical innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Hybrid solutions offer a rich plethora of interesting ideas regarding possible mechanisms of interactions between hierarchies and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, as all eclectic solutions, they do not have a solid theoretically grounding and tend to have internal contradictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Nothing illustrates this shortcoming better than the discussion of such 22 critical subject as the relationship between leaders/managers and networks/employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Opinions on this score vary widely: from a more activist role of leaders/ managers as enablers110 to a weaker role of regulators and filterers of external information,111 to an even weaker role as facilitators of critical discourse and enhancers of local interactions among network agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='112 Some even believe that the desired goal can be achieved without structural changes by merely modifying the rationale for the role of hierarchies and by educating managers in the values and merits of organizational democracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Martin Clarke and David Butcher, for example, see education and the principle of voluntarism they borrow from political philosophy as vehicles for reconciling hierarchies and networks in organizational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='113 There is no doubt that the literature on hybrid solutions certainly deserves serious attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It addresses many aspects of what is obviously a very complex and seemingly intractable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many of ideas articulated in hybrid solutions are undoubtedly very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' But even all together, they hardly measure up to the magnitude of the task, which leaves quite a few researchers dissatisfied and vying for a comprehensive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In their essay “Simplistic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Complex Organization: Markets, Hierarchies, and Networks in an Organizational Triangle,” Wolfram Elsner, Gero Hocker and Henning Schwardt make an argument for just such a comprehensive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In their view, “… pure market and hierarchy, including their potential formal hybrids, are an empirically void set.” Rather, “coordination forms” in the real world, they argue, “have to be conceptualized in a fundamentally different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' A relevant organizational space must reflect the dimensions of a complex world.”114 In making their appeal to complexity of the real world, Elsner, Hocker and Schwardt suggest that the division between hierarchical and non-hierarchical interactions is not real, it is merely conceptual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='115 that in reality, the two types of interactions are closely entangled with each other, although they fail to explain the nature of this entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Numerous other researchers support the approach that centers on the entanglement of hierarchical and non-hierarchical interactions and the complexity of their relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Antoine Danchin points to the ubiquity of networks and hierarchies in nature and their complementary relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='116 Joan Roelofs challenges the simplistic view of networks as spontaneously resistant to hierarchies and naturally prone to democracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As she maintains, …some participants in network governance are vastly more powerful than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As for “civil society” organizations, support from corporate or private foundations is essential to almost all civil rights, social justice or environmental organizations that wish to be viable and visible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' the funders exert control in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='117 While the above perspectives serve as a valuable source of insights, they ultimately do not resolve the problem of the relationship between networks and hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Despite their astute and nuanced observations on the nature of this relationship, they still see hierarchies and networks as ontologically separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In their view, tensions between networks118 and hierarchies can only be ameliorated, but they will ultimately always remain and be a potential source of conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='119 23 The perception that networks and hierarchies are polar opposites, perennially in tension and conflict with each other, contradicts what we know about systems in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As has been explained earlier, systems conserve themselves by forming bonds, or what Humberto Maturana and FranciscoVarela called structural coupling,120 with other systems in their environment as part of the process known as self-organization, thus creating new organized totalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='121 The process of creating a new organized totality gives rise to the operation that regulates the functioning of this totality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' And that is what a system is: an organized totality of coordinated operations with a common regulatory mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since the regulation of a system is a product of combining the capabilities of its constituent parts, the level of organization that supports regulation is more powerful than the level of organization of each subsystem or their sum total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The emergence of this more powerful level of organization creates a hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As one can see, regulation is a product of interactions among subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It supervenes on local interactions and vitally depends on them for its own existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The regulatory function also needs to be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For this reason, it has to form strong bonds that would activate it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and first and foremost, it should have strong bonds with the subsystems it regulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The process of forming bonds between the global level of regulation and the level of local interactions results in the integration of the two levels of the system and the adaptation of local interactions to the global operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Regulation can facilitate such adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When the weaker operations adapt to the more powerful level that sustains regulation, they change and gain in power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' the re-equilibration of these enhanced operations increases the power of the regulatory level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, the entire system evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This observation regarding systems in general suggests that in social systems the role of leaders and hierarchies, which also operate at a more powerful level of organization, must be very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By virtue of their position, leaders can enormously facilitate the integration of systems because they have access and can observe both the global and the local level of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to integrate the system they regulate, leaders must resort to reflective coding—the procedure that Gödel used in his famous proof of consistency and completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='122 It is a creative task because it creates a level of organization that can incorporate both global functions and local interactions as its particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This role leaders and hierarchies has nothing to do with command and control, that is, transmitting decisions from those above to those below and overseeing their implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Leaders must appreciate the enormous creative power of local interactions and be closely attuned to their variations and modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since they rely, or supervene, so much in what they do on interactions among network agents, or subsystems of the system, they should promote, regulate, and facilitate these interactions, not dominate them and impose on them their will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is a sensitive, delicate, and highly creative role that involves both cooperation and two-way adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Those who operate at the global level and those involved in local interactions are, in a way, equal participants in a common creative enterprise of ensuring the conservation and evolution of the system that they constitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Because of their location in the liminal space between the system and its environment, hierarchies and leaders are in a position to reflect critically (that is, 24 observing at the same time the system and also themselves in performing their function)123 on all interactions among all the local agents and subsystems of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The latter, by virtue of their position, can reflect only on local interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' For this reason, the position of leaders makes possible for them to see new and more powerful possibilities emerging in interactions within the system, as well as recognize, promote, and facilitate the utilization of these possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The creation of new and increasingly more powerful levels of organization that propels the system’s evolution is incompatible with the relationship of exclusion and domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It requires cooperation and close interaction in common creative work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such cooperation can only be effective if there is a balance between hierarchical and non- hierarchical interactions, between hierarchies and networks--managers and leaders, on one hand, and employees, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='124 Leaders should not see their role as that of ultimate arbiters whose word is decisive and final—far from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The notion of a leader as the ultimate arbiter without whom there will be chaos and instability is a result of a profoundly flawed view that excludes the process of creation from its frame of vision and, as a consequence, leads to a failure in understanding how systems function and evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This view makes impossible to have clear and rational validation criteria that can help choose the most powerful level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As has been argued elsewhere, the current approach largely relies on subjective choices of those at the top of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='125 The lack of such objective and rational criteria of validation is the main reason why we now tend to defer decisions to top managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In the absence of such criteria, all decisions are subjective and all are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Recognizing all decisions as equal is likely to lead to chaos and instability and nobody wants to argue for disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result, the common current default is to defer to the decision of those who are at the highest level in the hierarchy because even a bad decision that preserves order is deemed better than chaos and instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' How many businesses have paid a heavy price for limitations of those at the top of their hierarchies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Assessment of Economic Performance The current economic practice uses value as the most important criterion for assessing results of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, we greatly differ in our ways of understanding value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Some use market share in determining value, others profitability, still others prices of company’s shares on the stock market;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and there is more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The choice depends on how one defines value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There are many definitions of value currently in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' IGI Global, for example, identifies eight different definitions of value and its list is not the most extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='126 With this number of definitions, their differences and diversity do not come as a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Perhaps the most common formulation defines value as “a measure of the benefit provided by a good or service to an economic agent.” Currency is the most common measure of value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' consequently, money is the most common way to quantify value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, it is not the only measure of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In another formulation value is “the maximum amount of money a specific actor is willing and able to pay for the good or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' "127 Caroline Benton, for example, defines value in terms of preferences of 25 individuals who “determine the economic value of a good or service and the trade-offs that they will be willing to make to obtain it.”128 Although in the above formulations money usually objectifies value, the influence of subjectivity in valuation is unmistakable: value is based on preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Preferences do not have to be monetary either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yusuke Kuwayama, Justine Huetteman, and Bethany Mabee, for example, do not consider money as the only and even the most important way of assessing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Simply put,” they argue, “things that have ‘value’ are useful to you, improve your situation, or simply make you happy or more secure.”129 All these are more or less subjective factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There are some formulations that try to see value as something calculable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The emphasis on calculability is supposed to give value the appearance of something objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Yet even such formulations cannot escape subjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Market Business News defines value as “a calculation of the profits an asset has either produced or may produce in the future.” Value is “a measure of the benefit a product or service provides an economic agent (person or company).” Ultimately, the willingness of the agent to pay for something, his or her preferences, still appear in the estimation of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Finally, all formulations stress that economic value is not the same as market price that is regarded as more objective than individual preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='130 Finally, there is this gem of a definition that makes the determination of value a total exercise in futility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' A team from the Corporate Finance Institute offers the following formulation: The economic value of a business is the business’s contribution to the global gross domestic product (GDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The most common method of estimating economic value is the counter-factual method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The counter- factual method states that the economic value of a business is the difference between the current global GDP and the hypothetical global GDP if the business did not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='131 Even a brief overview of some of the examples of the current definitions of value shows their inadequacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They often contradict each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Also, and probably even more importantly, these definitions are to greater or lesser extent subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Basing our assessment of economic efficiency on such subjective understanding of value cannot result in an objective valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We need a uniform and objective understanding of what constitutes value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The economic practice that uses the process of creation as its main organizing principle offers such understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As the earlier discussion of the process of creation makes clear, efficiency depends on the use of resources and possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' A system—any system, including economic ones—is efficient if and only if it fully utilizes all resources and possibilities available to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Only by combining all these resources and possibilities, a system can create a new level of organization that will make it more powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such system will conserve itself and will make an evolutionary advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The new level of organization will offer access to new resources and new possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 26 In order to be efficient, a system must be inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to survive, business organizations must know their resources, both physical and mental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The latter are particularly important in our time because of the growing emphasis on creativity and the production of ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If resources are known, one can easily determine whether production uses all of them or just some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Also, production involves the creation of combinations of all available operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, each operation should establish connections with all other operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If a system that has n-number of operations functions efficiently, it will produce the number of operations that will be equal to n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, an efficiently functioning business organization will have an exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='132 Conversely, if a system demonstrates an exponential growth, it must be functioning with full efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Profits will necessarily reflect this efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Inclusivity and exponential growth are not subjective criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They do not depend on individual preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' These criteria are derivatives from the process of creation that is not our subjective construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This process is not our creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' On the contrary, this process has created the human race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This process is the source of value that is determined in objective terms of inclusivity and exponential growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The new conception of production and consumption The preceding discussion of the new economic practice focused primarily on production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, there is another dimension that is also relevant for determining economic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This dimension is consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Efficient economy is one in which all products are consumed and nothing goes to waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Economic efficiency includes both production and consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Despite the fact that efficient economy involves a close interrelationship between production and consumption, there is a strong tendency in our economic science to separate the two, even though it recognizes, at least to some extent, their close interrelationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our economists and business people accept the notion that ultimately we should strive for a full utilization of what we produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Otherwise, our production will be inefficient and sustain losses and waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, despite this recognition, our economic thinking also sees production and consumption as two ontological opposites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This thinking generally associates production with appreciation, or value creation, and consumption with depreciation or the erosion of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We know very well, for example, that when we buy a new car, the car will lose a significant part of what we paid for it as soon as it leaves the parking lot of the dealership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The analysis of the process of creation shows that production and consumption are two closely interrelated aspects of one integral whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are merely analytical categories, not ontological entities that exist separately from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When subsystems form bonds with each other, they assimilate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, they include each other in their own functional operations, which is essentially a form of consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, the inclusion of a different entity into the subsystem’s functional operations modifies these operations, thus producing something new that did not exist 27 prior to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Moreover, even to be open to a possibility of inclusion, a subsystem must create a construct that allows it to perceive another subsystem in its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such perception does not come automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' A system must have a capacity to perceive difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Regulation has this capacity to perceive something different form the entity it regulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our perception of reality offers a good illustration of this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In order to perceive, our mind has to create a mental construct that makes this perception possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The act of perceiving assimilates the perceived object and thus stabilizes and conserves the mental construct that makes the perception of this particular object possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Thus, the production of perceived reality goes hand in hand with consumption of this reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We cannot perceive what we have not first constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Infant’s experience of reality is defined by inherent sensory-motor operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Mental constructs that make possible for the infant to perceive permanent objects are a result of the combination of sensory-motor operations that create permanent mental images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='133 Thus, production and consumption go hand-in-hand together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are both aspects of the process that creates new properties and new possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Production and consumption are analytical, rather than ontological categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As the analysis of the process of creation shows, the conservation of a given level of organization creates a new and more powerful level of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The emerging level of organization supervenes on the level from which it emerges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' in other words, the level of organization that gives rise the new level of organization is a resource that is consumed in the process that creates the new level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can also represent this relationship between consumption and production as a balance between equilibration and the production of disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In this conception, disequilibrium is a resource for equilibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and equilibration results in a new disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' What practical consequences will the recognition of the unity of consumption and production?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' How will it affect our economic practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Organizing our economic activities around the process of creation will end the tendency to dissociate production from consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it will make the interrelationship between production and consumption effective and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By complementing each other, they both will be able to grow exponentially and will make possible an exponential growth of our entire economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can see the contours of this new economic organization in the comment of Alan Webber who concludes: “In the end, the location of the new economy is not in the technology, be it the microchip or the global telecommunications network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is in the human mind.”134 In contrast to ordinary goods, knowledge does not depreciate when used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' On the contrary, it appreciates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In other words, its value grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our assimilation of ideas creates new and increasingly more powerful levels of organization that give rise to even more innovative ideas, approaches, and decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By producing new and increasingly more powerful levels of organization we generate new knowledge and ideas that lead to economic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Knowledge, for example, is one important product that does not depreciate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It only appreciates when consumed, as its consumption leads to new and increasingly more powerful levels of organization that give rise to new knowledge and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As Thomas Davenport and Lawrence Prusak have noted, “ideas breed new ideas, and shared knowledge stays with the giver while it enriches the receiver.”135 28 The realization that production and consumption are intimately related will change our patterns of investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The economic practice that uses the process of creation as its main organizing principle will focus investments on the creation of new and increasingly more powerful levels of organization, not merely on putting new items on store shelves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such new pattern of investment will be becoming increasingly important with the on-going shift of the emphasis in our economy from production of things to production of knowledge and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The change in the pattern of investment will enhance our production and economic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In today’s economy many goods that reach the market often face no demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They are either drastically discounted or completely trashed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' With the new pattern of investment and the emphasis on the production of knowledge and ideas we will avoid this waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As our production grows exponentially, so will our consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' yet we will not be facing a situation that what we are not able to consume what we have produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Just like our production, our consumption can also grow infinitely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and this growth will not be posing any threats to our planet or the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' On the contrary, they will help us in solving numerous problems that we face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Moreover, they will also help us anticipate future problems and find their solutions even before these problems emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Finally, the new patterns of production and consumption will help us live better and have a more satisfying and happier life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Education The importance of knowledge for economic production is hard to overestimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Knowledge has always been a major contributor to economic progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Its contribution is particularly important in this day and age when our economy puts premium on knowledge production and creativity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' our economy is increasingly about knowledge production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='136 Education is one important sphere of our civilization that directly relates to knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Traditionally, storage and dissemination of the existing knowledge has been the main preoccupation of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' By transmitting knowledge to new generations of young men and women, the system of education prepared them for becoming productive members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Modernity self-consciously used education to set our civilization onto a new course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Since the very beginning of the industrialization, the connection between education and economic progress acquired particular importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Educational institutions became the main breeding ground where students were trained to become part of the labor force that propelled the industrial development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' They became the workers, technicians, managers, business organizers who built and operated the growing number of factories and plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The connection between education and economic progress has become particularly important in this day and age when the production of innovative ideas, approaches, and decisions has become the most important production factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Due to the close association between education and economic progress, economic needs have always shaped our educational system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' There is no doubt that the transition to the new economic practice will necessitate changes in our education and teaching methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 29 As has already been pointed out, educational institutions were traditionally preoccupied mostly with the dissemination of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our educational system exposed students to the vast body of information and skills that were accumulated in the course of the evolution of human civilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, it rarely, if at all, taught students the habits and skills required for producing new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, many talented young men and women engaged in knowledge production and became the source of radical innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, their success owed very little to the educational system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The manifestation of their genius occurred contrary to this system, rather than because of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our educational system does pitifully little to enhance and develop the creative capacity of young people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Generally, our educational institutions relegate creative type of activities to art and literature courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='137 There is very little room for creativity in mainstream courses in sciences, including social sciences, and math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result, more often than not, education stifles students’ creative impulses and suppresses their capacity to create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This situation is hardly an accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our knowledge of what makes creation possible is meager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The sad fact has been and remains that our understanding of the process of creation remains very rudimentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As a result, our ability to control this process is extremely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Margaret Boden, one of the pre-eminent researchers in the field, draws the following conclusion in her influential book on creativity: Our ignorance of our own creativity is very great.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We are not aware of all the structural constraints involved in particular domains, still less of the ways in which they can be creatively transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We use creative heuristics, but know very little about what they are or how they work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' If we do have any sense of these matters, it is very likely tacit rather than explicit: many people can be surprised by a novel harmony, but relatively few can explicitly predict even a plagal cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='138 This situation must change with the onset of the new economic practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Some radical innovations must take place in our theory of education and teaching methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is beyond the scope of this study to provide a detailed description of the kind of changes that are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The transformation of our education is a project that is only beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many of the required changes will become clear only in the course of this project’s implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, the description of some essential features is possible today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Whatever new forms our education will take, the process of creation should be their main model and inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Therefore, the recognition of the importance of this process, further study of this process, and a comprehensive understanding of it are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our educational system should use the process of creation as its main organizing principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The new teaching methodology must include important features that are known and characterize the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Universal inclusion and empowerment must constitute the basis of the new methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Teaching should also observe important balances that sustain the process of creation: the balance between equilibration and the production of disequilibrium, or the balance between equilibrium and disequilibrium, and the balance between hierarchical and non-hierarchical interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' One can find a more detailed description of the proposed innovations elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='139 30 Conclusion Economic production is, in more ways than one, a very important building block of our civilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It provides means for our sustenance, helps us control our environment, and makes our life more comfortable, enjoyable, and satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' But its importance transcends these utilitarian purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Economic production strengthens and develops our capacity to create, reinforcing our connection to the universal process of creation and, thus, to our entire universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This article argues that our economic problems, of which inflation is only one, are due to the fact that our economic production is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Its inefficiency is a result of the inadequate use of resources and possibilities available to us, most importantly human capacity to create an infinite number of new and increasingly more powerful levels of organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This inadequacy is not fortuitous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it is not a result of our mistakes or lack of industriousness and diligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is a result of our lack of understanding of the roots of our own existence—the universal process that led to the rise of humanity and has been driving the evolution of human civilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' To this day, we have not grasped the important role this process plays in our relationship with reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='140 The article shows the connection between economic production and the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It also shows that the reason why our economy is inefficient is the fact that we do not know what factors makes the process of creation so efficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' consequently, we cannot replicate these factors in our economic activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As the article shows, these factors include universal inclusion, the balance between hierarchical and non-hierarchical interactions, as well as the balance between equilibrium and disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Only full utilization of all available resources and possibilities can create new and increasingly more powerful levels of organization that can give rise to new ideas, decisions, and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such efficient use of resources will lead to constant, stable, and exponential economic growth without inflations, business cycles, and economic contractions and crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Understanding the process of creation is only the first step in addressing the problem of our economic inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' As this article has argued, we have to acquire knowledge of the process of creation and use this knowledge in establishing the new economic practice that will use the process of creation as its main organizing principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Such new practice will involve changes in investment patterns, conception of management and leadership, as well as re-conceptualization of consumption and the relationship between production and consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Indeed, some may make an argument that even though we do not understand the process of creation and its role in our relationship with reality, our civilization has been able to make a remarkable progress in the course of its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' This fact only reinforces the main point made in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It shows the enormous power of the process of creation that has helped our civilization to evolve to this point despite our disregard of this process and our failure to utilize fully its infinite possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In a way, we have been able to make a remarkable progress despite ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, our civilization can progress only so far if it continues to ignore the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The way we have pursued progress so far has limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='141 In order 31 to continue the progress of our civilization into the future, we need to embrace, understand, and use this process efficiently to transcend our self-imposed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our production has reached a limit of what it can achieve without availing ourselves fully of the enormous resources and possibilities that the process of creation offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our economic production has evolved to the point when our capacity to create is rapidly becoming the most important factor without which our production simply cannot evolve any further and help us in solving the problems we now face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our civilization badly needs new ideas,142 including new ideas in the way we run our economies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' So far, there have been two main approaches in organizing our production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Liberalism promotes one approach and socialism another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Both versions ignore the process of creation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and both have reached their limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The many disruptions of the world economy are an eloquent indication of the inadequacy of the current approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Capitalism is the term that has been much abused and maligned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' the meaning of capitalism has been perverted and ideologically misconstrued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Capitalism is essentially about the growth of capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Capitalists are people who focus their skills, energy, and resources on increasing capital, that is, on creating new value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The rise of capitalism represented a realization that making capital grow and creating value is so important that we should concentrate our will and resources on promoting constant growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The growth of capital has become a self-conscious goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In this sense, Western liberalism and Marxist socialism are not very different from each other, despite their assertions to the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Marx’s teaching that has probably done more than anything else to discredit capitalism is in this sense very capitalist in its spirit and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It is also about capital growth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' it is also about value creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' In fact, Marx’s main argument against capitalism is that capitalism has flaws that do not allow growing economy indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Socialism, according to Marx, was supposed to remove these fetters and to ensure unlimited growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' His argument for the capacity of socialism to accomplish this task has been a dismal failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' However, the passion and conviction that ring in this argument are genuine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' they reflect his aspiration for infinite growth that is essentially “capitalist” in its nature in the sense in which capitalism has been defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The similarity in the pursuit of economic growth between liberalism and socialism is perhaps one reason why Western liberalism has proven to be receptive to socialist ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Many even mainstream progressive liberals in the United States today see solutions to the current problems in socialist policies, such as redistribution of wealth and the involvement of the state in managing the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The rise of capitalism was an important evolutionary development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The idea that the survival of our civilization vitally depends on perennial creation of value and capital growth is our articulation of what we have inherited in the course of the evolution from the process of creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Our anthropocentric bias has prevented us from grasping the cosmic significance of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' We have failed in moving beyond our vague intuitions about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The anthropocentric attribution of the emergence of capitalism exclusively to humans and their decisions has overshadowed the intuition of early capitalists who believed that their ability to increase capital was divine in its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' These “secular monks” created much value and acquired enormous wealth, yet they did not use this wealth for the sake of hedonistic enjoyment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' they relished with the abandon 32 of religious ecstasy in their very capacity to create value that they felt brought them closer to God the Creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 33 ENDNOTES 1 David McMillan, “Inflation: There’s a Vital Way to Reduce It That Everyone Overlooks—Raise Productivity,” World Economic Forum, June 9, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='weforum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/agenda/2022/06/inflation-there-s-a-vital-way-to-reduce-it-that- everyone-overlooks-raise-productivity/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 2 Fabio Vighi, “A System on Life Support,” The Philosophical Salon (blog), September 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://thephilosophicalsalon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/a-system-on-life-support/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3 Vighi, “A System on Life Support.” 4 Alexander Barta, “Inflation in Economic Theory | Exploring Economics,” Exploring Economics, 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='exploring-economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/en/discover/inflation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 5 Kat Tretina, “Is Inflation Good Or Bad?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Forbes, September 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/advisor/investing/is-inflation-good-or-bad/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 6 Ceyda Oner, “Inflation: Prices on the Rise,” Finance and Development IMF, July 15, 2022, 30–31, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='imf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/en/Publications/fandd/issues/Series/Back-to- Basics/Inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 7 Joseph E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Stiglitz, “All Pain and No Gain from Higher Interest Rates,” Project Syndicate, December 8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='project-syndicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/commentary/fed- interest-rate-increases-counterproductive-all-pain-no-gain-by-joseph-e-stiglitz-2022-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 8 Spang, Rebecca L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “The Rise of Inflation | Rebecca L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Spang.” Cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Accessed November 27, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://cabinetmagazine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/issues/50/spang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 9 Friedman, Milton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The Counter-Revolution in Monetary Theory: First Wincott Memorial Lecture, Delivered at the Senate House, University of London, 16 September, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' London: The Institute of Economic Affairs, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 10 Barta, “Inflation in Economic Theory | Exploring Economics.” 11 Oner, “Inflation: Prices on the Rise,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 12 “Inflation—The ‘Cost Push’ Theory,” Britannica,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='britannica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/topic/inflation economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 13 Adam Tooze, “Chartbook Newsletter #22: How do you count inflation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=" Tracking Weimar's hyperinflation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=',” 2021, https://adamtooze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='substack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/p/chartbook- newsletter-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 34 14 “What Causes Inflation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Forbes, May 28, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/advisor/investing/what-causes-inflation/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “What Causes Inflation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Investopedia, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='investopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/ask/answers/111314/what- causes-inflation-and-does-anyone-gain-it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='asp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Understanding Inflation,” Pacific Investment Management Company LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', https://europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pimco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/en- eu/resources/education/understanding-inflation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “What Causes Inflation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Stanford News (blog), September 6, 2022, https://news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='edu/2022/09/06/what-causes-inflation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 15 Gerald P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Dwyer Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Economic Review (07321813), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 2 (Quarter 1999), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Peter Ireland, “The Classical Theory of Inflation and Its Uses Today,” Presented at the Shadow Open Market Committee Meeting, November 3, 2014, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 16 “Monetarist Theory of Inflation: Meaning & Examples,” StudySmarter US, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='studysmarter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='us/explanations/macroeconomics/economic- performance/monetarist-theory-of-inflation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 17 Milton Friedman, Counter-Revolution in Monetary Theory (London: The Institute of Economic Affairs, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 18 Ireland, “The Classical Theory of Inflation and Its Uses Today,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 19 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 20 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 21 David McMillan, “Inflation: There’s a Vital Way to Reduce It That Everyone Overlooks--Raise Productivity,” World Economic Forum, June 9, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='weforum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/agenda/2022/06/inflation-there-s-a-vital-way-to-reduce-it-that- everyone-overlooks-raise-productivity/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 22 Pradana M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Jayathileke and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Rathnayake, “Testing the Link between Inflation and Economic Growth: Evidence from Asia,” Modern Economy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 4 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 87-92, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 87, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='4236/me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='42011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 23 Jayathileke, and Rathnayake, “Testing the Link between Inflation and Economic Growth,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 91-92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 24 Hrushikesh Mallick, “Inflation and Growth Dynamics: The Indian Experience,” Journal of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1080/17487870802327249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 25 Max Gillman, Mark Harris, and László Mátyás, “Inflation and Growth: Some Theory and Evidence” (draft), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', https://econpapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='repec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/cpd/2002/42_Harris_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 35 26 Nathan Benefield, “Inflation’s Cause?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It’s the Spending, Stupid,” pennlive, May 28, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pennlive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/opinion/2022/05/inflations-cause-its-the-spending- stupid-opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 27 “Monetarist Theory of Inflation: Meaning & Examples,” StudySmarter US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='studysmarter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='us/explanations/macroeconomics/economic- performance/monetarist-theory-of-inflation/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Inflation--The ‘Cost-Push’ Theory,” Britannica,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='britannica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/topic/inflation-economics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Alexander Barta, “Inflation in Economic Theory,” Exploring Economics, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='exploring- economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/en/discover/inflation/;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Arthur Laffer and Stephen Moore, “Economic Growth, Not Austerity, Is the Answer to Inflation,” The Heritage Foundation, July 13, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/budget-and-spending/commentary/economic-growth-not- austerity-the-answer-inflation ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Oner, “Inflation: Prices on the Rise,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 30-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 28 Dwyer and Hafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 29 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Gregory Mankiw, “When the Scientist Is Also a Philosopher,” The New York Times, March 22, 2014, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/2014/03/23/business/economic-view-when- the-scientist-is-also-a-philosopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 30 Barta, “Inflation in Economic Theory | Exploring Economics.” 31 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 32 Barta, “Inflation in Economic Theory | Exploring Economics.” 33 Mohsin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Khan and Abdelhak S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Senhadji, “Threshold Effects in the Relationship Between Inflation and Growth,” IMF Staff Papers, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 48, no.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/topic/inflation economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 36 Tretina, “Is Inflation Good Or Bad?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/advisor/investing/is- inflation-good-or-bad/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 37 Barta, “Inflation in Economic Theory,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='exploring- economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/en/discover/inflation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 38 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 39 “Six Ways to Fight Inflation,” Committee for a Responsible Federal Budget, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='crfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/blogs/six-ways-fight-inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 36 40 “Six Ways to Fight Inflation,” Committee for a Responsible Federal Budget, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='crfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/blogs/six-ways-fight-inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 41 Joseph E Stiglitz, “All Pain and No Gain from Higher Interest Rates,” Project Syndicate, December 8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='project-syndicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/commentary/fed- interest-rate-increases-counterproductive-all-pain-no-gain-by-joseph-e-stiglitz-2022-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 42 Peter Suderman, “Inflation, Remixed.” Reason, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 6 (November 2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 24-30, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 29 43 Suderman, “Inflation, Remixed,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 44 Suderman, “Inflation, Remixed,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 45 Jim Tankersley and Alan Rappoport, “As Prices Rise, Biden Turns to Antitrust Enforcers,” The New York Times, December 25, 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/2021/12/25/business/biden-inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Suderman, “Inflation, Remixed,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 46 Suderman, “Inflation, Remixed,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 47 Hrushikesh Mallick, “Inflation and Growth Dynamics: The Indian Experience.” Journal of Economic Policy Reform, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3 (September 2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 163–72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 163 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 168, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1080/17487870802327249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 48 Dwyer and Hafer, “Are Money Growth and Inflation Still Related?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 49 David McMillan, “Inflation: There’s a Vital Way to Reduce It That Everyone Overlooks--Raise Productivity,” World Economic Forum, June 9, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='weforum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/agenda/2022/06/inflation-there-s-a-vital-way-to-reduce-it-that- everyone-overlooks-raise-productivity/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 50 Laffer and Moor, “Economic Growth, Not Austerity, Is the Answer to Inflation,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='wsj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/articles/growth-austerity-inflation-larry-summers-unemployment- prices-jobs-rates-11656596482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 51 Laffer and Moore “Economic Growth, Not Austerity, Is the Answer to Inflation.” 52 Nathan Benefield, “Inflation’s Cause?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It’s the Spending, Stupid,” pennlive, May 28, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pennlive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/opinion/2022/05/inflations-cause-its-the-spending- stupid-opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 53 Laffer and Moore, “Economic Growth, Not Austerity, Is the Answer to Inflation.” 37 54 Benefield, “Inflation’s Cause?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' It’s the Spending, Stupid.” 55 Tretina, “Is Inflation Good Or Bad?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 56 Woo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Jung and Peyton J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Marshall, “Inflation and Economic Growth: Some International Evidence on Structuralist and Distortionist Positions,” Journal of Money, Credit and Banking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 2 (1986), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 227–32, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 232, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2307/1992206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 57 Pierre L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Siklos, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', Great Inflations of the 20th Century: Theories, Policies, and Evidence, (Edward Elgar Publishing, 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Adam Fergusson, When Money Dies: The Nightmare of Deficit Spending, Devaluation, and Hyperinflation in Weimar Germany (Public Affairs, 2010), http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/details/when-money-dies-the-nightmare-of- deficit-spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 58 Abhijeet Awasthi, “Inflation, Philosophy and Wittgenstein’s Ruler,” September 15, 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='linkedin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/pulse/inflation-philosophy-wittgensteins-ruler-abhijeet- awasthi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Fergusson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' When Money Dies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 59 Fergusson, When Money Dies,” http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/details/when-money-dies-the- nightmare-of-deficit-spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 60 Oner, “Inflation: Prices on the Rise,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 61 Nouriel Roubini, “The Unavoidable Crash,” Project Syndicate, December 2, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='project-syndicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/commentary/stagflationary-economic-financial-and- debt-crisis-by-nouriel-roubini-2022-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 62 Tristan Bove, “Inflation May Be a Good Thing for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', Argues One Top Economist,” Fortune, September 17, 2022, https://fortune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/2022/09/17/why-inflation- good-economy-stagflation-recession-brad-delong-larry-summers/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 63 Paul Krugman, “Is the Inflation Storm Letting Up?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' The New York Times, December 26, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/2022/12/26/opinion/is-the-inflation-storm-letting- up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 64 Vighi, “A System on Life Support.” 65 Bove, “Inflation May Be a Good Thing for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=',” https://fortune.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 347, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1080/13547860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='516148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 82 Mirrlees, “Are Recessions Inevitable?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business- cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 84 Rothbard, “How the Business Cycle Happens,” https://mises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business- cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 85 Rothbard, “How the Business Cycle Happens,” https://mises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business- cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 86 Rothbard, “How the Business Cycle Happens,” https://mises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business- cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 87 Roger W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Garrison, “Reflections on Reflections: A Consensus about the Great Depression?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Independent Review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} 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https://mises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business-cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 40 89 Rothbard, “How the Business Cycle Happens,” https://mises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/library/how-business- cycle-happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 90 James A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Mirrlees, “Are Recessions Inevitable?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 7 (March 2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 990–1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 113 Martin Clarke and David Butcher, “Reconciling Hierarchy and Democracy: The Value of Management Learning,” Management Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3 (September 1, 2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 313–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 42 114 Wofram Elsner, Gero Hocker, and Henning Schwardt, “Simplistic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Complex Organization: Markets, Hierarchies, and Networks in an Organizational Triangle,” SSRN Scholarly Paper (Rochester, NY: Social Science Research Network, September 21, 2009), http://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/abstract=1476314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/2016/05/12/opinion/campaign-stops/as-west-virginia-goes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 115 Elsner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=', “Simplistic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Complex Organization.” 116 Danchin, “The Tree and the Ring.” 117 Roelofs, “Networks and Democracy,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 118 A network has been defined as “a set of interconnected nodes.” See Manuel Castells, The Rise of the Network Society;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Information Age, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 1 (Cambridge, Mass: Blackwell Publishers, 1996) and Lawrence Tshuma, “Hierarchies and Government Versus Networks and Governance: Competing Regulatory Paradigms in Global Economic Regulation,” Social & Legal Studies, vol.' metadata={'source': 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vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 33, issue 3 (May 11, 2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 452-470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 120 Humberto R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Maturana and Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Varela, The Tree of Knowledge: The Biological Roots of Human Understanding (Boston & London: Shambhala, 1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Humberto Maturana, “Autopoiesis, Structural Coupling and Cognition: A History of These and Other Notions in the Biology of Cognition,” Cybernetics & Human Knowing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3–4 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 5–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 121 Ilya Prigogine and Isabelle Stengers, Order out of Chaos: Man’s New Dialogue with Nature (Toronto ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='Y: Bantam Books, 1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Stuart A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Kauffman, The Origins of Order: Self-Organization and Selection in Evolution, 1 edition (New York: Oxford University Press, 1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Niklas Luhmann, Social Systems (Stanford: Stanford University Press, 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' John A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Buck and Gerard Endenburg, “The Creative Forces of Self-Organization,” Sociocratic Center, Rotterdam, Netherlands, 2016, https://sociocracyconsulting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/wp-content/uploads/2016/04/CreativeForces- updated2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 122 Ernest Nagel and James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Newman, Godel’s Proof (New York: New York University Press, 1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 43 123 Gennady Shkliarevsky, “The Paradox of Observing, Autopoiesis, and the Future of Social Sciences,” Systems Research and Behavioral Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 3 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 323–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='1002/sres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 124 Shkliarevsky “Rethinking Democracy.” 125 Shkliarevsky “Rethinking Democracy.” 126 “What is Economic Value?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' IGI Global, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='igi- global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/dictionary/economic-value/9060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 127 “Value (economics),” Wikipedia, https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/wiki/Value_(economics);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' see also Caroline Benton, “Economic Value: Definition, Examples, Ways To Estimate,” Investopedia, November 25, 2020, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='investopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/terms/e/economic- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='asp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 128 Caroline Benton, “Economic Value: Definition, Examples, Ways To Estimate,” Investopedia, November 25, 2020, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='investopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/terms/e/economic- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='asp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 129 Yusuke Kuwayama, Justine Huetteman, and Bethany Mabee, “What is Value?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Resources for the Future, March 5, 2019, https://media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='rff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/documents/Value_Explainer_19-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 130 “What is Economic Value?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Market Business News, https://marketbusinessnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/financial-glossary/economic-value/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 131 “Value,” Corporate Finance Institute, October 4, 2022, https://corporatefinanceinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/resources/valuation/value/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 132 Gennady Shkliarevsky, “Infinite Growth: A Curse or a Blessing?”' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' (London, New York: Routledge, 2004), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 139 Gennady Shkliarevsky, “Educating for Creativity.” 140 Gennady Shkliarevsky, “Living a Non-Anthropocentric Future,” SSRN, September 29, 2021, https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/abstract=3933108 or http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='3933108 141 Gennady Shkliarevsky, “Making Progress Work: A New Life for the Old Idea,” SSRN, September 27, 2022, https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/abstract=4231501 or http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='4231501 142 Shkliarevsky, The Civilization at a Crossroads, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 146-48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 45 REFERENCES “Business Cycles.” Columbia Electronic Encyclopedia, 6th Edition, March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Causes of Business Cycles.” Vedantu, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/advisor/investing/what-causes-inflation/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “What Causes Inflation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Investopedia, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='investopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/ask/answers/111314/what causes inflation and does anyone gain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='asp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='igi-global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/dictionary/economic- value/9060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “What is Economic Value?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Market Business News, https://marketbusinessnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/financial- glossary/economic-value/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' 46 “What Is Production?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Meaning, Types, Examples, Theory.” Carbon Collective, March 24, 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='carboncollective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='co/sustainable-investing/production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Aldrick, Philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Larry Summers Says US Needs 5% Jobless Rate for Five Years to Ease Inflation.” Bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='Com, June 20, 2022, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='com/news/articles/2022-06-20/summers-says-us-needs-5-jobless- rate-for-five-years-to-ease-cpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' Awasthi, Abhijeet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content=' “Inflation, Philosophy and Wittgenstein’s Ruler,” September 15, 2021, https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} +page_content='pdf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfSQNJ/content/2301.03063v1.pdf'} diff --git a/6dAzT4oBgHgl3EQff_zo/content/tmp_files/2301.01463v1.pdf.txt b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/2301.01463v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95aa218a5ce704f363513e2609326572168f7f47 --- /dev/null +++ b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/2301.01463v1.pdf.txt @@ -0,0 +1,1606 @@ +arXiv:2301.01463v1 [cond-mat.soft] 4 Jan 2023 +Mechanosensitive bonds induced complex cell motility patterns +Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 +1Department of Physics, National Central University, Jhongli 32001, Taiwan +2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan +3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan +(Dated: January 5, 2023) +The one-dimensional crawling movement of a cell is considered in this theoretical study. Our active +gel model shows that for a cell with weakly mechanosensitive adhesion complexes, as myosin contrac- +tility increases, a cell starts to move at a constant velocity. As the mechanosensitivity of the adhesion +complexes increases, a cell can exhibit stick-slip motion. Finally, a cell with highly mechanosensitive +adhesion complexes exhibits periodic back-and-forth migration. A simplified model which assumes +that the cell crawling dynamics are controlled by the evolution of the myosin density dipole and +the asymmetry of adhesion complex distribution captures the motility behaviors of crawling cells +qualitatively. It suggests that the complex cell crawling behaviors observed in the experiments could +result from the interplay between the distribution of contractile force and mechanosensitive bonds. +PACS numbers: 87.17.Jj,87.16.Uv +Introduction.– The crawling motion of eukaryotic cells +is ubiquitous in biology as it plays important roles in +processes such as embryogenesis, wound healing, cancer +metastasis, and immunology [1]. +Common if not uni- +versal features of a crawling cell include myosin motors +distributed mainly behind the center, dominant actin +polymerization in the leading edge, and higher density +of adhesion complexes in the leading region [2]. Such po- +larized molecular distribution enables protrusion in the +leading edge due to actin polymerization, treadmilling of +actomyosin cytoskeleton due to contractility, and traction +force pulling the cell body. These features, therefore, are +included in many theoretical models for crawling cells [3]. +Interestingly, besides non-motile resting and steady- +moving behaviors, cells crawling along a one-dimensional +track either on a substrate or in a three-dimensional en- +vironment also exhibit moving patterns that are non- +stationary in time. For example, stick-slip crawling mo- +tion due to slip between integrin and the extracellular +matrix in focal adhesions under the contractility provided +by myosin II has been observed in human osteosarcoma +cells [4]. Periodic back-and-forth migration has been ob- +served in crawling zyxin-depleted cells in a collagen ma- +trix [5] and dendritic cells crawling along microfabricated +channels [6]. +Several theoretical models have been proposed to ex- +plain some of these deterministic complex moving pat- +terns. A model that includes the mechanochemical cou- +pling of actin promotor dynamics and actin polymer- +ization to myosin kinetics was shown to produce peri- +odic back-and-forth migration [7]. On the other hand, a +purely mechanical model emphasizing the interplay be- +tween mechanosensitive bonds and membrane tension ex- +hibited stick-slip motion even for slip bonds [8]. Inter- +estingly, it has also been shown that stick-slip can re- +sult from the interplay between mechanosensitive bonds, +contractility, and a force that tends to restore a cell’s +preferred length [9]. +In this letter, we present a theoretical study to show +that the coupling between mechanosensitive adhesion +complexes and myosin contractility is sufficient to gener- +ate deterministic complex cell crawling behaviors, includ- +ing stick-slip, periodic back-and-forth movements, and +other complex moving patterns. We first construct an +active gel model with mechanosensitive adhesion com- +plexes and show that the distribution of myosin motors +and adhesion complexes computed from our model agree +with experimentally observed features. By exploring the +motility behavior with different strengths of contractil- +ity and mechanosensitivity, we show that this model can +lead to complex motility behaviors other than rest and +constant-velocity moving states. Among these complex +motility patterns, unidirectional stick-slip motion and pe- +riodic back-and-forth movement are the most common. +When the adhesion complexes are less mechanosensitive, +as myosin contractility increases, the cell performs con- +stant velocity motion. On the other hand, for cells with +highly mechanosensitive adhesion complexes, as myosin +contractility becomes sufficiently strong, periodic back- +and-forth crawling motion can be observed. +Finally, +stick-slip and other complex motility patterns can be ob- +served by increasing the mechanosensitivity of the adhe- +sion complexes for a cell moving at constant velocity. +To understand the physical mechanisms that produce +these complex motility patterns, a simplified model in- +spired by the active gel model is constructed. The sim- +plified model assumes that the dynamics of a sufficiently +slow-moving cell are dominated by the dipole moment +of myosin density and the difference in the total num- +ber of adhesion complexes near the two cell ends. Re- +markably, the motility behavior predicted by this simple +model agrees qualitatively with the motility behaviors +predicted by the active gel model. +These results sug- +gest that, in general, diverse complex motility behaviors +can result from the interplay between mechanosensitive +adhesions and the dynamical organization of contractile + +2 +myosin motors. +Model summary.– To focus on the contribution of cell +mechanics to motility behaviors, chemical signaling is not +included in our model. The cytoplasm of the cell is mod- +eled as an active gel [10, 11] enclosed by the cell mem- +brane, the adhesion complexes are treated as reversible +bonds with specific binding-unbinding rates, and actin +polymerization is assumed to happen only at the cell +ends. +The forces acting on the cell include the stress +in the cytoskeleton, the drag force from the substrate, +and the force due to the adhesion complexes. +Our model only considers one spatial direction, i.e., the +cell’s moving direction, and the stress in the cytoplasm +obeys the constitutive equation +σ = η ∂v +∂x + χc, +(1) +where η is the effective one-dimensional viscosity of the +cytoplasm, v is the flow field, χ is the strength of contrac- +tility provided by myosin motors (χ > 0), and c is the +concentration of myosin attached to the actin network. +For simplicity, compressibility is not included [11]. Thus +pressure does not appear in the constitutive relation. The +force exerted by the substrate is +Fdrag = −αnbv − ξv, +(2) +The first term on the right hand is the drag provided by +the adhesion complexes [12], α is a constant that char- +acterizes the resistance of the adhesion complexes to cell +movement, and nb is the number density of adhesion com- +plexes. The second term comes from the viscous drag of +the fluid between the cell and the substrate, and ξ is the +drag coefficient. Putting Eqs. (1)(2) together, the result- +ing force balance equation, ∂xσ + Fdrag = 0, takes the +following form +η ∂2v +∂x2 − (αnb + ξ)v = −χ ∂c +∂x. +(3) +Myosin motors attached to actin filaments move with +the cytoplasm, while those detached from actin filaments +diffuse freely. The attachment/detachment of motors is +reversible. On long-time scales, the density of the mo- +tors can be effectively described by an advection-diffusion +equation [13] +∂c +∂t = D ∂2c +∂x2 − ∂(cv) +∂x , +(4) +where D is the effective diffusion coefficient of myosin +motors. +Adhesion complexes providing anchorage to the extra- +cellular matrix are also physically coupled to the con- +tractile cytoplasm. +As a result, they are pulled when +the cytoplasm moves [8]. Once an adhesion complex is +formed, the adhesion site does not move, but the disso- +ciation rate of the adhesion complex is affected by the +motion of the cytoskeleton because the bond is stretched +or compressed. In our model, the evolution of the density +of adhesion complexes is assumed to obey +∂nb +∂t = −k0 e−k1∂xvnb + kon, +(5) +where kon is the binding rate, k0 is the unbinding rate at +∂xv = 0, and k1 tells us how unbinding rate is affected by +the cytoplasmic flow. Our model assumes that when the +strain rate is dilating, giving more space for the adhesion +complexes, the unbinding rate decreases. +Actin polymerization at the cell ends depends on the +distribution of actin activators [15][16]. In the presence +of environmental cues, a gradient of actin activator con- +centration within the cell is established, and actin poly- +merization is polarized due to this concentration gradi- +ent. In the absence of such external influence, the cell +can nevertheless polarize itself by spontaneous symmetry +breaking, and the net actin polymerization rate becomes +asymmetric. In our model, we consider a homogeneous +environment, and the net actin polymerization rate v± +p +at the ± end of the cell is assumed to be +v± +p = +2 e−v(1) +p +(L−L0) +1 + exp[∓ dl± +dt /v(2) +p ] +v(0) +p , +(6) +where v+ +p (v− +p ) is the net rate of extension due to actin +polymerization at the cell end located at x = l+(l−), +v(0) +p +comes from the base polymerization rate, v(1) +p +in the +exponent of the numerator comes from the effect of free +energy cost for polymerization when the cell length is +different from its natural length (L0 is the natural length +of the cell, and L = l+ − l−), and the term with v(2) +p +makes the net polymerization rate in a moving cell at +both ends different, with more polymerization events in +the leading end than the trailing end. +It will become +clear that the qualitative results of cell motility behavior +do not depend on the specific form we assumed for the +dissociation rate of the adhesion complexes and v± +p . +The evolution of cell-end positions is determined by +the velocity of cytoplasm and actin polymerization, +dl± +dt = v± ± v± +p , +(7) +where v+ (v−) is the velocity of cytoplasm at the + (−) +end. +Experimentally it has been shown that a cell tends to +restore its length L to a preferred magnitude L0 [17]. We +model this effect by the following force balance condition +at cell ends +σ± = +� +χc + η ∂v +∂x +� +l± += −γ(L − L0). +(8) +Here γ is a constant associated with the restoring force +that brings the cell length L to L0. + +3 +Because no myosin motors can leave or enter the cell, +the total flux of myosin motors across a cell end should +vanish. This leads to +[cv]l± − [c]l± +dl± +dt − D +� ∂c +∂x +� +l± += 0. +(9) +The first two terms on the left-hand side are the advective +flux relative to the moving cell end, and the last term is +the diffusive flux at the cell end. +It is convenient to introduce effective drag coefficient +ξeff = ξ + αkon/k0 and choose l0 = +� +η/ξeff as the unit +length, t0 = η/(ξeffD) as the unit time, σ0 = ξeffD as +the unit stress, n0 = ξ/α as the unit density for adhe- +sion complexes, and c0 = M/ +� +η/ξeff as the unit myosin +concentration, where M is the total number of myosin +motors in the cell. Therefore the dimensionless drag co- +efficient ˜ξ = ξ/(ξ + αkon/k0), contractility ˜χ = c0χ/σ0, +and cell elastic constant K = γl0/σ0 are used in the fol- +lowing discussion. +Simulation of motility behaviors.– From the point of +view of nonequilibrium thermodynamics, the drag force +between the cell and the substrate, the viscous force in +the cytoplasm, and the diffusion of myosin are passive +processes against cell movement. On the other hand, ac- +tive processes such as actin polymerization and myosin +contractility drive the movement of the cell, and the +binding/unbinding dynamics of the adhesion complexes +modulate cell movement. +As the myosin motors pro- +vide contractility against viscous and substrate drag, the +contractility-induced cytoplasmic flow drifts the motors +to aggregate and also affects the distribution of adhesion +complexes. Once the cell is in motion, the feedback in +the actin polymerization rate further enhances cell polar- +ization. The balance between these processes determines +the state of the cell. In general, there is no analytical so- +lution when all these effects are included. Therefore we +numerically integrate the equations of motion by a finite +difference method. The details of our numerical methods +and our choice of parameters are discussed in [20]. +Figure 1 shows the motility behaviors for a cell with +parameters chosen to be compatible with typical cells [20] +and a range of adhesion complex mechanosensitivity and +contractility strengths. The following motility behaviors +are found: rest, moving at a constant velocity, unidirec- +tional stick-slip movement, back-and-forth motion with +stick-slip, and periodic back-and-forth movement. For a +cell with weakly mechanosensitive adhesion complexes, as +contractility increases, a cell at rest starts to move at con- +stant velocity. As the adhesion complexes become more +mechanosensitive, a moving cell shows other complex +motility behaviors. For example, stick-slip motion and +(at high contractivity) back-and-forth motion with stick- +slip, and finally, the cell performs periodic back-and- +forth motion when the adhesion complexes are highly +mechanosensitive. Another motility phase diagram in the +Supplement Materials [20] shows that, within our model, +a cell with a high actin polymerization rate can exhibit +other complex motility behaviors between stick-slip and +periodic back-and-forth movements. +�� +�� +�� +�� +�� +�� +� +� +��� +��� +��� +��� +� +� +(a) +� +� +� +� +� +� +�� +� +(b) +� +� +�� +� +� +�� +�� +� +(c) +� +�� +�� +� +��� +��� +���� +� +(d) +� +� +�� +� +��� +��� +��� +� +(e) +FIG. 1. +(a) Motility phase diagram for a cell with dimension- +less parameters K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) +p += 0.2, +v(1) +p += 0.5, and v(2) +p += 2. +Rest state (squares), constant- +velocity motion (diamonds), stick-slip movement (triangles), +back-and-forth with stick-slip motion (empty circle), and pe- +riodic back-and-forth motion (filled circles) are found. +(b) +Trajectories of the cell ends for ˜χ = 18, k1 = 0.05, the cell +performs constant velocity motion. (c) Trajectories of the cell +ends for ˜χ = 17.5, k1 = 0.1, the cell performs stick-slip mo- +tion. (d) Trajectories of the cell ends for ˜χ = 19, k1 = 0.15, +the cell performs complex motility pattern which is periodic +back-and-forth with stick-slip. (e) Trajectories of the cell ends +for ˜χ = 18, k1 = 0.25, the cell performs periodic back-and- +forth motion. +Figure 2 shows that when the cell is at rest, myosin +motor distribution is symmetric around the center of the +cell, and the number of adhesion complexes near both +cell ends is the same; on the other hand, for a cell mov- +ing at constant velocity, myosin motors aggregate close +to the trailing end and adhesion complexes are mainly +close to the leading end. The distribution of myosin mo- +tors and adhesion complexes for a cell undergoes stick- +slip, and periodic back-and-forth movements are shown +in Fig. S2 of [20]. It is clear that whenever the cell has a +definite moving direction, myosin motors aggregate in a +regime behind the center of the cell, and more adhesion +complexes form near the leading end than the trailing +end. +Indeed, the adhesion complex binding/unbinding +rates Eq.(5) and net actin polymerization at the cell +ends Eq.(6) in our model lead to reasonable molecular +distributions in a cell. +Reduction to the simplified model.– To obtain an in- + +4 +���� +��� +��� +� +��� +��� +��� +� +� +� +(a) +���� +��� +��� +� +��� +��� +��� +(b) +FIG. 2. +Distribution of myosin motors and adhesion com- +plexes for a cell (a) at rest and (b) undergoes constant velocity +motion towards +x direction. The parameters are the same +as those in Fig. 1 and (a) k1 = 0.25, ˜χ = 16, (b) k1 = 0.1, +˜χ = 16. +tuitive physical picture of the complex motility behav- +iors, especially the transitions from the rest state to the +constant-velocity movement and periodic back-and-forth +movement, we construct a simplified model from the ac- +tive gel model. First, we consider a limiting situation in +which adhesion complexes only appear in a small region +close to the cell ends. In this regime, it is convenient to in- +troduce Nf(Nb), the total number of adhesion complexes +close to l+(l−), and N = Nf + Nb, ∆N = Nf − Nb to +describe the distribution of adhesion complexes. We also +introduce yc, the dipole moment of myosin motors den- +sity relative to the center of the cell [20], to characterize +the spatial distribution of myosin motors. The net actin +polymerization velocity at the cell ends v± +p ≡ vp±∆vp/2, +where ∆vp = v+ +p − v− +p ∝ Vcell in the limit of small cell +velocity. Therefore we write v± +p = vp ± βVcell/2. In this +regime, straightforward calculation shows that the veloc- +ity of the cell is [20] +Vcell ≈ +1 +1 − β/2(λν1vp∆N − ˜χλν2yc), +(10) +where λν1 and λν2 are positivet coefficients that depend +on N and L. Note that, from the definition, β/2 cannot +be greater than unity. +Therefore a cell with ∆N > 0 +and yc < 0 has positive Vcell. This is in agreement with +experimental observations. +We further simplify the analysis by considering the +limit of large K, i.e., L ≈ L0. The following equations +are constructed to describe the dynamics of N, ∆N, and +yc. First, the evolution equations for N and ∆N are +dN +dt = 2kon − k(0) +off N − k(1) +off yc ∆N, +d∆N +dt += −k(0) +off ∆N − k(1) +off Nyc, +(11) +where kon, k(0) +off , and k(1) +off are positive constants. Next, in +the spirit of Landau-type approximation, the evolution +of yc is assumed to obey the following equation, +dyc +dt = −Γ +� +−(˜χ − ˜χc)yc − a∆N∆N + a3y3 +c +� +, +(12) +where ˜χc and a∆N are treated as N-independent con- +stants for simplicity. +The simple model equations (11)(12) have a solution +with constant N and ∆N = yc = 0. This corresponds +to a cell at rest. Solutions with nonzero constant ∆N +and yc correspond to a cell moving at a constant veloc- +ity; solutions with time-periodic ∆N and yc are stick-slip +(periodic back-and-forth) movement if the time-average +of ∆N and yc are nonzero (zero). The linear stability +analysis of the rest state shows that as the contractil- +ity increases, a cell at rest starts to move as the system +undergoes a bifurcation, the moving state is the constant- +velocity state when k(1) +off is small, i.e., the adhesion com- +plexes are less mechanosensitive. When k(1) +off is sufficiently +large, the rest-to-moving transition leads to a periodic +back-and-forth moving state [20]. +As shown in Fig. S3 [20], the model Eqs. (11) (12) +exhibit a motility phase diagram qualitatively the same +as the numerical solutions of our active gel model. The +minor differences come from those simplifications made +when constructing the simplified model. +Furthermore, +from the simplified model, it is easy to see how the sym- +metry properties and the couplings of the key driving +variables lead to the observed cell motion. +For exam- +ple, yc(t) and ∆N(t) in Fig. 3(a) for a cell performing +periodic back-and-forth movement suggests the following +physical picture about how the coupling terms in the sim- +plified model lead to this motion. According to Eq. (11), +a small ∆N tend to increase when yc is sufficiently nega- +tive, and Eq. (12) states that when a∆N > 0, yc tends to +move toward the center of the cell when the magnitude of +∆N is sufficiently large. The result is that at sufficiently +large k(1) +off , the number of adhesion complexes in the lead- +ing end of a moving cell increases sufficiently fast such +that at some point, the myosins are pulled to the other +half of the cell, reversing the sign of yc, then reversing the +sign of ∆N, and eventually the direction of cell motion +is reversed. This is how rest/constant-velocity transition +becomes rest/back-and-forth transition as the adhesion +complexes are more mechanosensitive. In between the +constant velocity and periodic back-and-forth movement, +complex motility patterns with stick-slip can be observed. +As shown in Fig. 3(b), yc(t) and ∆N(t) in a cell that un- +dergoes stick-slip movement have nonzero time-average +values, and they oscillate with similar phase-relations as +a cell undergoes periodic back-and-forth movement. This +is because the mechanosensitivity of the adhesion com- +plexes is sufficiently strong to induce an oscillation of ∆N +and yc, but not sufficiently strong to change their signs. +Figure 3(c) and Fig. 3(d) show that yc(t) and ∆N(t) +(defined as the difference of the total number of adhe- +sion complexes in the leading and trailing halves of the +cell) in the numerical simulations of the active gel model +behave similarly, suggesting that the physical picture ob- +tained from studying the simplified model can be applied + +5 +to more detailed models. +� +�� +�� +� +���� +��� +��� +� +� +�� +(a) +� +�� +�� +� +���� +��� +��� +(b) +��� +��� +��� +� +���� +��� +��� +(c) +� +� +�� +� +���� +��� +��� +(d) +FIG. 3. +(a)(b): yc(t) and ∆N(t) in the simplified model with +kon/k(0) +off = 1, a∆N = 1, and a3 = 1. In (a), k(1) +off /k(0) +off = 0.6, +Γ(˜χ− ˜χc)/k(0) +off = 1.2; in (b), k(1) +off /k(0) +off = 0.62, Γ(˜χ− ˜χc)/k(0) +off = +5.5. (c)(d): yc(t) and ∆N(t) in the active gel model with K = +100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) +p += 0.2, v(1) +p += 0.5, v(2) +p += 2. +In (c), k1 = 0.55, ˜χ = 16; in (d), k1 = 0.1, ˜χ = 18. The cell +in (a)(c) performs oscillatory back-and-forth movement, and +the cell in (b)(d) performs stick-slip movement. +Discussion.– Within our active gel model, a cell with +highly mechanosensitive adhesion complexes can exhibit +periodic back-and-forth movement similar to what was +observed in zyxin-depleted cells in a collagen matrix. +Since zyxin proteins act as mechanosensors in mature +adhesion complexes [21], our study suggests that the dif- +ference in the mechanosensitivity of the adhesion com- +plexes in zyxin-depleted and wild-type cells could be the +origin of the periodic back-and-forth movement observed +in [5]. Future experiments can be designed to examine +this prediction. +The physical picture suggested by our simplified model +also implies the possibility that, in general, when some +of the simplifications are lifted, more complex one- +dimensional cell motility behaviors can be found. This +is indeed the case, as we explore the behavior of our ac- +tive gel model for a broader range of actin polymerization +rates, complex trajectories which come from further bi- +furcations are found. +This is shown in Fig. S1 in [20] +and Fig. 4. Further study of the physical mechanisms for +these behaviors will be our future work [22]. +Finally, +although +the +physical +mechanisms +for +symmetry-breaking transitions, such as rest/periodic +back-and-forth transition and rest/constant velocity +transition, can be understood from the dynamics of yc +and ∆N, it is interesting to study how other important +physical observables, such as the multipoles of the trac- +tion force [23][24], behave in cells with different moving +patterns. It is also important to check if the basic fea- +tures of these physical observables in different moving +patterns depend on the details of the binding/unbinding +dynamics of adhesion complexes, as it plays a significant +role in our understanding of many interesting features of +cell motility. +� +� +�� +� +� +� +� +� +(a) +� +� +�� +� +��� +��� +��� +� +(b) +FIG. 4. +Our active gel model predicts that at high actin +polymerization rates further complex motility behaviors, such +as (a) zig-zag with stick-slip, and (b) double-period back-and- +forth movement can be found. These trajectorjes are obtained +for K = 100, ˜χ = 14, kon = 6.0, k0 = 3.0, v(1) +p += 0.5, v(2) +p += +0.2. In (a), v(0) +p += 1.2, k1 = 0.9; in (b), v(0) +p += 1.1, k1 = 0.9. +The blue curves represent the trajectories of the cell ends. +Acknowledgments – +H.-Y. C. thanks Prof. +Jasnow (University of Pitts- +burgh) for stimulating discussions and encouragement in +the early stage of this work. H.-Y. C. is supported by +the Ministry of Science and Technology, Taiwan (MOST +108-2112-M-008-016 ). The authors also acknowledge the +support from National Center for Theoretical Sciences, +Taiwan. +[1] D. Bray, Cell Movements, 2nd ed. (Talyor and Francis, +New York, 2001). +[2] P.T. Yam, C.A. Wilson, L. Ji, B. Hebert, E.L. Barnhart, +N.A. Dye, P.W. Wiseman, G. Danuser, and J.A. Theriot, +Actin–myosin network reorganization breaks symmetry +at the cell rear to spontaneously initiate polarized cell +motility, J. Cell Biol., 178, 1207 (2007). +[3] I.S. Aranson ed., +Physical Models of Cell Motility, +Springer, 2016. +[4] Y. Aratyn-Schaus and M. L. Gardel, Transient frictional +slip between integrin and the ECM in focal adhesions +under myosin II tension, Curr. Biol. 20, 1145 (2010). +[5] S. I. Fraley, Y. Feng, A. Giri, G. D. Longmore, and +D.Wirtz, Dimensional and temporal controls of three- +dimensional cell migration by zyxin and binding partners, +Nat. 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SanoA simple model of cell +crawling, Physica D, 318, 3 (2016). + +arXiv:2301.01463v1 [cond-mat.soft] 4 Jan 2023 +Supplementary material for +“ Mechanosensitive bonds induced complex cell motility patterns” +Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 +1Department of Physics, National Central University, Jhongli 32001, Taiwan +2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan +3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan +(Dated: January 5, 2023) +S1. +DIMENSIONLESS EQUATIONS +We introduce effective drag coefficient ξeff = ξ + αkon/k0 and choose l0 = +� +η/ξeff as the unit length, t0 = +η/(ξeffD) as the unit time, σ0 = ξeffD as the unit stress, n0 = ξ/α (notice that, here ξ, not ξeff is used) as the +unit density for adhesion complexes, and c0 = M/ +� +η/ξeff as the unit myosin concentration, where M is the total +number of myosin motors in the cell. In the dimensionless form, the momentum equation is +∂2v +∂x2 − ˜ξ(1 + nb)v = −˜χ ∂c +∂x, +(S1) +myosin density evolution obeys +∂c +∂t = ∂2c +∂x2 − ∂(cv) +∂x , +(S2) +evolution of the density of the adhesion complexes is +∂nb +∂t = −¯k0e−¯k1∂xvnb + ¯kon +(S3) +and the positions of the cell ends obey +dl± +dt = [v]l± ± v± +p . +(S4) +The boundary condition for myosin current is +� +c(v − dl +dt) − ∂c +∂x +� +l± += 0, +(S5) +and the stress continuity at the cell ends leads to +σ± = −K (L − L0) = +�∂v +∂x +� +l± ++ ˜χcl±. +(S6) +Here all variables and x, t are dimensionless. The definitions of the parameters and important physical quantities are +listed in Table S1. The dimensionless parameters in our model are listed in Table S2. +S2. +NUMERICAL METHOD AND CHOICE OF PARAMETERS +In our numerical scheme, each iteration updates all dynamical variables by integrating the evolution equations +over a small time interval ∆t with a finite difference method. First, [v]l± and v± +p from the previous iteration were +substituted into Eq. (S4) to obtain the new positions of the cell ends. The densities of the adhesion complexes and +myosin motors are updated from the flow field of the previous iteration by integrating the evolution Eq. (S3) of nb +and the myosin advection-diffusion Eq. (S2). The force balance Eq. (S1) with the updated bond density and myosin +concentration is then solved to obtain the new flow field. +The numerics were carried out by dividing the cell into Nx = 100 segments and approximating the spatial derivatives +by the finite difference method with the size of a time step ∆t = 10−6. The typical material parameters are D ∼ +0.025 µm2/s [1], l0 ∼ 10 µm and unit time t0 ∼ 103 s [2]. k0 is the rate of dissociation of mature focal adhesion and + +S2 +Physical meaning +Symbol +effective drag coefficient +ξeff = ξ + αkon/k0 +unit length +l0 = +� +η/ξeff +unit time +t0 = η/ξeffD +unit stress +σ0 = ξeffD +unit myosin motors concentration +c0 = +M +√ +η/ξeff +unit density of cell-substrate bonds +n0 = ξ/α +TABLE S1. Definitions of physical parameters and characteristic quantities in our model. +Physical meaning +Symbol +unbinding rate +¯k0 = t0k0 +coefficient for strain-rate-dependent unbinding +¯k1 = k1/t0 +binding rate +¯kon = kont0/n0 +base actin polymerization speed +¯v(0) +p += v(0) +p l0/t0 +coefficient for stress-dependent actin polymerization +¯v(1) +p += v(1) +p l0 +coefficient for cell polarization effect on actin polymerization +¯v(2) +p += v(2) +p t0/l0 +contractility +˜χ = c0χ/σ0 +cell elastic constant +K = γl0/σ0 +drag coefficient +˜ξ = ξ/(ξ + αkon/k0) +TABLE S2. Definitions of dimensionless parameters in our model. +its typical value ∼ 1/min [3], and we chose ¯k0 = 3. ¯kon is chosen to be 6 for most simulations and the dimensionless +density of bonds in the absence of mechanosensitivity is of order unity. The dimensionless natural length of the cell +L0 = 1 for a typical cell. This means that in the absence of adhesion complexes, drag and viscous forces are of the +same order of magnitude. The dimensionless total number of myosin motors in the cell c0L0 = 1. + +S3 +S3. +SIMULATION RESULTS +The motility phase diagrams in the plane spanned by ˜χ and k1 are presented in the main text. Here Fig. S1 shows +the phase diagram in the plane spanned by v(0) +p +and k1. From this figure we can see how the actin polymerization +rate affects the motility behavior of the cell. It is clear that the moving state of a cell with small k1 is that with +a constant velocity, while the moving state of a cell with large k1 is periodic back-and-forth. This is in agreement +with the phase diagrams in the main text. Furthermore, there are several complex motility patterns between these +two states, including stick-slip motion and behaviors that can be seen as combinations of back-and-forth and stick- +slip movement. The trajectories for stick-slip movement and periodic back-and-forth movement with stick-slip are +shown in Fig. 1(c)(d) of the main text. +The trajectories for zig-zag movement with stick-slip and double-period +back-and-forth motion are shown in Fig. 4(a)(b) of the main text. +��� +��� +��� +��� +� +��� +� +��� +��� +��� +��� +��� +� +� +FIG. S1. Phase diagram for the motility behavior predicted by the reduced model. K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, ˜χ = 14, +v(1) +p += 0.5, v(2) +p += 0.2. The cells show the following motility behaviors: constant velocity motion (green diamonds), periodic +back-and-forth movement (blue circles), stick-slip movement (orange triangles), periodic back-and-forth movement with stick- +slip (empty gray circles), zig-zag movement with stick-slip (empty purple diamonds), and double-period back-and-forth motion +(brown squares). +Figure S2 shows the distribution of adhesion complexes and myosin motors for a cell that undergoes stick-slip +movement and periodic back-and-forth movement. Similar to the distributions shown in Fig. 2 of the main text, +for a moving cell, myosin motors are always located relatively close to the trailing end, and the density of adhesion +complexes is always higher in a region close to the leading end. + +S4 +� +�� +�� +� +� +� +�� +� +��� +��� +��� +� +(a) +� +�� +�� +� +� +� +�� +� +� +� +� +�� +� +� +(b) +��� +��� +��� +� +� +� +�� +� +��� +��� +��� +� +(c) +��� +��� +��� +� +� +� +�� +� +� +� +� +�� +� +� +(d) +FIG. S2. Distribution of adhesion complexes and myosin motors for K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) +p += 0.2, v(1) +p += 0.5, +v(2) +p += 2, and (a)(b) k1 = 0.12, ˜χ = 18, (c)(d) k1 = 0.25, ˜χ = 17. + +S5 +S4. +THE SIMPLIFIED MODEL +Since the adhesion complexes are concentrated close to the cell ends, to simplify the analysis, we assume +nb(x, t) = Nfδ(x − xf) + Nbδ(x − xb), +(S7) +where xf = l+ − ǫ, xb = l− + ǫ, and ǫ is a very small length. +A. +Stress field and flow field +Because there are no adhesion complexes in xb < x < xf, the dimensionless momentum equation in this region is +∂xσ = ˜ξv, +σ = ∂xv + ˜χc, +xb < x < xf. +(S8) +Integrating the full momentum equation from l+(−) to xf(b), we find +σf = −K(l − l0) − ˜ξNfvf, +σb = −K(l − l0) + ˜ξNbvb. +(S9) +Here vf = v(xf, t) ≈ dl+/dt − v+ +p , and vb = v(xb, t) ≈ dl−/dt + v− +p . The solution of stress from these equations is [4] +σ(x, t) = σf +sinh +� +˜ξ1/2(x − xb) +� +sinh +� +˜ξ1/2(xf − xb) +� + σb +sinh +� +˜ξ1/2(xf − x) +� +sinh +� +˜ξ1/2(xf − xb) +� + ˜χ˜ξ1/2 +� xf +xb +G(x, x′)c(x′, t)dx′, +(S10) +where +G(x, x′) = +sinh +� +˜ξ1/2(xf − x) +� +sinh +� +˜ξ1/2(x′ − xb) +� +sinh +� +˜ξ1/2(xf − xb) +� +− Θ(x′ − x) sinh +� +˜ξ1/2(x′ − x) +� +, +(S11) +Θ(x) is the Heaviside step function. This leads to the following expression for the flow field, +v(x, t) = +1 +˜ξ1/2 + + +σf +cosh +� +˜ξ1/2(x − xb) +� +sinh +� +˜ξ1/2(xf − xb) +� − σb +cosh +� +˜ξ1/2(xf − x) +� +sinh +� +˜ξ1/2(xf − xb) +� + ˜χ +� xf +xb +∂xG(x, x′)c(x′, t)dx′ + + + . +(S12) +B. +Myosin concentration +Substituting Eq. (S12) into the advection-diffusion for myosin concentration, one obtains the following equation +which does not have an explicit dependence on the velocity field. +∂tc(x, t) = D∂2 +xc − +1 +˜ξ1/2 ∂x + + + + +σf +cosh +� +˜ξ1/2(x − xb) +� +sinh +� +˜ξ1/2(xf − xb) +� − σb +cosh +� +˜ξ1/2(xf − x) +� +sinh +� +˜ξ1/2(xf − xb) +� + + c(x, t) ++˜χ +� xf +xb +c(x, t)∂xG(x, x′)c(x′, t)dx′ +� +. +(S13) + +S6 +C. +Velocity and length of the cell +The velocity of the cell Vcell = 1 +2 +� +dl+ +dt + dl− +dt +� +is +Vcell = +1 +2˜ξ1/2 +cosh +� +˜ξ1/2L +� ++ 1 +sinh +� +˜ξ1/2L +� +(σf − σb) + ˜χ +2 +� xf +xb +sinh +� +˜ξ1/2(xf − x′) +� +− sinh +� +˜ξ1/2(x′ − xb) +� +sinh +� +˜ξ1/2L +� +c(x′, t)dx′ ++v+ +p − v− +p +2 +. +(S14) +The evolution of the length of the cell dL +dt = dl+ +dt − dl− +dt obeys +dL +dt = +1 +˜ξ1/2 +cosh +� +˜ξ1/2L +� +− 1 +sinh +� +˜ξ1/2L +� +(σf + σb) − ˜χ +� xf +xb +sinh +� +˜ξ1/2(xf − x′) +� ++ sinh +� +˜ξ1/2(x′ − xb) +� +sinh +� +˜ξ1/2L +� +c(x′, t)dx′ ++(v+ +p + v− +p ). +(S15) +D. +Symmetries of the system +It is helpful to introduce the following variables +N = Nf + Nb, +∆N = Nf − Nb, +vp = v+ +p + v− +p +2 +, +∆vp = v+ +p − v− +p , +σS = σf + σb +2 += −K(L − L0) − +˜ξ +2 +� +N +�dL/dt +2 +− vp +� ++ ∆N +� +Vcell − ∆vp +2 +�� +, +σA = σf − σb +2 += − +˜ξ +2 +� +N +� +Vcell − ∆vp +2 +� ++ ∆N +�dL/dt +2 +− vp +�� +, +(S16) +and +y = x − l+ + l− +2 +. +(S17) +Notice that dL/dt, N, vp, and σS are symmetric under spatial inversion (y → −y), while Vcell, ∆N, ∆vp, and σA are +antisymmetric under spatial inversion. +The velocity of the cell Vcell can be expressed in terms of these parameters and variables that have clear parity +signatures. First, notice that only the part of c(y) that is anti-symmetric under y → −y contribute to the ˜χ-dependent +term of Eq. (S14), and for a slow-crawling cell this part should be significant only in the small |y| region. Thus by +expanding the ˜χ-dependent term of Vcell to the leading order in y, one finds that +Vcell = 1 +2 + +∆vp + +2 +˜ξ1/2 +cosh +� +˜ξ1/2L +� ++ 1 +sinh +� +˜ξ1/2L +� +σA − ˜χ˜ξ1/2 cosh +� +2˜ξ1/2L +� +sinh +� +˜ξ1/2L +� yc + ... + + , +where +yc ≡ +� L/2 +−L/2 +y c(y, t) dy, +(S18) +and “...” represents terms of higher order in this expansion. The expression for Vcell can be further simplified by +taking ∆vp ≈ βVcell (β is independent of Vcell) for a slow crawling cell and substituting Eq. (S16) for σA. Finally, one + +S7 +obtains +Vcell = − +˜ξ1/2 +1 − β/2 + +1 + +˜ξ1/2 +2 +cosh +� +˜ξ1/2L +� ++ 1 +sinh +� +˜ξ1/2L +� +N + + +−1 +× + +1 +2 +cosh +� +˜ξ1/2L +� ++ 1 +sinh +� +˜ξ1/2L +� +�1 +2 +dL +dt − vp +� +∆N + ˜χ +cosh +� +˜ξ1/2L/2 +� +sinh +� +˜ξ1/2L +� yc + ... + + . +(S19) +This expression tells us that Vcell is nonzero only when ∆N (asymmetry in the distribution of adhesion complexes) +or yc (asymmetry in the distribution of myosin motors) is nonzero. +Similar calculation leads to the following expression for the evolution of the length of the cell, +dL +dt = 2vp + +� +1 + +˜ξ1/2 +2 +cosh(˜ξ1/2L) − 1 +sinh(˜ξ1/2L) +N +�−1 +× + + + +˜ξ−1/2 cosh(˜ξ1/2L) − 1 +sinh(˜ξ1/2L) +� +−2K(L − L0) + ∆N +� +1 − β +2 +� +Vcell +� +− 2˜χ +sinh +� +˜ξ1/2L/2 +� +sinh +� +˜ξ1/2L +� Ctot + + + , +(S20) +where +Ctot ≡ +� L/2 +−L/2 +c(y, t)dy ≡ 1 +(S21) +is the total amount of myosin motors in the cell, which is unity in our dimensionless expression. Note that all terms +on the right-hand side of dL/dt are even under y → −y. +E. +The simplified model and bifurcations +From the previous analysis, it is clear that symmetry under y → −y plays an important role in Vcell and dL/dt. +Based on these observations, a simplified model is proposed for slow-crawling cells. +First, the evolution equations of Nf and Nb are +dNf +dt += kon − (k(0) +off + k(1) +off yc)Nf, +dNb +dt += kon − (k(0) +off − k(1) +off yc)Nb. +This leads to +dN +dt = 2kon − k(0) +off N − k(1) +off yc ∆N, +d∆N +dt += −k(0) +off ∆N − k(1) +off Nyc. +(S22) +We expect k(1) +off > 0 because a cell moving at constant velocity in the +x direction should have yc < 0 and ∆N > 0. +Next, the following evolution equation for yc is proposed +dyc +dt = −Γ +� +−(˜χ − ˜χc)yc − a∆N∆N + a3y3 +c +� +. +(S23) +The above equation describes a cell that becomes polarized (yc ̸= 0) when ˜χ is sufficiently large. Furthermore, a∆N +tells us how nonzero ∆N affects the evolution of yc. In general, ˜χc, a∆N, and a3 all depend on L and N. a3 > 0 such +that yc remains finite. +To focus on the physics that are most relevant to the transitions between different motility behaviors, we neglect the +N-dependencies in ˜χc and a∆N as they do not change the symmetry properties of the evolution equation of yc. This + +S8 +approximation is expected to be suitable for slow-moving cells. Furthermore, the L-dependencies of all coefficients in +our simplified model can be neglected by considering the large-K regime such that L → L0. In this regime, +Vcell = +˜ξ1/2 +(1 − β +2 ) +� +1 + +˜ξ1/2 +2 +cosh(˜ξ1/2L0)+1 +sinh(˜ξ1/2L0) N +� +�� +1 +2 +cosh(˜ξ1/2L0) + 1 +sinh(˜ξ1/2L0) +vp +� +∆N − +� +˜χcosh(˜ξ1/2L0/2) +sinh(˜ξ1/2L0) +� +yc +� +≡ +1 +1 − β/2(λν1vp∆N − ˜χλν2yc). +(S24) +Here λν1vp and λν2 are N-dependent parameters. +Many interesting features of the system described by Eqs. (S22)(S23) can be studied analytically. First, the steady- +state solutions include the rest-state solution +∆N = yc = 0, N = 2kon +k(0) +off +≡ N0, +(S25) +and solutions for a cell moving in the ± x-direction with a constant velocity +yc = ∓ +� +p4 + p2 +1p2 − +� +(p4 + p2 +1p2)2 − 4p2 +1p4(p2 − p1p3N0) +2p2 +1p4 +, +∆N = − +p1N0 +1 + (p1yc)2 yc, +N = +N0 +1 − (p1yc)2 , +(S26) +where p1 = k(1) +off /k(0) +off , p2 = Γ(˜χ− ˜χc)/k(0) +off , p3 = Γa∆N/k(0) +off , and p4 = Γa3/k(0) +off . Further checking the linear stability of +the rest state shows that the transition from the rest state to the state with constant velocity is a pitchfork bifurcation. +On the other hand, the transition from the rest state to the periodic back-and-forth movement is a Hopf bifurcation: +Pitchfork bifurcation (rest/constant-velocity transition) happens when +˜χ = ˜χc + 2a∆N +konk(1) +off +(k(0) +off )2 +(S27) +and +Γ(˜χ − ˜χc) − k(0) +off < 0. +(S28) +Hopf bifurcation (rest/back-and-forth-motion transition) occurs when +˜χ = ˜χc + k(0) +off /Γ +(S29) +and +˜χ − +� +˜χc + 2a∆N +konk(1) +off +(k(0) +off )2 +� +< 0. +(S30) +The phase diagram for the motility behavior predicted by this phenomenological model is shown in Fig. S3. It is +qualitatively similar to the phase diagrams of the active gel model. The differences are likely due to the approximations +we made when constructing the simplified model. For example, assuming a constant cell length and assuming that +the dynamics of yc are independent of N and L are likely to have some effects on the detailed shape of the phase +boundaries. + +S9 +� +� +� +� +� +� +� +�� +� +� +� +� +� +� +��� +��� +��� +��� +��� +��� +��� +��� +��� +� +��� +��� +�� +��� +��� +���� +���������� +�������������� +���������� +FIG. S3. Phase diagram for the motility behavior predicted by the simplified model. The following motility patterns are found: +a cell at rest (red squares), a cell moving at constant velocity (green diamonds), a cell performs stick-slip movement (orange +triangles), a cell performs back-and-forth movement with stick-slip (at k(1) +off /k(0) +off slightly greater than those orange triangles so +that we cannot show), and a cell performs periodic back-and-forth movement (blue circles). The boundary between the rest and +constant velocity movement is ˜χ = ˜χc + 2a∆N +konk(1) +off +� +k(0) +off +�2 . The boundary between the rest and periodic back-and-forth movement +states is ˜χ = ˜χc + k(0) +off /Γ. +[1] T. Luo, K. Mohan, V. Srivastava, Y. Ren, P.A. Iglesias, and D.N. Robinson, Understanding the cooperative interaction +between myosin II and actin cross-linkers mediated by actin filaments during mechanosensation, Biophys. J., 102, 238 +(2012). +[2] E.L. Barnhart, K-C Lee, K. Keren, A. Mogilner, and J.A. Theriot, An adhesion-dependent switch between mechanisms that +determine motile cell shape, PLoS Biol., 9, e1001059 (2011). +[3] Y-L Wang, Reorganization of actin filament bundles in living fibroblasts, J. Cell Biol., 99, 1478 (1984). +[4] P. Recho and L. Truskinovsky, “Cell locomotion in one dimension,” in Physical Models of Cell Motility, pp. 135–197, +Springer, 2016. + diff --git a/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..97d916306cfa45ee62aafe87a67c5c8bfae22121 --- /dev/null +++ b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt @@ -0,0 +1,559 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf,len=558 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='01463v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='soft] 4 Jan 2023 Mechanosensitive bonds induced complex cell motility patterns Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 1Department of Physics, National Central University, Jhongli 32001, Taiwan 2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan (Dated: January 5, 2023) The one-dimensional crawling movement of a cell is considered in this theoretical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our active gel model shows that for a cell with weakly mechanosensitive adhesion complexes, as myosin contrac- tility increases, a cell starts to move at a constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the mechanosensitivity of the adhesion complexes increases, a cell can exhibit stick-slip motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, a cell with highly mechanosensitive adhesion complexes exhibits periodic back-and-forth migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A simplified model which assumes that the cell crawling dynamics are controlled by the evolution of the myosin density dipole and the asymmetry of adhesion complex distribution captures the motility behaviors of crawling cells qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It suggests that the complex cell crawling behaviors observed in the experiments could result from the interplay between the distribution of contractile force and mechanosensitive bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' PACS numbers: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='Jj,87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='Uv Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– The crawling motion of eukaryotic cells is ubiquitous in biology as it plays important roles in processes such as embryogenesis, wound healing, cancer metastasis, and immunology [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Common if not uni- versal features of a crawling cell include myosin motors distributed mainly behind the center, dominant actin polymerization in the leading edge, and higher density of adhesion complexes in the leading region [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Such po- larized molecular distribution enables protrusion in the leading edge due to actin polymerization, treadmilling of actomyosin cytoskeleton due to contractility, and traction force pulling the cell body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These features, therefore, are included in many theoretical models for crawling cells [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Interestingly, besides non-motile resting and steady- moving behaviors, cells crawling along a one-dimensional track either on a substrate or in a three-dimensional en- vironment also exhibit moving patterns that are non- stationary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, stick-slip crawling mo- tion due to slip between integrin and the extracellular matrix in focal adhesions under the contractility provided by myosin II has been observed in human osteosarcoma cells [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Periodic back-and-forth migration has been ob- served in crawling zyxin-depleted cells in a collagen ma- trix [5] and dendritic cells crawling along microfabricated channels [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Several theoretical models have been proposed to ex- plain some of these deterministic complex moving pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A model that includes the mechanochemical cou- pling of actin promotor dynamics and actin polymer- ization to myosin kinetics was shown to produce peri- odic back-and-forth migration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, a purely mechanical model emphasizing the interplay be- tween mechanosensitive bonds and membrane tension ex- hibited stick-slip motion even for slip bonds [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Inter- estingly, it has also been shown that stick-slip can re- sult from the interplay between mechanosensitive bonds, contractility, and a force that tends to restore a cell’s preferred length [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this letter, we present a theoretical study to show that the coupling between mechanosensitive adhesion complexes and myosin contractility is sufficient to gener- ate deterministic complex cell crawling behaviors, includ- ing stick-slip, periodic back-and-forth movements, and other complex moving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We first construct an active gel model with mechanosensitive adhesion com- plexes and show that the distribution of myosin motors and adhesion complexes computed from our model agree with experimentally observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' By exploring the motility behavior with different strengths of contractil- ity and mechanosensitivity, we show that this model can lead to complex motility behaviors other than rest and constant-velocity moving states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Among these complex motility patterns, unidirectional stick-slip motion and pe- riodic back-and-forth movement are the most common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' When the adhesion complexes are less mechanosensitive, as myosin contractility increases, the cell performs con- stant velocity motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, for cells with highly mechanosensitive adhesion complexes, as myosin contractility becomes sufficiently strong, periodic back- and-forth crawling motion can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, stick-slip and other complex motility patterns can be ob- served by increasing the mechanosensitivity of the adhe- sion complexes for a cell moving at constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' To understand the physical mechanisms that produce these complex motility patterns, a simplified model in- spired by the active gel model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The sim- plified model assumes that the dynamics of a sufficiently slow-moving cell are dominated by the dipole moment of myosin density and the difference in the total num- ber of adhesion complexes near the two cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Re- markably, the motility behavior predicted by this simple model agrees qualitatively with the motility behaviors predicted by the active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These results sug- gest that, in general, diverse complex motility behaviors can result from the interplay between mechanosensitive adhesions and the dynamical organization of contractile 2 myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Model summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– To focus on the contribution of cell mechanics to motility behaviors, chemical signaling is not included in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cytoplasm of the cell is mod- eled as an active gel [10, 11] enclosed by the cell mem- brane, the adhesion complexes are treated as reversible bonds with specific binding-unbinding rates, and actin polymerization is assumed to happen only at the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The forces acting on the cell include the stress in the cytoskeleton, the drag force from the substrate, and the force due to the adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our model only considers one spatial direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', the cell’s moving direction, and the stress in the cytoplasm obeys the constitutive equation σ = η ∂v ∂x + χc, (1) where η is the effective one-dimensional viscosity of the cytoplasm, v is the flow field, χ is the strength of contrac- tility provided by myosin motors (χ > 0), and c is the concentration of myosin attached to the actin network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For simplicity, compressibility is not included [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Thus pressure does not appear in the constitutive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The force exerted by the substrate is Fdrag = −αnbv − ξv, (2) The first term on the right hand is the drag provided by the adhesion complexes [12], α is a constant that char- acterizes the resistance of the adhesion complexes to cell movement, and nb is the number density of adhesion com- plexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The second term comes from the viscous drag of the fluid between the cell and the substrate, and ξ is the drag coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Putting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (1)(2) together, the result- ing force balance equation, ∂xσ + Fdrag = 0, takes the following form η ∂2v ∂x2 − (αnb + ξ)v = −χ ∂c ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (3) Myosin motors attached to actin filaments move with the cytoplasm, while those detached from actin filaments diffuse freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The attachment/detachment of motors is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On long-time scales, the density of the mo- tors can be effectively described by an advection-diffusion equation [13] ∂c ∂t = D ∂2c ∂x2 − ∂(cv) ∂x , (4) where D is the effective diffusion coefficient of myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Adhesion complexes providing anchorage to the extra- cellular matrix are also physically coupled to the con- tractile cytoplasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As a result, they are pulled when the cytoplasm moves [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Once an adhesion complex is formed, the adhesion site does not move, but the disso- ciation rate of the adhesion complex is affected by the motion of the cytoskeleton because the bond is stretched or compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In our model, the evolution of the density of adhesion complexes is assumed to obey ∂nb ∂t = −k0 e−k1∂xvnb + kon, (5) where kon is the binding rate, k0 is the unbinding rate at ∂xv = 0, and k1 tells us how unbinding rate is affected by the cytoplasmic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our model assumes that when the strain rate is dilating, giving more space for the adhesion complexes, the unbinding rate decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Actin polymerization at the cell ends depends on the distribution of actin activators [15][16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the presence of environmental cues, a gradient of actin activator con- centration within the cell is established, and actin poly- merization is polarized due to this concentration gradi- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the absence of such external influence, the cell can nevertheless polarize itself by spontaneous symmetry breaking, and the net actin polymerization rate becomes asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In our model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' we consider a homogeneous environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and the net actin polymerization rate v± p at the ± end of the cell is assumed to be v± p = 2 e−v(1) p (L−L0) 1 + exp[∓ dl± dt /v(2) p ] v(0) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (6) where v+ p (v− p ) is the net rate of extension due to actin polymerization at the cell end located at x = l+(l−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' v(0) p comes from the base polymerization rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' v(1) p in the exponent of the numerator comes from the effect of free energy cost for polymerization when the cell length is different from its natural length (L0 is the natural length of the cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and L = l+ − l−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and the term with v(2) p makes the net polymerization rate in a moving cell at both ends different,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' with more polymerization events in the leading end than the trailing end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It will become clear that the qualitative results of cell motility behavior do not depend on the specific form we assumed for the dissociation rate of the adhesion complexes and v± p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The evolution of cell-end positions is determined by the velocity of cytoplasm and actin polymerization, dl± dt = v± ± v± p , (7) where v+ (v−) is the velocity of cytoplasm at the + (−) end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Experimentally it has been shown that a cell tends to restore its length L to a preferred magnitude L0 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We model this effect by the following force balance condition at cell ends σ± = � χc + η ∂v ∂x � l± = −γ(L − L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (8) Here γ is a constant associated with the restoring force that brings the cell length L to L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3 Because no myosin motors can leave or enter the cell, the total flux of myosin motors across a cell end should vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to [cv]l± − [c]l± dl± dt − D � ∂c ∂x � l± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (9) The first two terms on the left-hand side are the advective flux relative to the moving cell end, and the last term is the diffusive flux at the cell end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is convenient to introduce effective drag coefficient ξeff = ξ + αkon/k0 and choose l0 = � η/ξeff as the unit length, t0 = η/(ξeffD) as the unit time, σ0 = ξeffD as the unit stress, n0 = ξ/α as the unit density for adhe- sion complexes, and c0 = M/ � η/ξeff as the unit myosin concentration, where M is the total number of myosin motors in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore the dimensionless drag co- efficient ˜ξ = ξ/(ξ + αkon/k0), contractility ˜χ = c0χ/σ0, and cell elastic constant K = γl0/σ0 are used in the fol- lowing discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Simulation of motility behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– From the point of view of nonequilibrium thermodynamics, the drag force between the cell and the substrate, the viscous force in the cytoplasm, and the diffusion of myosin are passive processes against cell movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, ac- tive processes such as actin polymerization and myosin contractility drive the movement of the cell, and the binding/unbinding dynamics of the adhesion complexes modulate cell movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the myosin motors pro- vide contractility against viscous and substrate drag, the contractility-induced cytoplasmic flow drifts the motors to aggregate and also affects the distribution of adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Once the cell is in motion, the feedback in the actin polymerization rate further enhances cell polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The balance between these processes determines the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In general, there is no analytical so- lution when all these effects are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore we numerically integrate the equations of motion by a finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The details of our numerical methods and our choice of parameters are discussed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 1 shows the motility behaviors for a cell with parameters chosen to be compatible with typical cells [20] and a range of adhesion complex mechanosensitivity and contractility strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following motility behaviors are found: rest, moving at a constant velocity, unidirec- tional stick-slip movement, back-and-forth motion with stick-slip, and periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For a cell with weakly mechanosensitive adhesion complexes, as contractility increases, a cell at rest starts to move at con- stant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the adhesion complexes become more mechanosensitive, a moving cell shows other complex motility behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, stick-slip motion and (at high contractivity) back-and-forth motion with stick- slip, and finally, the cell performs periodic back-and- forth motion when the adhesion complexes are highly mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Another motility phase diagram in the Supplement Materials [20] shows that, within our model, a cell with a high actin polymerization rate can exhibit other complex motility behaviors between stick-slip and periodic back-and-forth movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' �� �� �� �� �� �� � � ��� ��� ��� ��� � � (a) � � � � � � �� � (b) � � �� � � �� �� � (c) � �� �� � ��� ��� ���� � (d) � � �� � ��� ��� ��� � (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (a) Motility phase diagram for a cell with dimension- less parameters K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, and v(2) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Rest state (squares), constant- velocity motion (diamonds), stick-slip movement (triangles), back-and-forth with stick-slip motion (empty circle), and pe- riodic back-and-forth motion (filled circles) are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (b) Trajectories of the cell ends for ˜χ = 18, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='05, the cell performs constant velocity motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (c) Trajectories of the cell ends for ˜χ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, the cell performs stick-slip mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (d) Trajectories of the cell ends for ˜χ = 19, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='15, the cell performs complex motility pattern which is periodic back-and-forth with stick-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (e) Trajectories of the cell ends for ˜χ = 18, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, the cell performs periodic back-and- forth motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 2 shows that when the cell is at rest, myosin motor distribution is symmetric around the center of the cell, and the number of adhesion complexes near both cell ends is the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' on the other hand, for a cell mov- ing at constant velocity, myosin motors aggregate close to the trailing end and adhesion complexes are mainly close to the leading end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The distribution of myosin mo- tors and adhesion complexes for a cell undergoes stick- slip, and periodic back-and-forth movements are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2 of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is clear that whenever the cell has a definite moving direction, myosin motors aggregate in a regime behind the center of the cell, and more adhesion complexes form near the leading end than the trailing end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Indeed, the adhesion complex binding/unbinding rates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (5) and net actin polymerization at the cell ends Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (6) in our model lead to reasonable molecular distributions in a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Reduction to the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– To obtain an in- 4 ���� ��� ��� � ��� ��� ��� � � � (a) ���� ��� ��� � ��� ��� ��� (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Distribution of myosin motors and adhesion com- plexes for a cell (a) at rest and (b) undergoes constant velocity motion towards +x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The parameters are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1 and (a) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, ˜χ = 16, (b) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, ˜χ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' tuitive physical picture of the complex motility behav- iors, especially the transitions from the rest state to the constant-velocity movement and periodic back-and-forth movement, we construct a simplified model from the ac- tive gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, we consider a limiting situation in which adhesion complexes only appear in a small region close to the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, it is convenient to in- troduce Nf(Nb), the total number of adhesion complexes close to l+(l−), and N = Nf + Nb, ∆N = Nf − Nb to describe the distribution of adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We also introduce yc, the dipole moment of myosin motors den- sity relative to the center of the cell [20], to characterize the spatial distribution of myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The net actin polymerization velocity at the cell ends v± p ≡ vp±∆vp/2, where ∆vp = v+ p − v− p ∝ Vcell in the limit of small cell velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore we write v± p = vp ± βVcell/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, straightforward calculation shows that the veloc- ity of the cell is [20] Vcell ≈ 1 1 − β/2(λν1vp∆N − ˜χλν2yc), (10) where λν1 and λν2 are positivet coefficients that depend on N and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Note that, from the definition, β/2 cannot be greater than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore a cell with ∆N > 0 and yc < 0 has positive Vcell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is in agreement with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We further simplify the analysis by considering the limit of large K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', L ≈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following equations are constructed to describe the dynamics of N, ∆N, and yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the evolution equations for N and ∆N are dN dt = 2kon − k(0) off N − k(1) off yc ∆N, d∆N dt = −k(0) off ∆N − k(1) off Nyc, (11) where kon, k(0) off , and k(1) off are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Next, in the spirit of Landau-type approximation, the evolution of yc is assumed to obey the following equation, dyc dt = −Γ � −(˜χ − ˜χc)yc − a∆N∆N + a3y3 c � , (12) where ˜χc and a∆N are treated as N-independent con- stants for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The simple model equations (11)(12) have a solution with constant N and ∆N = yc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This corresponds to a cell at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Solutions with nonzero constant ∆N and yc correspond to a cell moving at a constant veloc- ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' solutions with time-periodic ∆N and yc are stick-slip (periodic back-and-forth) movement if the time-average of ∆N and yc are nonzero (zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The linear stability analysis of the rest state shows that as the contractil- ity increases, a cell at rest starts to move as the system undergoes a bifurcation, the moving state is the constant- velocity state when k(1) off is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', the adhesion com- plexes are less mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' When k(1) off is sufficiently large, the rest-to-moving transition leads to a periodic back-and-forth moving state [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3 [20], the model Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (11) (12) exhibit a motility phase diagram qualitatively the same as the numerical solutions of our active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The minor differences come from those simplifications made when constructing the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, from the simplified model, it is easy to see how the sym- metry properties and the couplings of the key driving variables lead to the observed cell motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For exam- ple, yc(t) and ∆N(t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(a) for a cell performing periodic back-and-forth movement suggests the following physical picture about how the coupling terms in the sim- plified model lead to this motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (11), a small ∆N tend to increase when yc is sufficiently nega- tive, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (12) states that when a∆N > 0, yc tends to move toward the center of the cell when the magnitude of ∆N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The result is that at sufficiently large k(1) off , the number of adhesion complexes in the lead- ing end of a moving cell increases sufficiently fast such that at some point, the myosins are pulled to the other half of the cell, reversing the sign of yc, then reversing the sign of ∆N, and eventually the direction of cell motion is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is how rest/constant-velocity transition becomes rest/back-and-forth transition as the adhesion complexes are more mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In between the constant velocity and periodic back-and-forth movement, complex motility patterns with stick-slip can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(b), yc(t) and ∆N(t) in a cell that un- dergoes stick-slip movement have nonzero time-average values, and they oscillate with similar phase-relations as a cell undergoes periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is because the mechanosensitivity of the adhesion com- plexes is sufficiently strong to induce an oscillation of ∆N and yc, but not sufficiently strong to change their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 3(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(d) show that yc(t) and ∆N(t) (defined as the difference of the total number of adhe- sion complexes in the leading and trailing halves of the cell) in the numerical simulations of the active gel model behave similarly, suggesting that the physical picture ob- tained from studying the simplified model can be applied 5 to more detailed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' � �� �� � ���� ��� ��� � � �� (a) � �� �� � ���� ��� ��� (b) ��� ��� ��� � ���� ��� ��� (c) � � �� � ���� ��� ��� (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (a)(b): yc(t) and ∆N(t) in the simplified model with kon/k(0) off = 1, a∆N = 1, and a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (a), k(1) off /k(0) off = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='6, Γ(˜χ− ˜χc)/k(0) off = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (b), k(1) off /k(0) off = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='62, Γ(˜χ− ˜χc)/k(0) off = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (c)(d): yc(t) and ∆N(t) in the active gel model with K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (c), k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='55, ˜χ = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (d), k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, ˜χ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cell in (a)(c) performs oscillatory back-and-forth movement, and the cell in (b)(d) performs stick-slip movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– Within our active gel model, a cell with highly mechanosensitive adhesion complexes can exhibit periodic back-and-forth movement similar to what was observed in zyxin-depleted cells in a collagen matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Since zyxin proteins act as mechanosensors in mature adhesion complexes [21], our study suggests that the dif- ference in the mechanosensitivity of the adhesion com- plexes in zyxin-depleted and wild-type cells could be the origin of the periodic back-and-forth movement observed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Future experiments can be designed to examine this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The physical picture suggested by our simplified model also implies the possibility that, in general, when some of the simplifications are lifted, more complex one- dimensional cell motility behaviors can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is indeed the case, as we explore the behavior of our ac- tive gel model for a broader range of actin polymerization rates, complex trajectories which come from further bi- furcations are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1 in [20] and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Further study of the physical mechanisms for these behaviors will be our future work [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, although the physical mechanisms for symmetry-breaking transitions, such as rest/periodic back-and-forth transition and rest/constant velocity transition, can be understood from the dynamics of yc and ∆N, it is interesting to study how other important physical observables, such as the multipoles of the trac- tion force [23][24], behave in cells with different moving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is also important to check if the basic fea- tures of these physical observables in different moving patterns depend on the details of the binding/unbinding dynamics of adhesion complexes, as it plays a significant role in our understanding of many interesting features of cell motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' � � �� � � � � � (a) � � �� � ��� ��� ��� � (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our active gel model predicts that at high actin polymerization rates further complex motility behaviors, such as (a) zig-zag with stick-slip, and (b) double-period back-and- forth movement can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These trajectorjes are obtained for K = 100, ˜χ = 14, kon = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='0, k0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='0, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (a), v(0) p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (b), v(0) p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The blue curves represent the trajectories of the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Acknowledgments – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Jasnow (University of Pitts- burgh) for stimulating discussions and encouragement in the early stage of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' is supported by the Ministry of Science and Technology, Taiwan (MOST 108-2112-M-008-016 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The authors also acknowledge the support from National Center for Theoretical Sciences, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Bray, Cell Movements, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ingber, Mechanical forces alter zyxin unbind- ing kinetics within focal adhesions of living cells, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Cell Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 207, 187 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Lo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Chen, manuscript in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Tanimoto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Sano, A simple force-motion rela- tion for migrating cells revealed by multipole analysis of traction stress, Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 106, 16 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ohta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Tarama, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' SanoA simple model of cell crawling, Physica D, 318, 3 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='01463v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='soft] 4 Jan 2023 Supplementary material for “ Mechanosensitive bonds induced complex cell motility patterns” Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 1Department of Physics, National Central University, Jhongli 32001, Taiwan 2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan (Dated: January 5, 2023) S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' DIMENSIONLESS EQUATIONS We introduce effective drag coefficient ξeff = ξ + αkon/k0 and choose l0 = � η/ξeff as the unit length, t0 = η/(ξeffD) as the unit time, σ0 = ξeffD as the unit stress, n0 = ξ/α (notice that, here ξ, not ξeff is used) as the unit density for adhesion complexes, and c0 = M/ � η/ξeff as the unit myosin concentration, where M is the total number of myosin motors in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the dimensionless form, the momentum equation is ∂2v ∂x2 − ˜ξ(1 + nb)v = −˜χ ∂c ∂x, (S1) myosin density evolution obeys ∂c ∂t = ∂2c ∂x2 − ∂(cv) ∂x , (S2) evolution of the density of the adhesion complexes is ∂nb ∂t = −¯k0e−¯k1∂xvnb + ¯kon (S3) and the positions of the cell ends obey dl± dt = [v]l± ± v± p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S4) The boundary condition for myosin current is � c(v − dl dt) − ∂c ∂x � l± = 0, (S5) and the stress continuity at the cell ends leads to σ± = −K (L − L0) = �∂v ∂x � l± + ˜χcl±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S6) Here all variables and x, t are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The definitions of the parameters and important physical quantities are listed in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless parameters in our model are listed in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' NUMERICAL METHOD AND CHOICE OF PARAMETERS In our numerical scheme, each iteration updates all dynamical variables by integrating the evolution equations over a small time interval ∆t with a finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, [v]l± and v± p from the previous iteration were substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S4) to obtain the new positions of the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The densities of the adhesion complexes and myosin motors are updated from the flow field of the previous iteration by integrating the evolution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S3) of nb and the myosin advection-diffusion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The force balance Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S1) with the updated bond density and myosin concentration is then solved to obtain the new flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The numerics were carried out by dividing the cell into Nx = 100 segments and approximating the spatial derivatives by the finite difference method with the size of a time step ∆t = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The typical material parameters are D ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='025 µm2/s [1], l0 ∼ 10 µm and unit time t0 ∼ 103 s [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' k0 is the rate of dissociation of mature focal adhesion and S2 Physical meaning Symbol effective drag coefficient ξeff = ξ + αkon/k0 unit length l0 = � η/ξeff unit time t0 = η/ξeffD unit stress σ0 = ξeffD unit myosin motors concentration c0 = M √ η/ξeff unit density of cell-substrate bonds n0 = ξ/α TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Definitions of physical parameters and characteristic quantities in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Physical meaning Symbol unbinding rate ¯k0 = t0k0 coefficient for strain-rate-dependent unbinding ¯k1 = k1/t0 binding rate ¯kon = kont0/n0 base actin polymerization speed ¯v(0) p = v(0) p l0/t0 coefficient for stress-dependent actin polymerization ¯v(1) p = v(1) p l0 coefficient for cell polarization effect on actin polymerization ¯v(2) p = v(2) p t0/l0 contractility ˜χ = c0χ/σ0 cell elastic constant K = γl0/σ0 drag coefficient ˜ξ = ξ/(ξ + αkon/k0) TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Definitions of dimensionless parameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' its typical value ∼ 1/min [3], and we chose ¯k0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ¯kon is chosen to be 6 for most simulations and the dimensionless density of bonds in the absence of mechanosensitivity is of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless natural length of the cell L0 = 1 for a typical cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This means that in the absence of adhesion complexes, drag and viscous forces are of the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless total number of myosin motors in the cell c0L0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' SIMULATION RESULTS The motility phase diagrams in the plane spanned by ˜χ and k1 are presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Here Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1 shows the phase diagram in the plane spanned by v(0) p and k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' From this figure we can see how the actin polymerization rate affects the motility behavior of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is clear that the moving state of a cell with small k1 is that with a constant velocity, while the moving state of a cell with large k1 is periodic back-and-forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is in agreement with the phase diagrams in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, there are several complex motility patterns between these two states, including stick-slip motion and behaviors that can be seen as combinations of back-and-forth and stick- slip movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The trajectories for stick-slip movement and periodic back-and-forth movement with stick-slip are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1(c)(d) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The trajectories for zig-zag movement with stick-slip and double-period back-and-forth motion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4(a)(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ��� ��� ��� ��� � ��� � ��� ��� ��� ��� ��� � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Phase diagram for the motility behavior predicted by the reduced model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, ˜χ = 14, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cells show the following motility behaviors: constant velocity motion (green diamonds), periodic back-and-forth movement (blue circles), stick-slip movement (orange triangles), periodic back-and-forth movement with stick- slip (empty gray circles), zig-zag movement with stick-slip (empty purple diamonds), and double-period back-and-forth motion (brown squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure S2 shows the distribution of adhesion complexes and myosin motors for a cell that undergoes stick-slip movement and periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Similar to the distributions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 2 of the main text, for a moving cell, myosin motors are always located relatively close to the trailing end, and the density of adhesion complexes is always higher in a region close to the leading end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S4 � �� �� � � � �� � ��� ��� ��� � (a) � �� �� � � � �� � � � � �� � � (b) ��� ��� ��� � � � �� � ��� ��� ��� � (c) ��� ��� ��� � � � �� � � � � �� � � (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Distribution of adhesion complexes and myosin motors for K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 2, and (a)(b) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='12, ˜χ = 18, (c)(d) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, ˜χ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S5 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' THE SIMPLIFIED MODEL Since the adhesion complexes are concentrated close to the cell ends, to simplify the analysis, we assume nb(x, t) = Nfδ(x − xf) + Nbδ(x − xb), (S7) where xf = l+ − ǫ, xb = l− + ǫ, and ǫ is a very small length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Stress field and flow field Because there are no adhesion complexes in xb < x < xf, the dimensionless momentum equation in this region is ∂xσ = ˜ξv, σ = ∂xv + ˜χc, xb < x < xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S8) Integrating the full momentum equation from l+(−) to xf(b), we find σf = −K(l − l0) − ˜ξNfvf, σb = −K(l − l0) + ˜ξNbvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S9) Here vf = v(xf, t) ≈ dl+/dt − v+ p , and vb = v(xb, t) ≈ dl−/dt + v− p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The solution of stress from these equations is [4] σ(x, t) = σf sinh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � + σb sinh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � + ˜χ˜ξ1/2 � xf xb G(x, x′)c(x′, t)dx′, (S10) where G(x, x′) = sinh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2(xf − xb) � − Θ(x′ − x) sinh � ˜ξ1/2(x′ − x) � , (S11) Θ(x) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to the following expression for the flow field, v(x, t) = 1 ˜ξ1/2 \uf8f1 \uf8f2 \uf8f3σf cosh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � − σb cosh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � + ˜χ � xf xb ∂xG(x, x′)c(x′, t)dx′ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S12) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Myosin concentration Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S12) into the advection-diffusion for myosin concentration, one obtains the following equation which does not have an explicit dependence on the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ∂tc(x, t) = D∂2 xc − 1 ˜ξ1/2 ∂x \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0σf cosh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � − σb cosh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � \uf8f9 \uf8fb c(x, t) +˜χ � xf xb c(x, t)∂xG(x, x′)c(x′, t)dx′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S13) S6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Velocity and length of the cell The velocity of the cell Vcell = 1 2 � dl+ dt + dl− dt � is Vcell = 1 2˜ξ1/2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � (σf − σb) + ˜χ 2 � xf xb sinh � ˜ξ1/2(xf − x′) � − sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2L � c(x′, t)dx′ +v+ p − v− p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S14) The evolution of the length of the cell dL dt = dl+ dt − dl− dt obeys dL dt = 1 ˜ξ1/2 cosh � ˜ξ1/2L � − 1 sinh � ˜ξ1/2L � (σf + σb) − ˜χ � xf xb sinh � ˜ξ1/2(xf − x′) � + sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2L � c(x′, t)dx′ +(v+ p + v− p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S15) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Symmetries of the system It is helpful to introduce the following variables N = Nf + Nb, ∆N = Nf − Nb, vp = v+ p + v− p 2 , ∆vp = v+ p − v− p , σS = σf + σb 2 = −K(L − L0) − ˜ξ 2 � N �dL/dt 2 − vp � + ∆N � Vcell − ∆vp 2 �� , σA = σf − σb 2 = − ˜ξ 2 � N � Vcell − ∆vp 2 � + ∆N �dL/dt 2 − vp �� , (S16) and y = x − l+ + l− 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S17) Notice that dL/dt, N, vp, and σS are symmetric under spatial inversion (y → −y), while Vcell, ∆N, ∆vp, and σA are antisymmetric under spatial inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The velocity of the cell Vcell can be expressed in terms of these parameters and variables that have clear parity signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, notice that only the part of c(y) that is anti-symmetric under y → −y contribute to the ˜χ-dependent term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S14), and for a slow-crawling cell this part should be significant only in the small |y| region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Thus by expanding the ˜χ-dependent term of Vcell to the leading order in y, one finds that Vcell = 1 2 \uf8ee \uf8f0∆vp + 2 ˜ξ1/2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � σA − ˜χ˜ξ1/2 cosh � 2˜ξ1/2L � sinh � ˜ξ1/2L � yc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' \uf8f9 \uf8fb , where yc ≡ � L/2 −L/2 y c(y, t) dy, (S18) and “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..” represents terms of higher order in this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The expression for Vcell can be further simplified by taking ∆vp ≈ βVcell (β is independent of Vcell) for a slow crawling cell and substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S16) for σA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, one S7 obtains Vcell = − ˜ξ1/2 1 − β/2 \uf8ee \uf8f01 + ˜ξ1/2 2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � N \uf8f9 \uf8fb −1 × \uf8ee \uf8f01 2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � �1 2 dL dt − vp � ∆N + ˜χ cosh � ˜ξ1/2L/2 � sinh � ˜ξ1/2L � yc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S19) This expression tells us that Vcell is nonzero only when ∆N (asymmetry in the distribution of adhesion complexes) or yc (asymmetry in the distribution of myosin motors) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Similar calculation leads to the following expression for the evolution of the length of the cell, dL dt = 2vp + � 1 + ˜ξ1/2 2 cosh(˜ξ1/2L) − 1 sinh(˜ξ1/2L) N �−1 × \uf8f1 \uf8f2 \uf8f3 ˜ξ−1/2 cosh(˜ξ1/2L) − 1 sinh(˜ξ1/2L) � −2K(L − L0) + ∆N � 1 − β 2 � Vcell � − 2˜χ sinh � ˜ξ1/2L/2 � sinh � ˜ξ1/2L � Ctot \uf8fc \uf8fd \uf8fe , (S20) where Ctot ≡ � L/2 −L/2 c(y, t)dy ≡ 1 (S21) is the total amount of myosin motors in the cell, which is unity in our dimensionless expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Note that all terms on the right-hand side of dL/dt are even under y → −y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The simplified model and bifurcations From the previous analysis, it is clear that symmetry under y → −y plays an important role in Vcell and dL/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Based on these observations, a simplified model is proposed for slow-crawling cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the evolution equations of Nf and Nb are dNf dt = kon − (k(0) off + k(1) off yc)Nf, dNb dt = kon − (k(0) off − k(1) off yc)Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to dN dt = 2kon − k(0) off N − k(1) off yc ∆N, d∆N dt = −k(0) off ∆N − k(1) off Nyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S22) We expect k(1) off > 0 because a cell moving at constant velocity in the +x direction should have yc < 0 and ∆N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Next, the following evolution equation for yc is proposed dyc dt = −Γ � −(˜χ − ˜χc)yc − a∆N∆N + a3y3 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S23) The above equation describes a cell that becomes polarized (yc ̸= 0) when ˜χ is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, a∆N tells us how nonzero ∆N affects the evolution of yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In general, ˜χc, a∆N, and a3 all depend on L and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' a3 > 0 such that yc remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' To focus on the physics that are most relevant to the transitions between different motility behaviors, we neglect the N-dependencies in ˜χc and a∆N as they do not change the symmetry properties of the evolution equation of yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This S8 approximation is expected to be suitable for slow-moving cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, the L-dependencies of all coefficients in our simplified model can be neglected by considering the large-K regime such that L → L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, Vcell = ˜ξ1/2 (1 − β 2 ) � 1 + ˜ξ1/2 2 cosh(˜ξ1/2L0)+1 sinh(˜ξ1/2L0) N � �� 1 2 cosh(˜ξ1/2L0) + 1 sinh(˜ξ1/2L0) vp � ∆N − � ˜χcosh(˜ξ1/2L0/2) sinh(˜ξ1/2L0) � yc � ≡ 1 1 − β/2(λν1vp∆N − ˜χλν2yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S24) Here λν1vp and λν2 are N-dependent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Many interesting features of the system described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S22)(S23) can be studied analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the steady- state solutions include the rest-state solution ∆N = yc = 0, N = 2kon k(0) off ≡ N0, (S25) and solutions for a cell moving in the ± x-direction with a constant velocity yc = ∓ � p4 + p2 1p2 − � (p4 + p2 1p2)2 − 4p2 1p4(p2 − p1p3N0) 2p2 1p4 , ∆N = − p1N0 1 + (p1yc)2 yc, N = N0 1 − (p1yc)2 , (S26) where p1 = k(1) off /k(0) off , p2 = Γ(˜χ− ˜χc)/k(0) off , p3 = Γa∆N/k(0) off , and p4 = Γa3/k(0) off .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Further checking the linear stability of the rest state shows that the transition from the rest state to the state with constant velocity is a pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, the transition from the rest state to the periodic back-and-forth movement is a Hopf bifurcation: Pitchfork bifurcation (rest/constant-velocity transition) happens when ˜χ = ˜χc + 2a∆N konk(1) off (k(0) off )2 (S27) and Γ(˜χ − ˜χc) − k(0) off < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S28) Hopf bifurcation (rest/back-and-forth-motion transition) occurs when ˜χ = ˜χc + k(0) off /Γ (S29) and ˜χ − � ˜χc + 2a∆N konk(1) off (k(0) off )2 � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S30) The phase diagram for the motility behavior predicted by this phenomenological model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is qualitatively similar to the phase diagrams of the active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The differences are likely due to the approximations we made when constructing the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, assuming a constant cell length and assuming that the dynamics of yc are independent of N and L are likely to have some effects on the detailed shape of the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S9 � � � � � � � �� � � � � � � ��� ��� ��� ��� ��� ��� ��� ��� ��� � ��� ��� �� ��� ��� ���� ���������� �������������� ���������� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Phase diagram for the motility behavior predicted by the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following motility patterns are found: a cell at rest (red squares), a cell moving at constant velocity (green diamonds), a cell performs stick-slip movement (orange triangles), a cell performs back-and-forth movement with stick-slip (at k(1) off /k(0) off slightly greater than those orange triangles so that we cannot show), and a cell performs periodic back-and-forth movement (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The boundary between the rest and constant velocity movement is ˜χ = ˜χc + 2a∆N konk(1) off � k(0) off �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The boundary between the rest and periodic back-and-forth movement states is ˜χ = ˜χc + k(0) off /Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Luo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Mohan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Srivastava, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ren, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Iglesias, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Robinson, Understanding the cooperative interaction between myosin II and actin cross-linkers mediated by actin filaments during mechanosensation, Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 102, 238 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='L.' 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Box +15100, FI-00076 Aalto, Finland +Abstract +Ionic control of magnetism gives rise to high magneto-electric coupling efficiencies at low voltages, +which is essential for low-power magnetism-based non-conventional computing technologies. +However, for on-chip applications, magneto-ionic devices typically suffer from slow kinetics, poor +cyclability, impractical liquid architectures or strong ambient effects. As a route to overcoming these +problems, we demonstrate an LiPON-based solid-state ionic supercapacitor with a magnetic +Pt/Co40Fe40B20/Pt thin-film electrode which enables voltage control of a magnetic skyrmion state. +Skyrmion nucleation and annihilation are caused by Li ion accumulation and depletion at the magnetic +interface under an applied voltage. The skyrmion density can be controlled through dc applied fields +or through voltage pulses. The skyrmions are nucleated by single 60-µs voltage pulses and devices are +cycled 750,000 times without loss of electrical performance. Our results demonstrate a simple and +robust approach to ionic control of magnetism in spin-based devices. + + +2 + +Controlling magnetism through applied voltages would allow for the creation of a new class of low- +energy non-conventional computing devices. For technological applications the voltage-induced +changes need to be fast, reversible and have a strong impact on the magnetic system. The ability to +induce large magnetic effects at small voltages has led to an increasing interest in magneto-ionic +approaches [1-3]. Previous works have shown that magnetism can be altered ionically through redox +reactions [4-8], ion intercalation [9-14], or the formation of an electronic double layer at solid-ion liquid +interfaces [15,16]. Devices exploiting magneto-ionics have been shown to be able to control various +magnetic properties including the saturation magnetization [4-12], magnetic anisotropy [4-7,15], and +Dzyaloshinskii-Moriya interaction (DMI) [17,18]. The main technological bottleneck for ionically +controlled magnetism is the need to apply voltages for extended periods to create sizable effects at +room temperature. +Here we take a different approach to ionic control of magnetism by creating a solid-state +supercapacitor [19-21]. The large capacitance of supercapacitors is generated by ion adsorption on +the electrodes leading to the creation of an electrical double layer, surface redox reactions or ion +intercalation. Using a Li-enriched LiPON layer as the ion conduction layer we demonstrate fast, +reversible, and durable voltage control of magnetism. In particular, we control magnetic skyrmions — +topologically distinct quasiparticles of interest in magnetic data storage and non-conventional +computing devices [22-27]. Previously, voltage control of skyrmions has been shown through +interfacial charge modulation [17,28-33], strain transfer from piezoelectrics [34,35], and locally-applied +electric fields [36]. By integrating a skyrmion-hosting magnetic thin-film structure with a supercapacitor +we demonstrate nucleation and annihilation of skyrmions through sub-100 s voltage pulses, a +continuously controllable skyrmion density and the ability to extensively cycle the magnetic state +without degradation. The significant improvement in the ability to control skyrmions through applied +voltages demonstrated here is an important step towards technological applications, particularly +neuromorphic computing [24-27]. +As shown in Figure 1a, the magnetron-sputtered structure consists of an ionically conducting, 100 nm +thick Li-enriched lithium phosphorous oxynitride (LiPON) layer sandwiched between a 1 nm SiN/4 +nm Pt top gate electrode and a 2 nm Ta/4 nm Pt/0.9 nm CoFeB (40:40:20)/0.2 nm Pt bottom +electrode. This structure is patterned into 500 µm x 500 µm crossbar junctions shown in Figure 1b (see +Methods in SI). Magnetic hysteresis loops of one junction recorded under an applied bias voltage using +polar magneto-optical Kerr effect (MOKE) microscopy are shown in Figure 1c, with corresponding +MOKE images depicted in Figure 1d. At negative voltage the positively charged Li ions move away +from the Pt/CoFeB/Pt electrode leading to a square hysteresis loop and a fully saturated film +magnetization at 0 mT and +0.7 mT. The zero-voltage state shows a slanted hysteresis loop with + +3 + +magnetic stripe domains at 0 mT and a sparse skyrmion state at +0.7 mT. The application of a positive +voltage slants the hysteresis loop further and it increases the density of the stripe domains (0 mT) and +skyrmions +0.7 mT). At positive voltage the Li ions move towards the magnetic layer. + +Figure 1. Materials system and voltage control of magnetism. (a) Schematic of the magneto-ionic +heterostructure. Voltage is applied to the top gate electrode with the bottom electrode grounded. (b) +Image of the crossbar sample. Both top and bottom electrodes are 500 m wide. (c) Polar MOKE +hysteresis loops recorded under 0 V, +2 V and −2 V bias voltage. (d) MOKE microscopy images for +the same bias voltages under 0 mT and +0.7 mT perpendicular field. The scalebar indicates 10 m. +To investigate control of the skyrmion density, the voltage was stepped from –1.0 V to +2.0 V and +back to –1.0 V at 0.1 V intervals (Figure 2). MOKE microscopy images of the CoFeB film at selected +gate voltages recorded in +0.7 mT perpendicular field are shown in Figure 2a. Starting from a saturated +magnetization state at –1.0 V, inverse stripe domains form at +0.7 V, followed by the nucleation of +sparse skyrmions at +0.8 V. The density of the mixed stripe and skyrmion state increases with voltage +before morphing into a dense skyrmion lattice at +1.6 V. Hereafter, the skyrmion density increases +further up to +2.0 V. Sweeping the voltage in the opposite direction reduces the skyrmion density +gradually until all skyrmions are annihilated at –0.6 V. Figure 2b summarizes the skyrmion density +during the voltage sweep. The hysteresis demonstrates the existence of a memory effect in the device, +enabling access to a continuous range of skyrmion states, which is a requirement for neuromorphic +devices. Besides control over skyrmion nucleation and annihilation, the gate voltage also tunes the +skyrmion size (Figure 2c). The first skyrmions appearing at +0.8 V are large (800 nm) but their size +decreases continuously up to +2.0 V (600 nm) (see Methods in SI). Sweeping the voltage in the +negative direction only has a small effect on the skyrmion size. Full reversibility between a reproducible +skyrmion state and no skyrmions upon repeated voltage cycling is demonstrated in Figure 2d. + +a +O Li+ +b +d +-2 V +oV ++2 V +Pt4nm +SiNnm +0mT +MOKE signal (a.u.) +LiPON100nm +Pt20.2nm +CoFeB(40:40:20)0.9nm ++0.7 mT +Pt4nm +oV ++2 V +Ta2nm +-2 V +-2 +0 +2 +4 +Perpendicular magnetic field (mT4 + + +Figure 2. Voltage dependence of the skyrmion state. (a) Polar MOKE microscopy images recorded +while sweeping the applied voltage from –1.0 V to +2.0 V and back. The perpendicular magnetic field +is +0.7 mT. The scalebar corresponds to 10 µm. (b) Skyrmion density during the voltage sweep. (c) +Average skyrmion radius during the voltage sweep. The error bars show the standard error of the mean +size. (d) Reversible toggling of the skyrmion density by switching the voltage between –1.0 V and +1.5 +V. The voltage is applied for 1 min before data collection. +For applications, devices are likely to be controlled by voltage pulses, where the response to both the +application and removal of a voltage is relevant to the device operation. To investigate the decay of +the skyrmion state over time at zero-bias voltage, we applied +2.0 V for 1 min to a crossbar junction +followed by setting the voltage to zero. The skyrmion density as a function of time is shown in Figure +3a and MOKE microscopy images at different times are depicted in Figure 3b. The decay constant is +found to be approximately 8 min. +To assess the dynamic response of our magneto-ionic device under voltage pulsing, we applied 250 +ms long voltage pulses with magnitudes ranging from +1.7 V to +2.0 V at 500 ms intervals and +monitored the skyrmion density over time (Figure. 3c). The device was reset to a skyrmion-free state +between each series of pulses by applying –2 V for 5 s. MOKE microscopy images of the CoFeB film +taken after 3000 pulses are shown in Figure 3d for four different pulse voltages. Two clear features + +a +-1.0V +OV ++0.7V +2 +3 +5 +-0.7V +-0.5V +10 +9 +8 +7 +6 +b +c +0.8 +900 +Average skyrmion +tate +Skyrmion density +0.6 +radius (nm) +800 +S +D +forward +0.4 +forward +700 +0.2 +backward +10 +600 +0.0 +3 +S +1 +2 +backward +-1 +0 +1 +2 +-1 +0 +1 +2 +p +Voltage (V) +Voltage (V) +Skyrmion density +Voltage (V) +0.4 +0.0 +1 +2 +3 +4 +5 +6 +7 +Cycle5 + +stand out, firstly that the rate of approach to an equilibrium value is much faster at higher applied +voltage and secondly that the equilibrium skyrmion density is much higher at higher applied voltage. +The device shown here was cycled over 50,000 times whilst retaining the voltage control of the +skyrmion state. + +Figure 3. Control of skyrmions by voltage pulse number, pulse amplitude and pulse duration. (a) Time +evolution of the skyrmion density at 0 V after skyrmion nucleation at +2.0 V for 1 min. (b) MOKE +microscopy images recorded during the retention experiment shown in (a). The scalebar indicates 10 +m. (c) Skyrmion density as a function of voltage pulse number using a pulse duration of 250 ms with +a 50% duty cycle. The pulse amplitude is varied from +1.7 V to +2.0 V. (d) MOKE microscopy images +recorded after 3000 pulses for each of the applied voltages. The scalebar indicates 10 m. (e) Skyrmion +density after applying a single voltage pulse and a sequence of 100 voltage pulses to the uniform +magnetization state. The amplitude of the pulse is fixed at +10.0 V and the duration of the pulse is +varied. The 100-pulse sequence has a duty cycle of 10%. (f) MOKE microscopy images recorded after +60 µs, 80 µs, 100 µs for both single- and 100 pulses. The scalebar corresponds to 20 µm. All experiments +used a +0.7 mT perpendicular magnetic field. +We further exploit the dependence of the skyrmion density on voltage to probe the skyrmion +nucleation kinetics at shorter timescales. By applying a single pulse of +10 V, we show that the pulse +width required for skyrmion nucleation can be as low as 60 µs (Figure 3e, blue curve). In these +experiments, the device was reset by applying a –0.8 V gate voltage for 5 s before each voltage pulse +and the skyrmion density was recorded for a few seconds after the pulse. For a sequence of 100 + +a +e +0.25 +0.25 ++2 V +1 min +0.20 +0.20 +0.100 +1 pulse +100 pulses +0.15 +0.15 +decay constant +density +2 ~ 8 min +0.10 +0.010 +0.10 +skyrmion +0.05 +0.05 ++1.7 V ++1.8 V +initial state ++1.9 V +0.001 +0.00 + +2.0 V +S +H +0.00 +0 +60 +120 +180 +0 +1000 +2000 +3000 +0 +40 +80 +120 +160 +200 +b +Time (min) +d +pulse number +f +Pulse duration (μs) +0min +min +2.0AY +30min +80min6 + +identical pulses the number of nucleated skyrmions increase and a pulse duration of just 20 µs is +already sufficient to nucleate skyrmions (Figure 3e, red curve). MOKE microscopy images of the +crossbar junction after a pulse or pulse sequence are presented in Figure 3f for pulse durations between +60 µs and 100 µs. This is to our knowledge the fastest achieved ionically induced response in a voltage- +controlled magneto-ionic system at room temperature. + +Figure 4. Electrical characterization of supercapacitor junctions. (a) Cyclic voltammograms recorded +for different voltage ranges at 10 mV/s scan rate. (b) Junction current at 0 V as a function of voltage +range, derived from cyclic voltammetry with a 10 mV/s scan rate. The voltage range is symmetric +around 0 V. (c) Electrical impedance spectroscopy measurements on a junction using a 100 mV ac +driving voltage with bias voltages of −2 V, 0 V and +2 V. (d) Cyclic voltammograms at 50 mV/s scan +rate. An initial cyclic voltammogram (blue) was followed by cycling the junction between −2 V and ++2 V with a period of 250 ms for 750,000 cycles, after which a second cyclic voltammogram (red) was +recorded. +To understand the functioning of the devices we turn to electrical characterization. Cyclic +voltammograms (CVs) of the supercapacitor structure show a largely rectangular shape with no peaks +indicative of redox processes (Figure 4a). As shown in Figure 4b, for low voltage ranges the current at +0 V is a slowly increasing function of the voltage range with the junction current increasing notably for +larger voltage ranges. This suggests that both electric double layer and electrochemical mechanisms +are present, with the electrochemical mechanism dominating at higher voltages [8]. Given the material +system it is expected that the electrochemical mechanism is intercalation of the Li ions. The +capacitance of the junction is calculated to be 0.18 F at 1 V/s, which is equivalent to a capacity of 72 +F/cm2, showing large storage capability typical of supercapacitors. In Figure 4c electrical impedance + +a +b +25 +30 +EDLi +Intercalation +20 +Current (nA) +Current (nA) +15 +0 +10 +30 +±100 mV +±200 mV +5 +±300 mV +±400 mV +-60 +±500 mV +±1 V +±1.5 V +±2 V +0 +-2 +-1 +0 +1 +2 +0 +1 +2 +3 +4 +c +Voltage (V) +p +Voltage range (V) +10000 +300 +initial +after 750,000 cycles +7500 ++2 V +150 +urrent (nA) +(0) (z)wl- +oV +-2 V +5000 +0 +-150 +2500 +-300 +0 +2500 +5000 +7500 +10000 +-2 +-1 +0 +1 +2 +Re{Z) (2) +Voltage (V)7 + +spectroscopy is shown, giving a steep line at lower frequencies as expected from a capacitance- +dominated device. The supercapacitor system is highly cyclable, with Figure 4d showing the CV (at 50 +mV/s, giving squarer loops than in Figure 4a) before and after cycling 750,000 times with 250 ms pulses +at ±2V. Moreover, our supercapacitor is intrinsically fast with a characteristic charge/discharge time +of 560 s. Figure S1 in the SI provides additional information on the electrical properties of the +supercapacitor structure, including leakage current, open circuit voltage and its electrical impedance +as a function of frequency. +The combination of magnetic and electrical data shows that the accumulation or depletion of Li ions +at the CoFeB/Pt interface causes large changes to the magnetic state at low voltages. Values for the +perpendicular magnetic anisotropy (Ku) and the Dzyaloshinskii-Moriya interaction constant (D), along +with saturation magnetization (Ms) and exchange constant (Aex) were estimated from a thin film sample +with a similar structure (Figure S2 in the SI). Ku and D were found to be 9.96  105 J/m3 and 0.74 +mJ/m2, respectively, which is consistent with the creation of bubble-like magnetic skyrmions in this +sample at around the sizes seen in Figure 1 and Figure 2 (see Figure S3 in the SI). From our previous +work [14], the insertion of Li ions at the CoFeB/Pt interface is expected to reduce the perpendicular +magnetic anisotropy without reducing the magnetization [14], which reduces the energy barrier to +skyrmion nucleation and stabilizes skyrmions relative to the uniform state [28] (see also SI). One +interesting feature of the data in Figure 2c is the reduction in skyrmion size with increasing voltage. If +the system was simply undergoing a reduction in anisotropy, then the skyrmion size is expected to +increase [28,37]. Instead, a decrease in size is seen, which could either indicate that the skyrmions at +lower densities are preferentially found at defect sites [38] or that the DMI is also reduced by the +accumulation of Li ions at the CoFeB/Pt interface [17]. +The time-dependent experiments give insight into the timescales of the phenomena. The decay time +of the skyrmion state in Figure 3a corresponds to an energy barrier of around 0.38 eV, similar to that +expected for the thermally activated hopping motion of Li ions within LiPON [14]. To minimize the +internal electric field within the ion conduction layer there is a thermally activated redistribution of Li +ions within the layer, causing the skyrmions to consequently annihilate over time. This also explains +the results of the pulsed experiments in Figure 3c. Here the positive voltage pulses cause the +accumulation of Li ions at the interface, which decreases the skyrmion nucleation barrier, whilst during +the off state the accumulated ions decay. The concentration of interfacial Li ions increases with the +number of voltage pulses until the decay in the off state balances a further increase during the on state. +For the sub-ms pulses used in Fig. 3e, there is a further effect. Now the barrier for skyrmion nucleation +is lowered rapidly and then increases again as the Li accumulation decays. However, the nucleation of +skyrmions occurs on a timescale longer than the voltage pulses, leading to a peak in skyrmion density + +8 + +around a second after the pulse (Figure S4 in the SI). Therefore, the speed of the devices is also limited +by the thermally activated nucleation of the skyrmions. +Fast and durable voltage control of skyrmions in Li-ion supercapacitor structures, as shown here, offers +attractive pathways to the implementation of neuromorphic devices such as synapse-based neural +networks [24] and reservoir computers [25-27]. Proof-of-concepts demonstrating the suitability of +skyrmion dynamics for neuromorphic computing have thus far utilized magnetic fields or electric +currents to control the skyrmion state. Voltage gating of a skyrmion-hosting magnetic film provides +good scalability and energy efficiency in combination with deterministic accumulation/dissipation, +short-term memory, and nonlinearity. For instance, reversible nucleation and annihilation of skyrmions +through the application of positive and negative voltages (Figure 2) enables the emulation of synaptic +weights changes during potentiation and depression, while the decay of the skyrmion state after voltage +pulsing (Figure 3a,b) provides short-term memory to temporarily store information and trigger outputs +based on the time-dependent history of voltage inputs. Nonlinearity of voltage-driven skyrmion +dynamics, which is another key requirement for neuromorphic processing, is demonstrated in our +supercapacitors by varying the amplitude (Figure 3c,d) and duration (Figure 3e,f) of the voltage pulses. +Finally, we note that the complex interplay between the dynamics of Li ion migration in the solid-state +LiPON electrolyte and the ensuing nonlinear dynamics of skyrmions in the thin magnetic film offers +great flexibility in the design of functional responses and further device optimization. +In summary, we have shown that skyrmions in a Pt/CoFeB/Pt thin-film structure can be created and +annihilated in a fully voltage-controlled all-solid-state device via reversible Li ion migration at room +temperature. The hysteretic behavior of the device with respect to the voltage sweep direction, the +nonlinear effects observed as a function of voltage pulse number and pulse duration, along with the +decay behavior at zero-voltage constitute properties suitable for neuromorphic device architectures. +The use of a supercapacitor enables skyrmion nucleation with single voltage pulses down to 60 s, +combined with extensive cycling of the junctions. Further downscaling of the device from the 100 nm +thick solid-state electrolyte used here may allow access to sub-s functionality. + +Supporting Information +Methods and additional data, including electrical characterization, measurements to extract magnetic +parameters, micromagnetic simulations of the skyrmion energy and fast voltage pulsing experiments +(PDF) +Corresponding Authors + +9 + +rhodri.mansell@aalto.fi +sebastiaan.van.dijken@aalto.fi +Author Contributions +M.A., R.M. and S.v.D. conceived the research project. M.A. grew the supercapacitor heterostructures +and fabricated the crossbar junctions. M.A. conducted the electrical characterization and M.A., L.F. +and R.M. performed the magnetic measurements. J.H. and R.M. conducted the micromagnetic +simulations. R.M. and S.v.D. supervised the work. All authors discussed the results. M.A., R.M. and +S.v.D. wrote the manuscript. +Notes +The authors declare no competing interest. +Acknowledgments +This work was supported by the Academy of Finland (Grant No. 316857). Lithography was performed +at the OtaNano-Micronova Nanofabrication Centre, supported by Aalto University. Computational +resources were provided by the Aalto Science-IT project. + + +10 + +References +[1] +Nichterwitz, M.; Honnali, S.; Kutuzau, M.; Guo, S.; Zehner, J.; Nielsch, K.; Leistner, K. Advances +in magneto-ionic materials and perspectives for their application. APL Mater. 2021, 9, 030903. +[2] +Gu, Y.; Song, C.; Wang, Q.; Hu, W.; Liu, W.; Pan, F.; Zhang, Z. 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Phys. 2018, 1, 31. +[38] Zhou, Y.; Mansell, R.; Valencia, S.; Kronast, F.; van Dijken, S. Temperature dependence of the +Dzyaloshinskii-Moriya interaction in ultrathin films. Phys. Rev. B 2020, 101, 054433. + diff --git a/6tFJT4oBgHgl3EQflyzE/content/tmp_files/load_file.txt b/6tFJT4oBgHgl3EQflyzE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0574cd9b4cc203608c4a5c71fcc3bb8113632fb1 --- /dev/null +++ b/6tFJT4oBgHgl3EQflyzE/content/tmp_files/load_file.txt @@ -0,0 +1,1171 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf,len=1170 +page_content='1 Solid-state lithium-ion supercapacitor for voltage control of skyrmions Maria Ameziane, Joonatan Huhtasalo, Lukáš Flajšman, Rhodri Mansell and Sebastiaan van Dijken NanoSpin, Department of Applied Physics, Aalto University School of Science, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Box 15100, FI-00076 Aalto, Finland Abstract Ionic control of magnetism gives rise to high magneto-electric coupling efficiencies at low voltages, which is essential for low-power magnetism-based non-conventional computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' However, for on-chip applications, magneto-ionic devices typically suffer from slow kinetics, poor cyclability, impractical liquid architectures or strong ambient effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' As a route to overcoming these problems, we demonstrate an LiPON-based solid-state ionic supercapacitor with a magnetic Pt/Co40Fe40B20/Pt thin-film electrode which enables voltage control of a magnetic skyrmion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Skyrmion nucleation and annihilation are caused by Li ion accumulation and depletion at the magnetic interface under an applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The skyrmion density can be controlled through dc applied fields or through voltage pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The skyrmions are nucleated by single 60-µs voltage pulses and devices are cycled 750,000 times without loss of electrical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Our results demonstrate a simple and robust approach to ionic control of magnetism in spin-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 2 Controlling magnetism through applied voltages would allow for the creation of a new class of low- energy non-conventional computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' For technological applications the voltage-induced changes need to be fast, reversible and have a strong impact on the magnetic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The ability to induce large magnetic effects at small voltages has led to an increasing interest in magneto-ionic approaches [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Previous works have shown that magnetism can be altered ionically through redox reactions [4-8], ion intercalation [9-14], or the formation of an electronic double layer at solid-ion liquid interfaces [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Devices exploiting magneto-ionics have been shown to be able to control various magnetic properties including the saturation magnetization [4-12], magnetic anisotropy [4-7,15], and Dzyaloshinskii-Moriya interaction (DMI) [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The main technological bottleneck for ionically controlled magnetism is the need to apply voltages for extended periods to create sizable effects at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Here we take a different approach to ionic control of magnetism by creating a solid-state supercapacitor [19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The large capacitance of supercapacitors is generated by ion adsorption on the electrodes leading to the creation of an electrical double layer, surface redox reactions or ion intercalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Using a Li-enriched LiPON layer as the ion conduction layer we demonstrate fast, reversible, and durable voltage control of magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' In particular, we control magnetic skyrmions — topologically distinct quasiparticles of interest in magnetic data storage and non-conventional computing devices [22-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Previously, voltage control of skyrmions has been shown through interfacial charge modulation [17,28-33], strain transfer from piezoelectrics [34,35], and locally-applied electric fields [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' By integrating a skyrmion-hosting magnetic thin-film structure with a supercapacitor we demonstrate nucleation and annihilation of skyrmions through sub-100 \uf06ds voltage pulses, a continuously controllable skyrmion density and the ability to extensively cycle the magnetic state without degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The significant improvement in the ability to control skyrmions through applied voltages demonstrated here is an important step towards technological applications, particularly neuromorphic computing [24-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' As shown in Figure 1a, the magnetron-sputtered structure consists of an ionically conducting, 100 nm thick Li-enriched lithium phosphorous oxynitride (LiPON) layer sandwiched between a 1 nm SiN/4 nm Pt top gate electrode and a 2 nm Ta/4 nm Pt/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='9 nm CoFeB (40:40:20)/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='2 nm Pt bottom electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' This structure is patterned into 500 µm x 500 µm crossbar junctions shown in Figure 1b (see Methods in SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Magnetic hysteresis loops of one junction recorded under an applied bias voltage using polar magneto-optical Kerr effect (MOKE) microscopy are shown in Figure 1c, with corresponding MOKE images depicted in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' At negative voltage the positively charged Li ions move away from the Pt/CoFeB/Pt electrode leading to a square hysteresis loop and a fully saturated film magnetization at 0 mT and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The zero-voltage state shows a slanted hysteresis loop with 3 magnetic stripe domains at 0 mT and a sparse skyrmion state at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The application of a positive voltage slants the hysteresis loop further and it increases the density of the stripe domains (0 mT) and skyrmions +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' At positive voltage the Li ions move towards the magnetic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Materials system and voltage control of magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (a) Schematic of the magneto-ionic heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Voltage is applied to the top gate electrode with the bottom electrode grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (b) Image of the crossbar sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Both top and bottom electrodes are 500 \uf06dm wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (c) Polar MOKE hysteresis loops recorded under 0 V, +2 V and −2 V bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (d) MOKE microscopy images for the same bias voltages under 0 mT and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT perpendicular field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The scalebar indicates 10 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' To investigate control of the skyrmion density, the voltage was stepped from –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V and back to –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='1 V intervals (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' MOKE microscopy images of the CoFeB film at selected gate voltages recorded in +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT perpendicular field are shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Starting from a saturated magnetization state at –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V, inverse stripe domains form at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 V, followed by the nucleation of sparse skyrmions at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The density of the mixed stripe and skyrmion state increases with voltage before morphing into a dense skyrmion lattice at +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Hereafter, the skyrmion density increases further up to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Sweeping the voltage in the opposite direction reduces the skyrmion density gradually until all skyrmions are annihilated at –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Figure 2b summarizes the skyrmion density during the voltage sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The hysteresis demonstrates the existence of a memory effect in the device, enabling access to a continuous range of skyrmion states, which is a requirement for neuromorphic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Besides control over skyrmion nucleation and annihilation, the gate voltage also tunes the skyrmion size (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The first skyrmions appearing at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='8 V are large (\uf07e800 nm) but their size decreases continuously up to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V (\uf07e600 nm) (see Methods in SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Sweeping the voltage in the negative direction only has a small effect on the skyrmion size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Full reversibility between a reproducible skyrmion state and no skyrmions upon repeated voltage cycling is demonstrated in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' a O Li+ b d 2 V oV +2 V Pt4nm SiNnm 0mT MOKE signal (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=') LiPON100nm Pt20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='2nm CoFeB(40:40:20)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='9nm +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT Pt4nm oV +2 V Ta2nm 2 V 2 0 2 4 Perpendicular magnetic field (mT4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Voltage dependence of the skyrmion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (a) Polar MOKE microscopy images recorded while sweeping the applied voltage from –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The perpendicular magnetic field is +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The scalebar corresponds to 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (b) Skyrmion density during the voltage sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (c) Average skyrmion radius during the voltage sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The error bars show the standard error of the mean size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (d) Reversible toggling of the skyrmion density by switching the voltage between –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The voltage is applied for 1 min before data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' For applications, devices are likely to be controlled by voltage pulses, where the response to both the application and removal of a voltage is relevant to the device operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' To investigate the decay of the skyrmion state over time at zero-bias voltage, we applied +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V for 1 min to a crossbar junction followed by setting the voltage to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The skyrmion density as a function of time is shown in Figure 3a and MOKE microscopy images at different times are depicted in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The decay constant is found to be approximately 8 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' To assess the dynamic response of our magneto-ionic device under voltage pulsing, we applied 250 ms long voltage pulses with magnitudes ranging from +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 V to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V at 500 ms intervals and monitored the skyrmion density over time (Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The device was reset to a skyrmion-free state between each series of pulses by applying –2 V for 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' MOKE microscopy images of the CoFeB film taken after 3000 pulses are shown in Figure 3d for four different pulse voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Two clear features a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0V OV +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7V 2 3 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='5V 10 9 8 7 6 b c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='8 900 Average skyrmion tate Skyrmion density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='6 radius (nm) 800 S D forward 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='4 forward 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='2 backward 10 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 3 S 1 2 backward 1 0 1 2 1 0 1 2 p Voltage (V) Voltage (V) Skyrmion density Voltage (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 1 2 3 4 5 6 7 Cycle5 stand out, firstly that the rate of approach to an equilibrium value is much faster at higher applied voltage and secondly that the equilibrium skyrmion density is much higher at higher applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The device shown here was cycled over 50,000 times whilst retaining the voltage control of the skyrmion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Control of skyrmions by voltage pulse number, pulse amplitude and pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (a) Time evolution of the skyrmion density at 0 V after skyrmion nucleation at +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V for 1 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (b) MOKE microscopy images recorded during the retention experiment shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The scalebar indicates 10 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (c) Skyrmion density as a function of voltage pulse number using a pulse duration of 250 ms with a 50% duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The pulse amplitude is varied from +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 V to +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (d) MOKE microscopy images recorded after 3000 pulses for each of the applied voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The scalebar indicates 10 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (e) Skyrmion density after applying a single voltage pulse and a sequence of 100 voltage pulses to the uniform magnetization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The amplitude of the pulse is fixed at +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V and the duration of the pulse is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The 100-pulse sequence has a duty cycle of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (f) MOKE microscopy images recorded after 60 µs, 80 µs, 100 µs for both single- and 100 pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The scalebar corresponds to 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' All experiments used a +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 mT perpendicular magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' We further exploit the dependence of the skyrmion density on voltage to probe the skyrmion nucleation kinetics at shorter timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' By applying a single pulse of +10 V, we show that the pulse width required for skyrmion nucleation can be as low as 60 µs (Figure 3e, blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' In these experiments, the device was reset by applying a –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='8 V gate voltage for 5 s before each voltage pulse and the skyrmion density was recorded for a few seconds after the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' For a sequence of 100 a e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='25 +2 V 1 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='100 1 pulse 100 pulses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='15 decay constant density 2 ~ 8 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='10 skyrmion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='05 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='7 V +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='8 V initial state +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='9 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='00 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0 V S H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='00 0 60 120 180 0 1000 2000 3000 0 40 80 120 160 200 b Time (min) d pulse number f Pulse duration (μs) 0min min 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='0AY 30min 80min6 identical pulses the number of nucleated skyrmions increase and a pulse duration of just 20 µs is already sufficient to nucleate skyrmions (Figure 3e, red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' MOKE microscopy images of the crossbar junction after a pulse or pulse sequence are presented in Figure 3f for pulse durations between 60 µs and 100 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' This is to our knowledge the fastest achieved ionically induced response in a voltage- controlled magneto-ionic system at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Electrical characterization of supercapacitor junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (a) Cyclic voltammograms recorded for different voltage ranges at 10 mV/s scan rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (b) Junction current at 0 V as a function of voltage range, derived from cyclic voltammetry with a 10 mV/s scan rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The voltage range is symmetric around 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (c) Electrical impedance spectroscopy measurements on a junction using a 100 mV ac driving voltage with bias voltages of −2 V, 0 V and +2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' (d) Cyclic voltammograms at 50 mV/s scan rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' An initial cyclic voltammogram (blue) was followed by cycling the junction between −2 V and +2 V with a period of 250 ms for 750,000 cycles, after which a second cyclic voltammogram (red) was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' To understand the functioning of the devices we turn to electrical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Cyclic voltammograms (CVs) of the supercapacitor structure show a largely rectangular shape with no peaks indicative of redox processes (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' As shown in Figure 4b, for low voltage ranges the current at 0 V is a slowly increasing function of the voltage range with the junction current increasing notably for larger voltage ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' This suggests that both electric double layer and electrochemical mechanisms are present, with the electrochemical mechanism dominating at higher voltages [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Given the material system it is expected that the electrochemical mechanism is intercalation of the Li ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The capacitance of the junction is calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='18 \uf06dF at 1 V/s, which is equivalent to a capacity of 72 \uf06dF/cm2, showing large storage capability typical of supercapacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' In Figure 4c electrical impedance a b 25 30 EDLi Intercalation 20 Current (nA) Current (nA) 15 0 10 30 ±100 mV ±200 mV 5 ±300 mV ±400 mV 60 ±500 mV ±1 V ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='5 V ±2 V 0 2 1 0 1 2 0 1 2 3 4 c Voltage (V) p Voltage range (V) 10000 300 initial after 750,000 cycles 7500 +2 V 150 urrent (nA) (0) (z)wl- oV 2 V 5000 0 150 2500 300 0 2500 5000 7500 10000 2 1 0 1 2 Re{Z) (2) Voltage (V)7 spectroscopy is shown, giving a steep line at lower frequencies as expected from a capacitance- dominated device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The supercapacitor system is highly cyclable, with Figure 4d showing the CV (at 50 mV/s, giving squarer loops than in Figure 4a) before and after cycling 750,000 times with 250 ms pulses at ±2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Moreover, our supercapacitor is intrinsically fast with a characteristic charge/discharge time of 560 \uf06ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Figure S1 in the SI provides additional information on the electrical properties of the supercapacitor structure, including leakage current, open circuit voltage and its electrical impedance as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The combination of magnetic and electrical data shows that the accumulation or depletion of Li ions at the CoFeB/Pt interface causes large changes to the magnetic state at low voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Values for the perpendicular magnetic anisotropy (Ku) and the Dzyaloshinskii-Moriya interaction constant (D), along with saturation magnetization (Ms) and exchange constant (Aex) were estimated from a thin film sample with a similar structure (Figure S2 in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Ku and D were found to be 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='96 \uf0b4 105 J/m3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='74 mJ/m2, respectively, which is consistent with the creation of bubble-like magnetic skyrmions in this sample at around the sizes seen in Figure 1 and Figure 2 (see Figure S3 in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' From our previous work [14], the insertion of Li ions at the CoFeB/Pt interface is expected to reduce the perpendicular magnetic anisotropy without reducing the magnetization [14], which reduces the energy barrier to skyrmion nucleation and stabilizes skyrmions relative to the uniform state [28] (see also SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' One interesting feature of the data in Figure 2c is the reduction in skyrmion size with increasing voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' If the system was simply undergoing a reduction in anisotropy, then the skyrmion size is expected to increase [28,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Instead, a decrease in size is seen, which could either indicate that the skyrmions at lower densities are preferentially found at defect sites [38] or that the DMI is also reduced by the accumulation of Li ions at the CoFeB/Pt interface [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The time-dependent experiments give insight into the timescales of the phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The decay time of the skyrmion state in Figure 3a corresponds to an energy barrier of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='38 eV, similar to that expected for the thermally activated hopping motion of Li ions within LiPON [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' To minimize the internal electric field within the ion conduction layer there is a thermally activated redistribution of Li ions within the layer, causing the skyrmions to consequently annihilate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' This also explains the results of the pulsed experiments in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Here the positive voltage pulses cause the accumulation of Li ions at the interface, which decreases the skyrmion nucleation barrier, whilst during the off state the accumulated ions decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The concentration of interfacial Li ions increases with the number of voltage pulses until the decay in the off state balances a further increase during the on state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' For the sub-ms pulses used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 3e, there is a further effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Now the barrier for skyrmion nucleation is lowered rapidly and then increases again as the Li accumulation decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' However, the nucleation of skyrmions occurs on a timescale longer than the voltage pulses, leading to a peak in skyrmion density 8 around a second after the pulse (Figure S4 in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Therefore, the speed of the devices is also limited by the thermally activated nucleation of the skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Fast and durable voltage control of skyrmions in Li-ion supercapacitor structures, as shown here, offers attractive pathways to the implementation of neuromorphic devices such as synapse-based neural networks [24] and reservoir computers [25-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Proof-of-concepts demonstrating the suitability of skyrmion dynamics for neuromorphic computing have thus far utilized magnetic fields or electric currents to control the skyrmion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Voltage gating of a skyrmion-hosting magnetic film provides good scalability and energy efficiency in combination with deterministic accumulation/dissipation, short-term memory, and nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' For instance, reversible nucleation and annihilation of skyrmions through the application of positive and negative voltages (Figure 2) enables the emulation of synaptic weights changes during potentiation and depression, while the decay of the skyrmion state after voltage pulsing (Figure 3a,b) provides short-term memory to temporarily store information and trigger outputs based on the time-dependent history of voltage inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Nonlinearity of voltage-driven skyrmion dynamics, which is another key requirement for neuromorphic processing, is demonstrated in our supercapacitors by varying the amplitude (Figure 3c,d) and duration (Figure 3e,f) of the voltage pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Finally, we note that the complex interplay between the dynamics of Li ion migration in the solid-state LiPON electrolyte and the ensuing nonlinear dynamics of skyrmions in the thin magnetic film offers great flexibility in the design of functional responses and further device optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' In summary, we have shown that skyrmions in a Pt/CoFeB/Pt thin-film structure can be created and annihilated in a fully voltage-controlled all-solid-state device via reversible Li ion migration at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The hysteretic behavior of the device with respect to the voltage sweep direction, the nonlinear effects observed as a function of voltage pulse number and pulse duration, along with the decay behavior at zero-voltage constitute properties suitable for neuromorphic device architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' The use of a supercapacitor enables skyrmion nucleation with single voltage pulses down to 60 \uf06ds, combined with extensive cycling of the junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Further downscaling of the device from the 100 nm thick solid-state electrolyte used here may allow access to sub-\uf06ds functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Supporting Information Methods and additional data, including electrical characterization, measurements to extract magnetic parameters, micromagnetic simulations of the skyrmion energy and fast voltage pulsing experiments (PDF) Corresponding Authors 9 rhodri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='mansell@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='fi sebastiaan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='van.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='dijken@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='fi Author Contributions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' conceived the research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' grew the supercapacitor heterostructures and fabricated the crossbar junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' conducted the electrical characterization and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' performed the magnetic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' conducted the micromagnetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' supervised the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' All authors discussed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Notes The authors declare no competing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Acknowledgments This work was supported by the Academy of Finland (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 316857).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Lithography was performed at the OtaNano-Micronova Nanofabrication Centre, supported by Aalto University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Computational resources were provided by the Aalto Science-IT project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 10 References [1] Nichterwitz, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Avci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Büttner, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Mann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Valvidares, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Sheffels, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Churikova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Büttner, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Yildiz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Beach, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Voltage control of ferrimagnetic order and voltage-assisted writing of ferrimagnetic spin textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 2021, 16, 981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' [14] Ameziane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Mansell, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Fähler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Marty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Souche, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Poinsignon, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Hahn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Kruk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Voltage-control of magnetism in all-solid-state and solid/liquid magnetoelectric composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Mater.' metadata={'source': 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S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Litzius, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Jakob, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Virnau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Pinna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Everschor-Sitte, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Rózsa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Donges, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Finizio, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Bourianoff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Everschor-Sitte, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Reservoir computing with random skyrmion textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Kasai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Otani, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Pattern recognition with neuromorphic computing using magnetic field-induced dynamics of skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 2022, 8, eabq5652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' [27] Raab, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Brems, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Beneke, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 2022, 13, 6982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' [28] Schott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Bernand-Mantel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Ranno, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Pizzini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Vogel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Béa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' A theory on skyrmion size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' 2018, 1, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' [38] Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Mansell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Valencia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Kronast, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' van Dijken, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Temperature dependence of the Dzyaloshinskii-Moriya interaction in ultrathin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} +page_content=' B 2020, 101, 054433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFJT4oBgHgl3EQflyzE/content/2301.11585v1.pdf'} diff --git a/79AzT4oBgHgl3EQfgfxi/content/tmp_files/2301.01469v1.pdf.txt b/79AzT4oBgHgl3EQfgfxi/content/tmp_files/2301.01469v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff182f902d418be40ab257526f19cad030176c79 --- /dev/null +++ b/79AzT4oBgHgl3EQfgfxi/content/tmp_files/2301.01469v1.pdf.txt @@ -0,0 +1,1864 @@ +Machine Learning-based Signal Quality Assessment +for Cardiac Volume Monitoring in Electrical +Impedance Tomography +Chang Min Hyun1, Tae Jun Jang1, Jeongchan Nam2, +Hyeuknam Kwon3, Kiwan Jeon4, and Kyunghun Lee5¶ +1School of Mathematics and Computing (Computational Science and Engineering), +Yonsei University, Seoul, Republic of Korea. +2BiLab, Pangyo, Republic of Korea. +3Division of Software, Yonsei University, Wonju, Republic of Korea. +4National Institute for Mathematical Sciences, Daejeon, Republic of Korea. +5Kyung Hee University, Seoul, Republic of Korea. +Abstract. +Owing to recent advances in thoracic electrical impedance tomography, +a patient’s hemodynamic function can be noninvasively and continuously estimated +in real-time by surveilling a cardiac volume signal associated with stroke volume and +cardiac output. In clinical applications, however, a cardiac volume signal is often of low +quality, mainly because of the patient’s deliberate movements or inevitable motions +during clinical interventions. +This study aims to develop a signal quality indexing +method that assesses the influence of motion artifacts on transient cardiac volume +signals. The assessment is performed on each cardiac cycle to take advantage of the +periodicity and regularity in cardiac volume changes. +Time intervals are identified +using the synchronized electrocardiography system. +We apply divergent machine- +learning methods, which can be sorted into discriminative-model and manifold-learning +approaches. The use of machine-learning could be suitable for our real-time monitoring +application that requires fast inference and automation as well as high accuracy. In +the clinical environment, the proposed method can be utilized to provide immediate +warnings so that clinicians can minimize confusion regarding patients’ conditions, +reduce clinical resource utilization, and improve the confidence level of the monitoring +system. +Numerous experiments using actual EIT data validate the capability of +cardiac volume signals degraded by motion artifacts to be accurately and automatically +assessed in real-time by machine learning. The best model achieved an accuracy of 0.95, +positive and negative predictive values of 0.96 and 0.86, sensitivity of 0.98, specificity +of 0.77, and AUC of 0.96. +¶ To whom correspondence should be addressed (imlkh84@gmail.com) +arXiv:2301.01469v1 [eess.SP] 4 Jan 2023 + +2 +1. Introduction +Over several decades, continued advances in electrical impedance tomography (EIT) +have expanded the clinical capability of real-time cardiopulmonary monitoring systems +by overcoming the limitations of traditional methods, such as cardiac catheterization +through blood vessels [3, 8, 18, 19, 28, 29, 37, 41, 57]. Recently, based on thoracic EIT, +a patient’s hemodynamic function can be noninvasively and continuously estimated +in real-time by surveilling a signal extracted using EIT, the so-called cardiac volume +signal (CVS), which has a strong relationship with key hemodynamic factors such +as stroke volume and cardiac output [4, 27, 51]. +In clinical applications, however, a +cardiac volume signal is often of low quality, mainly because of the patient’s deliberate +movements or inevitable motions during clinical interventions such as medical treatment +and nursing. +Because postural change causes movement of the chest boundary to +which existing EIT solvers are highly sensitive owing to time-difference-reconstruction +characteristics [1, 7, 32, 36, 44], motion-induced artifacts are generated in the CVS, as +shown in Figure 1. +Figure 1. +Motion-induced artifacts in cardiac volume monitoring using electrical +impedance tomography. Patient’s deliberate movements or inevitable motions during +clinical intervention cause severe artifacts in a cardiac volume signal. +CVS extraction is to separate a cardiogenic component from the EIT voltage data, +resulting from current injections at electrodes attached across a human chest. In recent +studies [27, 39], effective CVS extraction was successful in motion-free measurements +where voltage data are mainly influenced by air and blood volume changes in the lungs, +heart, and blood vessels comprehensively, but not by motions. In contrast, achieving +the cardiogenic component separation in motion-influenced measurements is still a long- +term challenge. Postural changes in EIT measurements cause strong distortion of the +voltage data [1, 56] and easily disturb the extraction of relatively weak cardiogenic +signals [6,33,40]. +Handling motion interference has been a huge challenge in most EIT-based +techniques for enhancing clinical capability, but not researched much yet [54]. Adler +et al. [1] and Zhang et al. [56] investigated the negative motion effect in the EIT. + +w3 +Soleimani et al. [43] and Dai et al. [17] proposed a motion-induced artifact reduction +method by reconstructing electrode movements along with conductivity changes. Lee et +al. [36] analyzed motion artifacts in EIT measurements and proposed a subspace-based +artifact rejection method. Yang et al. [54] suggested the discrete wavelet transform- +based approach that reduces motion artifacts of three specific types. However, clinical +motion artifacts are still not effectively addressed because of practical motion’s immense +diversity and complexity. Accordingly, for the time being, the EIT-based hemodynamic +monitoring system attempts to be preferentially developed toward filtering motion- +influenced CVSs rather than recovering them. In the clinical environment, this filtration +can provide immediate warnings so that clinicians can minimize confusion regarding the +patient’s condition, reduce clinical resource utilization, and improve the confidence level +of the monitoring system [16]. +This study aims to develop a signal quality indexing (SQI) method that assesses +whether motion artifacts influence transient CVSs. +To take advantage of the +periodicity and regularity in cardiac volume changes, the assessment is performed +on each cardiac cycle, whose time intervals are identified using the synchronized +electrocardiography (ECG) system. +We leverage machine learning (ML), which +has provided effective solutions for various biosignal-related tasks through feature +disentanglement of complicated signals [5,9,14,25,34,47,48,52]. The use of ML could +be suitable for our real-time monitoring application that requires fast inference and +automation as well as high accuracy. +We apply divergent ML methods, which can be sorted into discriminative-model and +manifold-learning approaches. The discriminative-model approach is first considered, +where an SQI map is directly trained using a paired dataset of CVS and its label +[12, 22, 45]. Although this approach provides a high performance on a fixed dataset, +owing to the class imbalance problem, there is a risk of overfitting on motion-influenced +CVS data in the scope of generalization or stability [10, 15, 23, 49]. Motion artifacts +can vary considerably in real circumstances, whereas collecting CVS data in numerous +motion-influenced cases is practically limited because of the high cost, intensive labor, +security, and ambiguity in clinical data acquisition and annotation [13, 46, 50, 58]. +To handle this conceivable difficulty, the manifold-learning approach [2, 24, 26, 30] is +examined as an alternative. +It does not learn irregular and capricious patterns of +motion-influenced CVSs and only takes advantage of the learned features from motion- +free CVSs. +Numerous experiments have been conducted using actual EIT data. +Empirical +results demonstrate that discriminative and manifold-learning models provide accurate +and automatic detection of motion-influenced CVS in real-time. The best discriminative +model achieved an accuracy of 0.95, positive and negative predictive values of 0.96 and +0.86, sensitivity of 0.98, specificity of 0.77, and AUC of 0.96. The best manifold-learning +model achieved accuracy of 0.93, positive and negative predictive values of 0.97 and +0.71, sensitivity of 0.95, specificity of 0.80, and AUC of 0.95. The discriminative models +yielded a more powerful SQI performance; in contrast, the manifold-learning models + +4 +provided stable outcomes between the training and test sets. Regarding to practical +applications, the choice of two models relies on what should be emphasized in the +monitoring system in terms of performance and stability. +2. Methods +0 +Figure 2. 16-channel system of thoracic EIT and CVS extraction. The EIT machine +measures voltage differences by injecting currents via electrodes attached along human +chest. +A cardiac volume signal xt is extracted by taking suitable weighting w to +the time-difference transconductance 9gt, which is defined by measured voltage data. +Here, w is called as a leadforming vector, which is designed to separate a cardiogenic +trans-conductance change from superposed data 9gt [39]. +This study considers the 16-channel system of the thoracic EIT, where 16 electrodes +are attached along the human chest (see Figure 2). The EIT system is assumed to be +synchronized with the ECG system, which provides the time interval for each cardiac +cycle. The EIT device measures a set of voltage differences by injecting an alternative +current of I (mA) through pairs of adjacent electrodes while keeping all other electrodes +insulated. At sampling time t, the following voltages are acquired: +tV j,k +t +: V j,k +t +“ U j,k +t +´ U j,k`1 +t +, j P I, k P Iztj, j ` 1uu +(1) +where I is an index set defined by I “ t1, 2, ¨ ¨ ¨ , 16u, Ek is the k-th electrode, and U j,k +t +is the electrical potential on Ek subject to the current injection from Ej to Ej`1. For +notational convenience, E0 and E17 can be understood as E16 and E1, respectively. Once +the current is injected from Ej to Ej`1 for some j P I, the voltage is measured at each of +the 16 adjacent electrode pairs pEk, Ek`1qkPI. Among the 16 voltages, V j,j´1 +t +, V j,j +t +, and +V j,j`1 +t +are discarded to reduce the influence of the skin-electrode contact impedance [44]. +Because we perform 16 independent current injections, in total, 208 p“ 16ˆ13q voltages +are obtained and used to produce the CVS. + +Electrode +positionElectrode +positionElectrode +positionCardiac +Volt +ume +Sig +gna +&t5 +2.1. CVS Extraction Using EIT and Influence of Motion +A transconductance (column) vector gt P R208 can be defined using the voltage data (1) +as follows: +gt “ +„ +I +RpV 1,3 +t +q +, ¨ ¨ ¨ , +I +RpV 1,15 +t +q +, ¨ ¨ ¨ , +I +RpV 16,2 +t +q +, ¨ ¨ ¨ +I +RpV 16,14 +t +q +ȷT +(2) +where T represents the vector transpose and R is an operation for extracting the real +part of a complex number. Here, gt is updated every 10ms. +A CVS, denoted by xt P R, is obtained by +xt “ wT 9gt +(3) +where w P R208 is a weighting (so-called leadforming) vector and 9gt is time difference +of gt given by +9gt “ gt ´ gt0 for reference time t0 +(4) +In the absence of motion, the transconductance 9gt can be expressed by +9gt “ 9gair +t +` 9gblood +t +(5) +where gair +t +and gblood +t +are transconductance vectors related to air and blood volume +changes in the lungs and heart, respectively. The weighting vector w is designed to +provide +wT 9gt “ wTp 9gair +t +` 9gblood +t +q “ wT 9gblood +t +(6) +See Figure 2. +Kindly refer to [39] for details on determining w. +Even though the +cardiogenic signal gblood +t +is weak, it can be accurately decomposed from the data gt. +In light of the previous analysis in [36], the following explains why the quality of the +CVS is degraded by motion, as shown in the middle part of Figure 1. In the presence +of motion, the transconductance 9gt can be approximated by +9gt « 9gnormal +t +` 9gmotion +t +(7) +where 9gnormal +t +“ 9gair +t ` 9gblood +t +and 9gmotion +t +is the motion-induced effect. Appendix Appendix +A presents details of (7). Determining the vector w itself can be considerably affected +by motion artifacts [39]. Moreover, even if w satisfies (6), we have +xt “ wT 9gt « xnormal +t +` xmotion +t +(8) +where xnormal +t +“ wT 9gblood +t +and xmotion +t +“ wT 9gmotion +t +. The last term xmotion +t +describes +motion artifacts in the CVS. +2.2. CVS Quality Assessment and Data Preprocessing +This study aims to assess the CVS (xt) for detecting motion-induced signal quality +degradation. +See Figure 3. +This can be accomplished by developing an SQI map +f : xt ÞÑ yt such that +fpxtq “ yt “ +# +1 +if xmotion +t +« 0 +0 +if xmotion +t +ff 0 +(9) + +6 +Figure 3. Schematic description of machine learning-based signal quality assessment +for cardiac volume monitoring in electrical impedance tomography. +However, it is arduous to achieve (9), where the assessment is conducted on an individual +CVS at every sampling time. Instead, we take advantage of the periodicity and regularity +of cardiac volume changes according to the heartbeat. The time interval of each cardiac +cycle is identified using a synchronized ECG system. +Our quality assessment is conducted on every cardiac cycle of CVS, where a cardiac +cycle is defined by the time interval consisting of two consecutive ECG R-wave peaks +as the end points. +For a given time tcyc, let the interval rtcyc, tcyc ` ∆tcycs be the +corresponding cardiac cycle, where ∆tcyc is assumed to be ∆tcycle “ 10ms ˆ pv ´ 1q for +some v P Nzt1u. Here, N denotes the set of positive integers. A vector gathering all +CVSs during the cycle, denoted by Xtcyc P Rv, is defined as +Xtcyc “ +” +xtcyc, xtcyc`10ms, ¨ ¨ ¨ , xtcyc`10msˆpv´1q +ıT +(10) +The map f in (9) can be modified into +fpXtcycq “ yt “ +# +1 +for normal Xtcyc +0 +for motion-influenced Xtcyc +(11) +To find f in (11), we leverage ML, which can learn the domain knowledge of normal +and motion-influenced CVSs from a training dataset of N data pairs tXpiq, ypiquN +i“1. +Prior to ML applications, the following issues need to be addressed in the CVS data. +First, CVSs have significant inter-subject and intra-subject variability. This is because +cardiac volume varies depending on various factors, including sex, age, condition, +time, and body temperature. +Therefore, scale normalization is required to enhance +the stability and performance of ML while mitigating the high learning complexity +associated with scale-invariant feature extraction [20,53]. Second, the dimensions of the +input CVS data in (11) do not match each other (i.e., v is not constant) owing to heart +rate variability [11]. Because most existing ML methods are based on an input with +consistent dimensions, size normalization is required. Figure 4 schematically illustrates +the overall process. + +diac +C +ar +ycle+202107.21 +500M +73 +37 +5.5 +2.7 +13 +2726 +75 +97 +BiLab +HemoVistaSignal +Cardiac +Volume +CVsHighcycleCVSAcguisitionHigh +SQ1LoW +TOScycle7 +Figure 4. From the monitoring system, electrocadiography and cardiac volume signals +are obtained. By identifying a cardiac cycle through electrocadiography data (R-wave +peak detection), we extract cardiac volume signals at the corresponding cycle and then +lastly apply normalization in terms of scale and size. +2.2.1. Scale normalization +A simple method of normalizing the scale is to rescale the +CVS data for individual cardiac cycles. Specifically, for a given CVS vector Xtcyc P Rv, +the scaling factor S is obtained using +S “ max +iPV |xtcyc`10ˆi(ms)| +(12) +where the index set V is given by V “ t0, 1, ¨ ¨ ¨ , v ´ 1u. Normalized CVS data, denoted +by Xtcyc, are obtained by +Xtcyc “ Xtcyc +S +(13) +However, this scaling may not be appropriate to our application for the following reason. +Abnormalities in CVS data include sudden increases or decreases in signal amplitude +as well as irregular deformations of the shape profile. The normalization in (13) can +contribute to ignoring rapid amplitude changes. +This study uses the following subject-specific scale normalization strategy. When +the EIT device is used to monitor a certain subject, it is supposed that during the initial +20s calibration process, the device measures the normal CVS data available for scale +normalization. Let X subject be a set of corresponding CVSs given by +X subject “ tx10ˆi(ms) : i “ ´1999, ´1998, ¨ ¨ ¨ , ´1u +(14) +Using the set X subject, a subject-specific scaling factor Ssubject is obtained by +Ssubject “ +max +xPXsubject|x| +(15) +This scale factor Ssubject is used for the normalization in (13) instead of the naive factor +S in (12). + +C +Vol +Si +ardiac +ume +gnalectrocardiography +Si +gnaardiac +CycleAt +ycle1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +0 +50 +100 +150tX +1f +cyc202107.21 +500M +73 +37 +5.5 +2.7 +13 +2726 +75 +97 +BiLab +HemoVista8 +2.2.2. Size normalization +To make the dimensions of the CVS data consistent, a CVS +vector Xtcyc is embedded into Rν for a fixed constant ν. In the empirical experiment, +the embedding space dimension was to be larger than any dimension of the CVS data +in our dataset (ν “ 150). +Two normalization methods are considered. The first approach is to resample ν +points using linear interpolation with v data points in Xt. For the stationary interval +r0, 1s, the following linear interpolation function L is constructed: +Lp +i +v ´ 1q “ xtcyc`10(ms)ˆpi´1q for i “ 0, ¨ ¨ ¨ , v ´ 1 +(16) +Subsequently, we obtain the normalized vector Xtcyc P Rν using +Xtcyc “ +„ +Lp0q, Lp +1 +ν ´ 1q, Lp +2 +ν ´ 1q, ¨ ¨ ¨ , Lp1q +ȷT +(17) +This method normalizes the signal profile of CVS data into the desired length (ν) +with no significant loss, but loses sampling time information. Second, the last value in +Xtcyc (i.e., xtcyc`10(ms)ˆpv´1q) is padded up to the desired length. This constant padding +provides a vector Xtcyc P Rν, expressed by +Xtcyc “r xtcyc, ¨ ¨ ¨ , xtcyc`10(ms)ˆpv´2q, xtcyc`10(ms)ˆpv´1q, +(18) +xtcyc`10(ms)ˆpv´1q, ¨ ¨ ¨ , xtcyc`10(ms)ˆpv´1q sT +(19) +where the part (19) corresponds to the padding. In contrast to the first method, this +normalization can preserve time information regarding sampling frequency, whereas the +core profile of the CVS is supported at different time intervals. +2.3. Machine Learning Application +At this point, we are ready to apply ML for determining the SQI function (11). Collected +from various subjects and cardiac cycles, the following dataset is used: +tX +piq, ypiquN +i“1 +(20) +where ypiq is the SQI label corresponding to X +piq. We note that X is the CVS data for +a cardiac cycle of some subjects and is normalized for both scale and size. In practice, +the available training dataset (20) was highly imbalanced, where there were relatively +few negative samples (motion-influenced CVSs). +2.3.1. Discriminative-model approach +The discriminative-model approach trains the +SQI map f : X ÞÑ y in the following sense: +f “ argmin +fPF +1 +N +N +ÿ +i“1 +distpfpX +piqq, ypiqq +(21) + +9 +(b) Manifold-learning Approach +(a) Discriminative-model Approach +Figure 5. (a) Discriminative-model approach learns a signal quality indexing map +f by using CVS and label data. (b) Manifold-learning approach first learns common +features of normal CVS data by finding a low dimensional manifold Mnormal. Signal +quality assessment is based on computing the residual between original CVS data and +projected one onto or near the learned manifold. +where F is a set of learnable functions for a given ML model and dist is a metric that +measures the difference between the ML output fpXq and label y. See Figure 5 (a). In +our application with high class-imbalance, the following weighted cross-entropy can be +used: +distpfpXq, yq “ ´ζposylogpfpXqq ´ ζnegp1 ´ yqlogp1 ´ fpXqq +(22) +where ζpos and ζneg are the relative ratios of the positive and negative samples, +respectively. Various classification models can be used, such as the logistic regression +model (LR) [12], multi-layer perceptron (MLP) [22], and convolutional neural networks +(CNN) [45]. Detailed models used in this study are explained in Appendix Appendix +B.1. +The discriminative model approach is a powerful method to guarantee high +performance in a fixed dataset. However, it might suffer from providing stable SQI +results in clinical practice because of highly variable negative samples. This is because +these methods take advantage of learned information using only a few negative samples +[10, 15, 23, 49]. +To achieve stable prediction, the manifold-learning approach can be +alternatively used [13,50,58]. +2.3.2. +Manifold-learning approach The manifold-learning approach learns common +features from positive samples (i.e., normal CVS) and uses them to develop an SQI +map. The remaining negative samples are utilized as auxiliary means for selecting a +hyperparameter. Figure 5 (b) shows a schematic description of this process. + +Not +uisectrainingP +rojectionCITraininganifoldmrma.R. +eso +ta10 +A set of positive samples is denoted by tX +piq +posu +Npos +i“1 , where Npos denotes the number +of positive samples. In the first step, we learn a low-dimensional representation of Xpos +by training an encoder E : Xpos ÞÑ z and decoder D : z ÞÑ Xpos in the following +sense [21,26]: +pD, Eq “ argmin +pD,Eq +1 +Npos +Npos +ÿ +i“1 +}D ˝ EpX +piq +posq ´ X +piq +pos}2 +2 +(23) +where z is a low dimensional latent vector and } ¨ }2 is the standard Euclidean norm. +The architectures D and E can be used in PCA [26], VAE [30], and β-VAE [24]. See +more details in Appendix Appendix B.2. +Borrowing the idea from [2], an SQI map f is constructed as follows: For a given +CVS data X in any class, a residual r is computed by +r “ }X ´ D ˝ EpXq}2 +(24) +The decoder D is trained to generate normal CVS-like output. In other words, operation +D˝E transforms X to lie in or near the learned manifold using normal CVS data [44,55]. +Therefore, the residual r can be viewed as an anomaly score, where r is small if X is +normal CVS data, and large if X is motion-influenced CVS data. For some non-negative +constant d, an SQI map f can be constructed using +fpXtq “ +# +1 +if r ď d +0 +if r ą d +(25) +The remainder of this subsection explains how the thresholding value d is +determined by utilizing negative samples as well as positive. +By varying d from 0 +to 8, a receiver operating characteristic (ROC) curve is calculated, where a point in the +ROC curve is obtained using a fixed d. We choose d such that maximizing Youden’s J +statistics, which is known as an unbiased metric in the class imbalance case [42]. The +value J is given by +J d “ Sensitivityd ` Specificityd ´ 1 +(26) +where +Sensitivityd “ +N d +TP +N d +TP ` N d +FN +and Specificityd “ +N d +TN +N d +TN ` N d +FP +(27) +Here, N d +TP, N d +TN, N d +FP, and N d +FN respectively represent the number of true positives, true +negatives, false positives, and false negatives for predictions depending on a selected +threshold value d. +3. Results +3.1. Data Acquisition and Experimental Setting +Our dataset was obtained from healthy volunteers using an EIT-based hemodynamic +monitoring device (HemoVista, BiLab, South Korea). Synchronized ECG data were + +11 +obtained with EIT and used to identify the cardiac cycles. While lying in a hospital +bed, each subject was requested to make intentional motions mimicking postural changes +in the clinical ward. A total of 16140 CVS data were obtained regarding the cardiac +cycle. +Manual labeling was individually performed by two- and ten- years bio-signal +experts (Nam and Lee). Subsequently, they reviewed the results and made the final +decision about CVS abnormality through an agreement between them. The final labels +were annotated into three classes: normal, ambiguous, and motion-influenced. When +classified as normal or abnormal by both experts with an agreement, CVS data were +annotated as normal or motion-influenced classes. The ambiguous class stands for CVS +data in which motion artifacts were included with high possibility, but the experts did +not reach an explicit agreement about motion influence. The assigned label is y “ 1 +for the normal class and y “ 0 for the other classes. As a result, 12928 (80.09%), 1526 +(9.45%), and 1686 (10.45%) samples were labeled as normal, ambiguous, and motion- +influenced classes, respectively. +For ML applications, a total of 16372 CVS data were divided into 13100 (80%), +1520 (10%), and 1520 (10%), which were used for training, validation, and testing, +respectively. The data split was performed such that CVS data obtained from a common +subject did not exist between the three sets. For the training dataset, labels for the +ambiguous class were reassigned to y “ 0.25. +This was done to prevent the over- +classification of ambiguous classes. +ML experiments were conducted in a computer system with GeForce RTX 3080 +Ti, Intel® Core™ X-series Processors i9-10900X, and 128GB DDR4 RAM. Python with +scikit-learn and Pytorch packages were used for the ML implementation. When training +the ML models, the Adam optimizer was consistently employed, which is an effective +adaptive stochastic gradient descent method [31]. Hyperparameters such as epoch and +learning rate were heuristically chosen based on the validation results. +3.2. Results of CVS Quality Assessment +We compared the performance of the ML-based CVS quality assessment results by +using six metrics: accuracy, positive and negative predictive values (PPV and NPV), +sensitivity, specificity, and AUC. Accuracy, PPV, and NPV were defined by +Accuracy “ +NTP ` NTN +NTP ` NTN ` NFP ` NFN +, PPV “ +NTP +NTP ` NFP +, and NPV “ +NTN +NTN ` NFN +(28) +and AUC was the area under the ROC curve. NPV, specificity, and AUC should be +emphasized in our evaluation owing to the high-class imbalance (small negative samples). +3.2.1. Discriminative Models +The first and second rows of Tables 1 (a) and (b) show +the quantitative evaluations of CVS quality assessment using various discriminative +models: LR, MLPs, and CNNs. +The results in Tables 1 (a) and (b) differ in size +normalization: (a) linear interpolation and (b) constant padding. + +12 +(a) SQI with scale and size normalization using linear interpolation. +Discriminative Model +LR +MLP1 +MLP2 +VGG16-3 +VGG16-4 +VGG16-5 +Test +Accuracy +0.8665 +0.9323 +0.9348 +0.9468 +0.9468 +0.9437 +PPV +1.0000 +0.9790 +0.9747 +0.9525 +0.9605 +0.9679 +NPV +0.1097 +0.7241 +0.7445 +0.9047 +0.8591 +0.8083 +Sensitivity +0.8643 +0.9404 +0.9479 +0.9866 +0.9776 +0.9657 +Specificity +1.0000 +0.8860 +0.8607 +0.7215 +0.7721 +0.8185 +AUC +0.6615 +0.9506 +0.9558 +0.9709 +0.9645 +0.9653 +Manifold-learning Model +PCA +VAE +β-VAE +CVAE +β-CVAE +- +Test +Accuracy +0.8468 +0.9066 +0.9221 +0.9292 +0.9298 +PPV +0.9510 +0.9687 +0.9672 +0.9688 +0.9739 +NPV +0.4573 +0.6181 +0.6900 +0.7100 +0.7011 +Sensitivity +0.8675 +0.9218 +0.9439 +0.9486 +0.9441 +- +Specificity +0.7142 +0.8095 +0.7952 +0.8047 +0.8380 +AUC +0.8735 +0.9513 +0.9489 +0.9528 +0.9603 +(b) SQI with scale and size normalization using constant padding. +Discriminative Model +LR +MLP1 +MLP2 +VGG16-3 +VGG16-4 +VGG16-5 +Test +Accuracy +0.8664 +0.9487 +0.9518 +0.9455 +0.9487 +0.9500 +PPV +1.0000 +0.9745 +0.9767 +0.9533 +0.9655 +0.9731 +NPV +0.0826 +0.7851 +0.8065 +0.8870 +0.8433 +0.8185 +Sensitivity +0.8648 +0.9651 +0.9666 +0.9844 +0.9748 +0.9681 +Specificity +1.0000 +0.8521 +0.8652 +0.7173 +0.7956 +0.8434 +AUC +0.6628 +0.9725 +0.9669 +0.9782 +0.9683 +0.9757 +Manifold-learning Model +PCA +VAE +β-VAE +CVAE +β-CVAE +- +Test +Accuracy +0.8809 +0.8918 +0.9214 +0.9015 +0.8861 +PPV +0.9590 +0.9660 +0.9679 +0.9731 +0.9636 +NPV +0.5333 +0.5629 +0.6694 +0.5882 +0.5467 +Sensitivity +0.9014 +0.9074 +0.9407 +0.9118 +0.9029 +- +Specificity +0.7450 +0.7892 +0.7941 +0.8333 +0.7745 +AUC +0.9150 +0.9206 +0.9412 +0.9170 +0.9041 +Table 1. Machine learning-based CVS quality assessment results +MLPs and CNNs performed better than LR, which provided miserable NPV and +AUC. MLPs and CNNs outperformed each other in specificity and NVP respectively, +while achieving comparable levels for the other metrics. +There was no significant +performance gap depending on the size normalization. +One interesting observation was as follows: In our experiments, there seems to be +a compensation between specificity and NPV, depending on the emphasis on locality +and globality. Enriching global information on CVS data positively affected specificity; +in contrast, local information helped improve NPV. As the receptive field size in +VGG16 increased (see Appendix Appendix B.1), specificity tended to increase and NPV +decrease. In MLP, which is more flexible for catching global information than CNNs, +specificity was highest, and NPV lowest. In other words, the local information of CVS +data is likely to play a crucial role in reducing false negatives rather than false positives. +From a practical point of view, reducing false negatives is more desirable; therefore, using +VGG16-3 or VGG16-4, which have the powerful ability to take advantage of locality, +can be an excellent option. +3.2.2. Manifold-learning Models +Positive samples in the validation set were used for +hyperparameter selection in training the encoder and decoder. A threshold value was +determined by using data from all the training and validation sets. +Figure 6 shows manifold projection results of test samples in normal and motion- + +13 +Figure 6. Test samples and VAE-based projection results for (a) normal and (b) +motion-influenced CVS data, where the red line is original CVS data and the blue line +is the correspondent CVS data projected by VAE. By the way, ROC curves for (c) +VGG 16-4 and (d) β-VAE are provided, where the blue and red lines correspond to +the curves calculated using training and test sets, respectively. +influenced classes. An input CVS is projected onto or near a manifold learned by positive +samples. As desired, the residual (24) tends to be small for normal samples and high +for motion-influenced samples. +The third and fourth rows of Tables 1 (a) and (b) show the final assessment +results using manifold-learning models. +The performance was comparable to that +of discriminative models. +We note that the manifold-learning models never learned +negative samples for classifier development. As shown in Figure 6 (d), the manifold- +learning model’s performance gap between training and test sets was very small. +There was a slight difference in performance for the manifold-learning models +depending on the size normalization. Linear interpolation promised a slightly better +assessment of accuracy, NPV, and AUC than the other. For the case of constant padding, +because core profiles of CVS data are supported at different intervals, the learning +complexity can be increased, which is associated with invariant feature extraction to +the intervals. This may cause a slight drop in performance. +In our dataset, both discriminative and manifold learning models provided accurate +detection of motion-influenced CVS. The discriminative model yielded a more powerful +SQI performance; in contrast, the manifold-learning model provided stable outcomes +between the training and test sets. +Regarding practical applications, the choice of +two models relies on what should be emphasized in the monitoring system in terms of +performance and stability. Their ensemble is also worth considering. +3.2.3. +Impact of Scale Normalization Table 2 shows the worst case when scale +normalization was not applied. +In CNNs, network training was very unstable, and +assessment performance was considerably degraded, especially regarding accuracy, NPV, +sensitivity, and AUC. In VAEs, large-scale variability of CVS data highly affected +the loss of accuracy in manifold projection; therefore, the performance significantly +deteriorated in terms of accuracy, NPV, sensitivity, and AUC. This verifies the impact +of scale normalization. + +3 - +Real Cycle +Projected Cycle +2 +1 +0- +-1 +0 +20 +40 +60 +80 +100 +120 +1401.0 +0.8 +True Positive Rate +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False PositiveRate1.0 +0.8 +True Positive Rate +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False PositiveRate0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +-0.4 +Real Cycle +-0.6 +Projected Cycle +0 +20 +40 +60 +80 +100 +120 +140Real Cycle +Projected Cycle14 +With Scaling +Without Scaling +Model +VGG16-3 +VAE +VGG16-3 +VAE +Accuracy +0.9468 +0.9066 +0.7862 +0.7509 +PPV +0.9525 +0.9687 +0.9763 +0.9668 +NPV +0.9047 +0.6181 +0.4038 +0.3327 +Sensitivity +0.9886 +0.9218 +0.7671 +0.7373 +Specificity +0.7215 +0.8095 +0.8945 +0.8380 +AUC +0.9709 +0.9513 +0.9067 +0.8906 +Table 2. Results of machine learning-based CVS quality assessment with and without +scale normalization. +Model +LR +MLP1 +MLP2 +VGG16-3 +VGG16-4 +Time +0.633µs +1.265µs +0.700µs +1.897µs +3.162µs +Model +VGG16-5 +PCA +VAE +CVAE +- +Time +3.562µs +48.412µs +3.703µs +15.192µs +- +Table 3. +Test inference time of machine learning-based CVS quality assessment +methods. +3.2.4. +Inference Time In real-time monitoring, assessment should be performed +quickly. The input for the proposed method was updated for every heartbeat in the +EIT system. Assuming a subject with a constant 80bpm, the CVS input is updated +every 0.75s. Roughly, the assessment should be faster than approximately 10´2s. Table +3 shows the inference time for the test data, calculated by taking the average over the +entire test data. The ML models provided a test outcome with inference times between +100µs (10´4s) and 0.1µs (10´7s). This confirms that the proposed method meets the +speed requirements for real-time monitoring. +4. Conclusion and Discussion +We developed a novel automated SQI method using two machine learning techniques, the +discriminative model and manifold learning, to detect abnormal CVS caused by motion- +induced artifacts. We discussed how body movement influences the transconductance +data and how the resulting CVS is degraded by movement. +Numerous experiments +support the idea that the proposed method can successfuly filter motion-induced +unrealistic variations in CVS data. +To the best of our knowledge, this is the first attempt to assess CVS quality to +enhance the clinical capability of an EIT-based cardiopulmonary monitoring system. +From a practical point of view, the proposed method can alert clinicians about CVS +corruption to minimize misinformation about patient safety and facilitate adequate +management of patients and medical resources. The proposed method can be combined +with a software system for existing EIT devices. +The use of only healthy subject data in the training process did not fully consider +possible influence of the subject’s illness on CVS. SQI performance might be degraded +in patients with illnesses such as arrhythmias, in which irregular deformation may occur + +15 +in CVS due to premature ventricular contraction and lead to be classified as low signal +quality. +However, when ill patient data are available and appended in the training +process, a slightly modified SQI can detect the illness and motion by adding another +label class. Meanwhile, arrhythmia can be easily detected using ECG signals. +A further collection of CVS data could be a strategy for enhancing model +generalization or stability toward being equipped with an actual monitoring system. +In discriminative models, even with additional data collection, generalization or +stability might not be meaningfully improved because the class imbalance problem +remains or increases. In contrast, the manifold-learning models can accurately infer +common features (i.e., data manifolds) as the total number of normal CVS data grows +regardless of class imbalance. In addition, it can be extended into a semi-supervised +or unsupervised learning framework [2, 46], which reduces the requirement for labeled +datasets. Thus, manifold-learning models might be favorable. +Data Availability +The data that support the findings of this study are available from the corresponding +author, K. Lee, upon reasonable request. +Acknowledgements +This work was supported by the Ministry of Trade, Industry and Energy (MOTIE) in +Korea through the Industrial Strategic Technology Development Program under Grant +20006024. Hyun was supported by Samsung Science & Technology Foundation (No. +SRFC-IT1902-09). We are deeply grateful to BiLab (Pangyo, South Korea) for their +help and collaboration. +Conflict of Interest +The authors have no conflicts to disclose. +Appendix A. Motion-induced Effect on Trans-conductance +In the 16 channel EIT system, the voltage data tV j,k +t +uj,k in (1) are governed by the +following complete electrode model [44]: At time t, the electric potential distribution + +16 +(uj +t) and electric potential on an electrode (U j,k +t ) satisfy +$ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +% +∇ ¨ pγt∇uj +tq +“ 0 +in Ω Ă R3 +γt∇uj +t ¨ n +“ 0 +on BΩz Ť16 +i Ek +ˆ +Ek γt∇uj +t ¨ n +“ 0 +for k P Iztj, j ` 1u +uj +t ` zkpγt∇uj +t ¨ nq +“ U j,k +t +on Ek for k P I +ˆ +Ej γt∇uj +t ¨ nds +“ ´ +ˆ +Ej`1 γt∇uj +t ¨ nds “ I +(A.1) +where γt is a conductivity distribution in a human chest Ω at t, n is an unit normal +vector outward BΩ, ds is a surface element, and zk is a skin-electrode contact impedance +on Ek. The amount of electric current I, which is injected to the domain Ω, can be scaled +and, thus, assumed to be I “ 1. +In the case that the human chest Ω is time-varying owing to motions, Reynolds +transport theorem yields the following approximation [39]: +9V j,k +t +« 9V j,k,normal +t +` 9V j,k,motion +t +(A.2) +where +9V j,k,normal +t +“ ´ +ˆ +Ω +9γtprq∇uj +tprq ¨ ∇uk +t prqdr +(A.3) +9V j,k,motion +t +“ ´ +ˆ +BΩ +vnpr, tqγtprq∇uj +tprq ¨ ∇uk +t prqds +(A.4) +Here, vn is an outward-normal directional velocity of BΩ and r P Ω is a position vector +in Ω. The term 9V j,k,normal +t +and 9V j,k,motion +t +can be viewed as voltage data acquirable in +normal EIT measurement and motion-induced inference, respectively. +A similar relation to (A.2) for trans-conductance can be derived as follows: Let us +define a trans-conductance-related value gj,k +t +by +gj,k +t +“ +I +RpV j,k +t +q +(A.5) +By differentiating gj,k +t +with respect to t, we obtain +9gj,k +t +“ ´IRp 9V j,k +t +q +´ +RpV j,k +t +q +¯2 « ´IpRp 9V j,k,normal +t +q ` Rp 9V j,k,motion +t +qq +´ +RpV j,k +t +q +¯2 +(A.6) +The approximation (A.6) can be expressed as +9gj,k +t +« 9gj,k,normal +t +` 9gj,k,motion +t +(A.7) +where +9gj,k,normal +t +“ ´IRp 9V j,k,normal +t +q +pRpV j,k +t +qq2 +and 9gj,k,motion +t +“ ´IRp 9V j,k,motion +t +q +pRpV j,k +t +qq2 +(A.8) + +17 +We note that, in the case of vn “ 0 in (A.4) (i.e., EIT measurement is not affected by +motions), the relation (A.6) becomes 9gj,k +t +“ 9gj,k,normal +t +by the reason of V j,k,motion +t +“ 0. +In the form of trans-conductance vector, the following approximation holds: +9gt « 9gnormal +t +` 9gmotion +t +(A.9) +where +9gnormal +t +“ +” +9g1,3,normal +t +, ¨ ¨ ¨ , 9g16,14,normal +t +ı +and 9gmotion +t +“ +” +9g1,3,motion +t +, ¨ ¨ ¨ , 9g16,14,motion +t +ı +(A.10) +If 9gnormal +t +satisfies the relation (5), we consequently obtain +9gt « 9gair +t +` 9gblood +t +` 9gmotion +t +(A.11) +Here, we note that 9gmotion +t +becomes more significant as motion (i.e., |vn| in (A.4)) is +large. +Appendix B. Machine Learning Models +Appendix B.1. Discriminative Models +Logistic Regression (LR) +A LR model fLR consists of linear transformation and sigmoid +as follows: +fLRpXq “ σpwTX ` bq +(B.1) +where w P R150 and b P R are learnable weight and bias, and σ is a sigmoid function +given by σpxq “ p1 ` exppxqq´1. +Multilayer Perceptron (MLP) +A MLP model fMLP has a hierarchical structure with +nonlinearity compared to LR. Each layer consists of linear transformation and nonlinear +activation. In our MLP models, ReLU is used in all layers except the last to avoid +gradient vanishing [20]. Table B1 shows the architectures of the MLPs used in this +study. +Convolutional Neural Network (CNN) +A CNN model fCNN consists of two paths; 1) +feature extraction and 2) classification paths. In this study, the feature extraction path +is based on VGG16 [45], as shown in Table B1. The resultant feature map is flattened +and then forwarded to the classification path, which is a MLP. +The feature extraction path is a series of two convolutional and maxpooling (or +flatten) layers, whose depth is associated with receptive field (RF) size of a unit in the +last convolutional layer [35]. According to the length of this series, VGG16-3, -4, and -5 +are defined, where 3, 4, and 5 represent the iteration number of the layers in the series. +Here, RFs are given by 32, 68, and 140, respectively. + +18 +(a) MLP1 (MLP2) +Layer +Input Dim +Output Dim +Activation +Linear +150 (150) +150 (150) +ReLU +Linear +150 (150) +300 (150) +ReLU +Linear +300 (150) +300 (100) +ReLU +Linear +300 (100) +150 (50) +ReLU +Linear +150 (50) +150 (25) +ReLU +Linear +150 (25) +150 (10) +ReLU +Linear +150 (10) +1 (1) +Sigmoid +(b) VGG16-5; [1] Feature extraction and [2] Classification networks +Layer +Input Dim +Output Dim +Kernel +Activation +RF +[1] +Conv1D +150ˆ1 +150ˆ4 +3ˆ4 +ReLU +3 +Conv1D +150ˆ4 +150ˆ4 +3ˆ4 +ReLU +5 +MaxPool1D +150ˆ4 +75ˆ4 +2 +ReLU +6 +Conv1D +75ˆ4 +75ˆ8 +3ˆ8 +ReLU +10 +Conv1D +75ˆ8 +75ˆ8 +3ˆ8 +ReLU +14 +MaxPool1D +75ˆ8 +37ˆ8 +2 +ReLU +16 +Conv1D +37ˆ8 +37ˆ16 +3ˆ16 +ReLU +24 +Conv1D +37ˆ16 +37ˆ16 +3ˆ16 +ReLU +32 +MaxPool1D +37ˆ16 +18ˆ16 +2 +ReLU +36 +Conv1D +18ˆ16 +18ˆ32 +3ˆ32 +ReLU +52 +Conv1D +18ˆ32 +18ˆ32 +3ˆ32 +ReLU +68 +MaxPool1D +18ˆ32 +9ˆ32 +2 +ReLU +76 +Conv1D +9ˆ32 +9ˆ64 +3ˆ64 +ReLU +108 +Conv1D +9ˆ64 +9ˆ64 +3ˆ64 +ReLU +140 +Flatten +9ˆ64 +576ˆ1 +- +- +- +[2] +Linear +576ˆ1 +576ˆ1 +- +ReLU +- +Linear +576ˆ1 +1ˆ1 +- +Sigmoid +- +Table B1. Network architectures; MLP and VGG16-6. +Appendix B.2. Manifold-learning Models +This subsection explains structures of an encoder E and a decoder D in (23), which were +used for the manifold-learning approach described in Section 2.3.2. The dimension of +the latent vector z was constantly set as 10 in our experiments. +Principal Component Analysis (PCA) +PCA learns principal vectors tvi P R150u10 +i“1 in +the following sense: For i “ 1, ¨ ¨ ¨ , 10, +vi “ argmax +}v}“1 +}Xiv}2 +2 and Xi “ Xi´1 ´ vi´1vT +i´1 +(B.2) +where X1 :“ rX +p1q +pos, X +p2q +pos, ¨ ¨ ¨ , X +pNposq +pos +sT. For ease of explanation, X1 is assumed to be +zero-mean. An encoder Epca and a decoder Dpca are given by +EpcapXq “ z :“ +” +xX, v1y, ¨ ¨ ¨ , xX, v10y +ı +and Dpcapzq “ +10 +ÿ +j“1 +zivi +(B.3) +where zi is i-th component of z. +Variational Auto-encoder (VAE) +Table B2 shows encoder-decoder models for VAE, +whose network architecture is based on either MLP or CNN. In VAE, z is given by the +following sampling procedure: z “ µ ` σ d znoise and znoise „ Np0, Iq, where µ and σ +are substantial outputs generated by a neural network, d is the element-wise product, + +19 +(a) VAE +Encoder +Layer +Input Dim +Output Dim +Activation +Linear +150 +125 +ReLU +Linear +125 +75 +ReLU +Linear +75 +50 +ReLU +Linear +50 +10ˆ2 +- +Sampling +10ˆ2 +10 +- +Decoder +Linear +10 +50 +ReLU +Linear +50 +75 +ReLU +Linear +75 +125 +ReLU +Linear +125 +150 +- +(b) Convolutional VAE +Encoder +Layer +Input Dim +Output Dim +Kernel +Activation +Conv1D +150ˆ1 +75ˆ8 +3ˆ8 +ReLU +Conv1D +75ˆ8 +38ˆ16 +3ˆ16 +ReLU +Conv1D +38ˆ16 +19ˆ24 +3ˆ24 +ReLU +Conv1D +19ˆ24 +10ˆ32 +3ˆ32 +ReLU +Flattening +10ˆ32 +320ˆ1 +- +- +Linear +320ˆ1 +10ˆ2 +- +- +Sampling +10ˆ2 +10ˆ1 +- +- +Decoder +Linear +10ˆ1 +320ˆ1 +- +- +Reshaping +320ˆ1 +10ˆ32 +- +- +DeConv1D +10ˆ32 +19ˆ24 +3ˆ24 +ReLU +DeConv1D +19ˆ24 +38ˆ16 +3ˆ16 +ReLU +DeConv1D +38ˆ16 +75ˆ8 +3ˆ8 +ReLU +DeConv1D +75ˆ8 +150ˆ8 +3ˆ8 +ReLU +Conv1D +150ˆ8 +150ˆ1 +1ˆ1 +ReLU +Linear +150ˆ1 +150ˆ1 +- +Table B2. VAE network architectures. +and Np0, Iq is the normal distribution of mean 0 and covariance I. Here, 0 is the zero +vector and I is the identity matrix of 10 ˆ 10. +For VAE training, the following term is added to the loss function (23): +KLpNpµ, Σq}Np0, Iqq “ 1 +2 +10 +ÿ +i“1 +pµ2 +i ` σ2 +i ´ log σi ´ 1q +(B.4) +where KL is Kullback-Leibler divergence and Σ is a 10ˆ10 diagonal matrix whose pi, iq +entry is σi. This term enables VAE to learn dense and smooth latent space embedding +in or near Np0, Iq [30,47,55]. +β-Variational Auto-encoder (β-VAE) +β-VAE differs with VAE in terms of loss function +while sharing a model architecture. For some β P R, β ˆ KL is added to the loss (23) +instead of (B.4) (i.e., VAE is the case of β “ 1). This simple weighting is known to +be advantageous on disentangled representation learning of underlying factors [24]. 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Goldberg, “Introduction to semi-supervised learning,” Synthesis lectures on +artificial intelligence and machine learning, 3(1), 1-130, 2009. + diff --git a/79AzT4oBgHgl3EQfgfxi/content/tmp_files/load_file.txt b/79AzT4oBgHgl3EQfgfxi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e84e66a91980e9e7c48746128332e5114533969 --- /dev/null +++ b/79AzT4oBgHgl3EQfgfxi/content/tmp_files/load_file.txt @@ -0,0 +1,1257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf,len=1256 +page_content='Machine Learning-based Signal Quality Assessment for Cardiac Volume Monitoring in Electrical Impedance Tomography Chang Min Hyun1, Tae Jun Jang1, Jeongchan Nam2, Hyeuknam Kwon3, Kiwan Jeon4, and Kyunghun Lee5¶ 1School of Mathematics and Computing (Computational Science and Engineering), Yonsei University, Seoul, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2BiLab, Pangyo, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3Division of Software, Yonsei University, Wonju, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 4National Institute for Mathematical Sciences, Daejeon, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 5Kyung Hee University, Seoul, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Owing to recent advances in thoracic electrical impedance tomography, a patient’s hemodynamic function can be noninvasively and continuously estimated in real-time by surveilling a cardiac volume signal associated with stroke volume and cardiac output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In clinical applications, however, a cardiac volume signal is often of low quality, mainly because of the patient’s deliberate movements or inevitable motions during clinical interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This study aims to develop a signal quality indexing method that assesses the influence of motion artifacts on transient cardiac volume signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The assessment is performed on each cardiac cycle to take advantage of the periodicity and regularity in cardiac volume changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Time intervals are identified using the synchronized electrocardiography system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We apply divergent machine- learning methods, which can be sorted into discriminative-model and manifold-learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The use of machine-learning could be suitable for our real-time monitoring application that requires fast inference and automation as well as high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the clinical environment, the proposed method can be utilized to provide immediate warnings so that clinicians can minimize confusion regarding patients’ conditions, reduce clinical resource utilization, and improve the confidence level of the monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Numerous experiments using actual EIT data validate the capability of cardiac volume signals degraded by motion artifacts to be accurately and automatically assessed in real-time by machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The best model achieved an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='95, positive and negative predictive values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='96 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='86, sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='98, specificity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='77, and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' ¶ To whom correspondence should be addressed (imlkh84@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='com) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='01469v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='SP] 4 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Introduction Over several decades, continued advances in electrical impedance tomography (EIT) have expanded the clinical capability of real-time cardiopulmonary monitoring systems by overcoming the limitations of traditional methods, such as cardiac catheterization through blood vessels [3, 8, 18, 19, 28, 29, 37, 41, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Recently, based on thoracic EIT, a patient’s hemodynamic function can be noninvasively and continuously estimated in real-time by surveilling a signal extracted using EIT, the so-called cardiac volume signal (CVS), which has a strong relationship with key hemodynamic factors such as stroke volume and cardiac output [4, 27, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In clinical applications, however, a cardiac volume signal is often of low quality, mainly because of the patient’s deliberate movements or inevitable motions during clinical interventions such as medical treatment and nursing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Because postural change causes movement of the chest boundary to which existing EIT solvers are highly sensitive owing to time-difference-reconstruction characteristics [1, 7, 32, 36, 44], motion-induced artifacts are generated in the CVS, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Motion-induced artifacts in cardiac volume monitoring using electrical impedance tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Patient’s deliberate movements or inevitable motions during clinical intervention cause severe artifacts in a cardiac volume signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' CVS extraction is to separate a cardiogenic component from the EIT voltage data, resulting from current injections at electrodes attached across a human chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In recent studies [27, 39], effective CVS extraction was successful in motion-free measurements where voltage data are mainly influenced by air and blood volume changes in the lungs, heart, and blood vessels comprehensively, but not by motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In contrast, achieving the cardiogenic component separation in motion-influenced measurements is still a long- term challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Postural changes in EIT measurements cause strong distortion of the voltage data [1, 56] and easily disturb the extraction of relatively weak cardiogenic signals [6,33,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Handling motion interference has been a huge challenge in most EIT-based techniques for enhancing clinical capability, but not researched much yet [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Adler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [1] and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [56] investigated the negative motion effect in the EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' w3 Soleimani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [43] and Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [17] proposed a motion-induced artifact reduction method by reconstructing electrode movements along with conductivity changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [36] analyzed motion artifacts in EIT measurements and proposed a subspace-based artifact rejection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [54] suggested the discrete wavelet transform- based approach that reduces motion artifacts of three specific types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' However, clinical motion artifacts are still not effectively addressed because of practical motion’s immense diversity and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Accordingly, for the time being, the EIT-based hemodynamic monitoring system attempts to be preferentially developed toward filtering motion- influenced CVSs rather than recovering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the clinical environment, this filtration can provide immediate warnings so that clinicians can minimize confusion regarding the patient’s condition, reduce clinical resource utilization, and improve the confidence level of the monitoring system [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This study aims to develop a signal quality indexing (SQI) method that assesses whether motion artifacts influence transient CVSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' To take advantage of the periodicity and regularity in cardiac volume changes, the assessment is performed on each cardiac cycle, whose time intervals are identified using the synchronized electrocardiography (ECG) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We leverage machine learning (ML), which has provided effective solutions for various biosignal-related tasks through feature disentanglement of complicated signals [5,9,14,25,34,47,48,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The use of ML could be suitable for our real-time monitoring application that requires fast inference and automation as well as high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We apply divergent ML methods, which can be sorted into discriminative-model and manifold-learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The discriminative-model approach is first considered, where an SQI map is directly trained using a paired dataset of CVS and its label [12, 22, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Although this approach provides a high performance on a fixed dataset, owing to the class imbalance problem, there is a risk of overfitting on motion-influenced CVS data in the scope of generalization or stability [10, 15, 23, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Motion artifacts can vary considerably in real circumstances, whereas collecting CVS data in numerous motion-influenced cases is practically limited because of the high cost, intensive labor, security, and ambiguity in clinical data acquisition and annotation [13, 46, 50, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' To handle this conceivable difficulty, the manifold-learning approach [2, 24, 26, 30] is examined as an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' It does not learn irregular and capricious patterns of motion-influenced CVSs and only takes advantage of the learned features from motion- free CVSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Numerous experiments have been conducted using actual EIT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Empirical results demonstrate that discriminative and manifold-learning models provide accurate and automatic detection of motion-influenced CVS in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The best discriminative model achieved an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='95, positive and negative predictive values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='96 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='86, sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='98, specificity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='77, and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The best manifold-learning model achieved accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='93, positive and negative predictive values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='97 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='71, sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='95, specificity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='80, and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The discriminative models yielded a more powerful SQI performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' in contrast, the manifold-learning models 4 provided stable outcomes between the training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Regarding to practical applications, the choice of two models relies on what should be emphasized in the monitoring system in terms of performance and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Methods 0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 16-channel system of thoracic EIT and CVS extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The EIT machine measures voltage differences by injecting currents via electrodes attached along human chest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A cardiac volume signal xt is extracted by taking suitable weighting w to the time-difference transconductance 9gt, which is defined by measured voltage data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Here, w is called as a leadforming vector, which is designed to separate a cardiogenic trans-conductance change from superposed data 9gt [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This study considers the 16-channel system of the thoracic EIT, where 16 electrodes are attached along the human chest (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The EIT system is assumed to be synchronized with the ECG system, which provides the time interval for each cardiac cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The EIT device measures a set of voltage differences by injecting an alternative current of I (mA) through pairs of adjacent electrodes while keeping all other electrodes insulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' At sampling time t, the following voltages are acquired: tV j,k t : V j,k t “ U j,k t ´ U j,k`1 t , j P I, k P Iztj, j ` 1uu (1) where I is an index set defined by I “ t1, 2, ¨ ¨ ¨ , 16u, Ek is the k-th electrode, and U j,k t is the electrical potential on Ek subject to the current injection from Ej to Ej`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For notational convenience, E0 and E17 can be understood as E16 and E1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Once the current is injected from Ej to Ej`1 for some j P I, the voltage is measured at each of the 16 adjacent electrode pairs pEk, Ek`1qkPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Among the 16 voltages, V j,j´1 t , V j,j t , and V j,j`1 t are discarded to reduce the influence of the skin-electrode contact impedance [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Because we perform 16 independent current injections, in total, 208 p“ 16ˆ13q voltages are obtained and used to produce the CVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Electrode positionElectrode positionElectrode positionCardiac Volt ume Sig gna &t5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' CVS Extraction Using EIT and Influence of Motion A transconductance (column) vector gt P R208 can be defined using the voltage data (1) as follows: gt “ „ I RpV 1,3 t q , ¨ ¨ ¨ , I RpV 1,15 t q , ¨ ¨ ¨ , I RpV 16,2 t q , ¨ ¨ ¨ I RpV 16,14 t q ȷT (2) where T represents the vector transpose and R is an operation for extracting the real part of a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Here, gt is updated every 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A CVS, denoted by xt P R, is obtained by xt “ wT 9gt (3) where w P R208 is a weighting (so-called leadforming) vector and 9gt is time difference of gt given by 9gt “ gt ´ gt0 for reference time t0 (4) In the absence of motion, the transconductance 9gt can be expressed by 9gt “ 9gair t ` 9gblood t (5) where gair t and gblood t are transconductance vectors related to air and blood volume changes in the lungs and heart, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The weighting vector w is designed to provide wT 9gt “ wTp 9gair t ` 9gblood t q “ wT 9gblood t (6) See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Kindly refer to [39] for details on determining w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Even though the cardiogenic signal gblood t is weak, it can be accurately decomposed from the data gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In light of the previous analysis in [36], the following explains why the quality of the CVS is degraded by motion, as shown in the middle part of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the presence of motion, the transconductance 9gt can be approximated by 9gt « 9gnormal t ` 9gmotion t (7) where 9gnormal t “ 9gair t ` 9gblood t and 9gmotion t is the motion-induced effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Appendix Appendix A presents details of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Determining the vector w itself can be considerably affected by motion artifacts [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Moreover, even if w satisfies (6), we have xt “ wT 9gt « xnormal t ` xmotion t (8) where xnormal t “ wT 9gblood t and xmotion t “ wT 9gmotion t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The last term xmotion t describes motion artifacts in the CVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' CVS Quality Assessment and Data Preprocessing This study aims to assess the CVS (xt) for detecting motion-induced signal quality degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This can be accomplished by developing an SQI map f : xt ÞÑ yt such that fpxtq “ yt “ # 1 if xmotion t « 0 0 if xmotion t ff 0 (9) 6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Schematic description of machine learning-based signal quality assessment for cardiac volume monitoring in electrical impedance tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' However, it is arduous to achieve (9), where the assessment is conducted on an individual CVS at every sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Instead, we take advantage of the periodicity and regularity of cardiac volume changes according to the heartbeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The time interval of each cardiac cycle is identified using a synchronized ECG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Our quality assessment is conducted on every cardiac cycle of CVS, where a cardiac cycle is defined by the time interval consisting of two consecutive ECG R-wave peaks as the end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For a given time tcyc, let the interval rtcyc, tcyc ` ∆tcycs be the corresponding cardiac cycle, where ∆tcyc is assumed to be ∆tcycle “ 10ms ˆ pv ´ 1q for some v P Nzt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Here, N denotes the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A vector gathering all CVSs during the cycle, denoted by Xtcyc P Rv, is defined as Xtcyc “ ” xtcyc, xtcyc`10ms, ¨ ¨ ¨ , xtcyc`10msˆpv´1q ıT (10) The map f in (9) can be modified into fpXtcycq “ yt “ # 1 for normal Xtcyc 0 for motion-influenced Xtcyc (11) To find f in (11), we leverage ML, which can learn the domain knowledge of normal and motion-influenced CVSs from a training dataset of N data pairs tXpiq, ypiquN i“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Prior to ML applications, the following issues need to be addressed in the CVS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' First, CVSs have significant inter-subject and intra-subject variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This is because cardiac volume varies depending on various factors, including sex, age, condition, time, and body temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Therefore, scale normalization is required to enhance the stability and performance of ML while mitigating the high learning complexity associated with scale-invariant feature extraction [20,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Second, the dimensions of the input CVS data in (11) do not match each other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', v is not constant) owing to heart rate variability [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Because most existing ML methods are based on an input with consistent dimensions, size normalization is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Figure 4 schematically illustrates the overall process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' diac C ar ycle+202107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='21 500M 73 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7 13 2726 75 97 BiLab HemoVistaSignal Cardiac Volume CVsHighcycleCVSAcguisitionHigh SQ1LoW TOScycle7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' From the monitoring system, electrocadiography and cardiac volume signals are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' By identifying a cardiac cycle through electrocadiography data (R-wave peak detection), we extract cardiac volume signals at the corresponding cycle and then lastly apply normalization in terms of scale and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Scale normalization A simple method of normalizing the scale is to rescale the CVS data for individual cardiac cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Specifically, for a given CVS vector Xtcyc P Rv, the scaling factor S is obtained using S “ max iPV |xtcyc`10ˆi(ms)| (12) where the index set V is given by V “ t0, 1, ¨ ¨ ¨ , v ´ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Normalized CVS data, denoted by Xtcyc, are obtained by Xtcyc “ Xtcyc S (13) However, this scaling may not be appropriate to our application for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Abnormalities in CVS data include sudden increases or decreases in signal amplitude as well as irregular deformations of the shape profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The normalization in (13) can contribute to ignoring rapid amplitude changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This study uses the following subject-specific scale normalization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' When the EIT device is used to monitor a certain subject, it is supposed that during the initial 20s calibration process, the device measures the normal CVS data available for scale normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Let X subject be a set of corresponding CVSs given by X subject “ tx10ˆi(ms) : i “ ´1999, ´1998, ¨ ¨ ¨ , ´1u (14) Using the set X subject, a subject-specific scaling factor Ssubject is obtained by Ssubject “ max xPXsubject|x| (15) This scale factor Ssubject is used for the normalization in (13) instead of the naive factor S in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' C Vol Si ardiac ume gnalectrocardiography Si gnaardiac CycleAt ycle1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75 0 50 100 150tX 1f cyc202107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='21 500M 73 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7 13 2726 75 97 BiLab HemoVista8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Size normalization To make the dimensions of the CVS data consistent, a CVS vector Xtcyc is embedded into Rν for a fixed constant ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the empirical experiment, the embedding space dimension was to be larger than any dimension of the CVS data in our dataset (ν “ 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Two normalization methods are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The first approach is to resample ν points using linear interpolation with v data points in Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For the stationary interval r0, 1s, the following linear interpolation function L is constructed: Lp i v ´ 1q “ xtcyc`10(ms)ˆpi´1q for i “ 0, ¨ ¨ ¨ , v ´ 1 (16) Subsequently, we obtain the normalized vector Xtcyc P Rν using Xtcyc “ „ Lp0q, Lp 1 ν ´ 1q, Lp 2 ν ´ 1q, ¨ ¨ ¨ , Lp1q ȷT (17) This method normalizes the signal profile of CVS data into the desired length (ν) with no significant loss, but loses sampling time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Second, the last value in Xtcyc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', xtcyc`10(ms)ˆpv´1q) is padded up to the desired length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This constant padding provides a vector Xtcyc P Rν, expressed by Xtcyc “r xtcyc, ¨ ¨ ¨ , xtcyc`10(ms)ˆpv´2q, xtcyc`10(ms)ˆpv´1q, (18) xtcyc`10(ms)ˆpv´1q, ¨ ¨ ¨ , xtcyc`10(ms)ˆpv´1q sT (19) where the part (19) corresponds to the padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In contrast to the first method, this normalization can preserve time information regarding sampling frequency, whereas the core profile of the CVS is supported at different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Machine Learning Application At this point, we are ready to apply ML for determining the SQI function (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Collected from various subjects and cardiac cycles, the following dataset is used: tX piq, ypiquN i“1 (20) where ypiq is the SQI label corresponding to X piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We note that X is the CVS data for a cardiac cycle of some subjects and is normalized for both scale and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In practice, the available training dataset (20) was highly imbalanced, where there were relatively few negative samples (motion-influenced CVSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Discriminative-model approach The discriminative-model approach trains the SQI map f : X ÞÑ y in the following sense: f “ argmin fPF 1 N N ÿ i“1 distpfpX piqq, ypiqq (21) 9 (b) Manifold-learning Approach (a) Discriminative-model Approach Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' (a) Discriminative-model approach learns a signal quality indexing map f by using CVS and label data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' (b) Manifold-learning approach first learns common features of normal CVS data by finding a low dimensional manifold Mnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Signal quality assessment is based on computing the residual between original CVS data and projected one onto or near the learned manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' where F is a set of learnable functions for a given ML model and dist is a metric that measures the difference between the ML output fpXq and label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' See Figure 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In our application with high class-imbalance, the following weighted cross-entropy can be used: distpfpXq, yq “ ´ζposylogpfpXqq ´ ζnegp1 ´ yqlogp1 ´ fpXqq (22) where ζpos and ζneg are the relative ratios of the positive and negative samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Various classification models can be used, such as the logistic regression model (LR) [12], multi-layer perceptron (MLP) [22], and convolutional neural networks (CNN) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Detailed models used in this study are explained in Appendix Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The discriminative model approach is a powerful method to guarantee high performance in a fixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' However, it might suffer from providing stable SQI results in clinical practice because of highly variable negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This is because these methods take advantage of learned information using only a few negative samples [10, 15, 23, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' To achieve stable prediction, the manifold-learning approach can be alternatively used [13,50,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Manifold-learning approach The manifold-learning approach learns common features from positive samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', normal CVS) and uses them to develop an SQI map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The remaining negative samples are utilized as auxiliary means for selecting a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Figure 5 (b) shows a schematic description of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Not uisectrainingP rojectionCITraininganifoldmrma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' eso ta10 A set of positive samples is denoted by tX piq posu Npos i“1 , where Npos denotes the number of positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the first step, we learn a low-dimensional representation of Xpos by training an encoder E : Xpos ÞÑ z and decoder D : z ÞÑ Xpos in the following sense [21,26]: pD, Eq “ argmin pD,Eq 1 Npos Npos ÿ i“1 }D ˝ EpX piq posq ´ X piq pos}2 2 (23) where z is a low dimensional latent vector and } ¨ }2 is the standard Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The architectures D and E can be used in PCA [26], VAE [30], and β-VAE [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' See more details in Appendix Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Borrowing the idea from [2], an SQI map f is constructed as follows: For a given CVS data X in any class, a residual r is computed by r “ }X ´ D ˝ EpXq}2 (24) The decoder D is trained to generate normal CVS-like output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In other words, operation D˝E transforms X to lie in or near the learned manifold using normal CVS data [44,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Therefore, the residual r can be viewed as an anomaly score, where r is small if X is normal CVS data, and large if X is motion-influenced CVS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For some non-negative constant d, an SQI map f can be constructed using fpXtq “ # 1 if r ď d 0 if r ą d (25) The remainder of this subsection explains how the thresholding value d is determined by utilizing negative samples as well as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' By varying d from 0 to 8, a receiver operating characteristic (ROC) curve is calculated, where a point in the ROC curve is obtained using a fixed d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We choose d such that maximizing Youden’s J statistics, which is known as an unbiased metric in the class imbalance case [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The value J is given by J d “ Sensitivityd ` Specificityd ´ 1 (26) where Sensitivityd “ N d TP N d TP ` N d FN and Specificityd “ N d TN N d TN ` N d FP (27) Here, N d TP, N d TN, N d FP, and N d FN respectively represent the number of true positives, true negatives, false positives, and false negatives for predictions depending on a selected threshold value d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Data Acquisition and Experimental Setting Our dataset was obtained from healthy volunteers using an EIT-based hemodynamic monitoring device (HemoVista, BiLab, South Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Synchronized ECG data were 11 obtained with EIT and used to identify the cardiac cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' While lying in a hospital bed, each subject was requested to make intentional motions mimicking postural changes in the clinical ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A total of 16140 CVS data were obtained regarding the cardiac cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Manual labeling was individually performed by two- and ten- years bio-signal experts (Nam and Lee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Subsequently, they reviewed the results and made the final decision about CVS abnormality through an agreement between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The final labels were annotated into three classes: normal, ambiguous, and motion-influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' When classified as normal or abnormal by both experts with an agreement, CVS data were annotated as normal or motion-influenced classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The ambiguous class stands for CVS data in which motion artifacts were included with high possibility, but the experts did not reach an explicit agreement about motion influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The assigned label is y “ 1 for the normal class and y “ 0 for the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' As a result, 12928 (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='09%), 1526 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='45%), and 1686 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='45%) samples were labeled as normal, ambiguous, and motion- influenced classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For ML applications, a total of 16372 CVS data were divided into 13100 (80%), 1520 (10%), and 1520 (10%), which were used for training, validation, and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The data split was performed such that CVS data obtained from a common subject did not exist between the three sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For the training dataset, labels for the ambiguous class were reassigned to y “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This was done to prevent the over- classification of ambiguous classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' ML experiments were conducted in a computer system with GeForce RTX 3080 Ti, Intel® Core™ X-series Processors i9-10900X, and 128GB DDR4 RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Python with scikit-learn and Pytorch packages were used for the ML implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' When training the ML models, the Adam optimizer was consistently employed, which is an effective adaptive stochastic gradient descent method [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Hyperparameters such as epoch and learning rate were heuristically chosen based on the validation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Results of CVS Quality Assessment We compared the performance of the ML-based CVS quality assessment results by using six metrics: accuracy, positive and negative predictive values (PPV and NPV), sensitivity, specificity, and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Accuracy, PPV, and NPV were defined by Accuracy “ NTP ` NTN NTP ` NTN ` NFP ` NFN , PPV “ NTP NTP ` NFP , and NPV “ NTN NTN ` NFN (28) and AUC was the area under the ROC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' NPV, specificity, and AUC should be emphasized in our evaluation owing to the high-class imbalance (small negative samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Discriminative Models The first and second rows of Tables 1 (a) and (b) show the quantitative evaluations of CVS quality assessment using various discriminative models: LR, MLPs, and CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The results in Tables 1 (a) and (b) differ in size normalization: (a) linear interpolation and (b) constant padding.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8861 PPV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9636 NPV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5467 Sensitivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9029 Specificity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7745 AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9041 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Machine learning-based CVS quality assessment results MLPs and CNNs performed better than LR, which provided miserable NPV and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' MLPs and CNNs outperformed each other in specificity and NVP respectively, while achieving comparable levels for the other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' There was no significant performance gap depending on the size normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' One interesting observation was as follows: In our experiments, there seems to be a compensation between specificity and NPV, depending on the emphasis on locality and globality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Enriching global information on CVS data positively affected specificity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' in contrast, local information helped improve NPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' As the receptive field size in VGG16 increased (see Appendix Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1), specificity tended to increase and NPV decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In MLP, which is more flexible for catching global information than CNNs, specificity was highest, and NPV lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In other words, the local information of CVS data is likely to play a crucial role in reducing false negatives rather than false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' From a practical point of view, reducing false negatives is more desirable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' therefore, using VGG16-3 or VGG16-4, which have the powerful ability to take advantage of locality, can be an excellent option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Manifold-learning Models Positive samples in the validation set were used for hyperparameter selection in training the encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A threshold value was determined by using data from all the training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Figure 6 shows manifold projection results of test samples in normal and motion- 13 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Test samples and VAE-based projection results for (a) normal and (b) motion-influenced CVS data, where the red line is original CVS data and the blue line is the correspondent CVS data projected by VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' By the way, ROC curves for (c) VGG 16-4 and (d) β-VAE are provided, where the blue and red lines correspond to the curves calculated using training and test sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' influenced classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' An input CVS is projected onto or near a manifold learned by positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' As desired, the residual (24) tends to be small for normal samples and high for motion-influenced samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The third and fourth rows of Tables 1 (a) and (b) show the final assessment results using manifold-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The performance was comparable to that of discriminative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We note that the manifold-learning models never learned negative samples for classifier development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' As shown in Figure 6 (d), the manifold- learning model’s performance gap between training and test sets was very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' There was a slight difference in performance for the manifold-learning models depending on the size normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Linear interpolation promised a slightly better assessment of accuracy, NPV, and AUC than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For the case of constant padding, because core profiles of CVS data are supported at different intervals, the learning complexity can be increased, which is associated with invariant feature extraction to the intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This may cause a slight drop in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In our dataset, both discriminative and manifold learning models provided accurate detection of motion-influenced CVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The discriminative model yielded a more powerful SQI performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' in contrast, the manifold-learning model provided stable outcomes between the training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Regarding practical applications, the choice of two models relies on what should be emphasized in the monitoring system in terms of performance and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Their ensemble is also worth considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Impact of Scale Normalization Table 2 shows the worst case when scale normalization was not applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In CNNs, network training was very unstable, and assessment performance was considerably degraded, especially regarding accuracy, NPV, sensitivity, and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In VAEs, large-scale variability of CVS data highly affected the loss of accuracy in manifold projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' therefore, the performance significantly deteriorated in terms of accuracy, NPV, sensitivity, and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This verifies the impact of scale normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3 - Real Cycle Projected Cycle 2 1 0- 1 0 20 40 60 80 100 120 1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8 True Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 False PositiveRate1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8 True Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 False PositiveRate0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4 Real Cycle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6 Projected Cycle 0 20 40 60 80 100 120 140Real Cycle Projected Cycle14 With Scaling Without Scaling Model VGG16-3 VAE VGG16-3 VAE Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7509 PPV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9668 NPV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3327 Sensitivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7373 Specificity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8380 AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8906 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Results of machine learning-based CVS quality assessment with and without scale normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Model LR MLP1 MLP2 VGG16-3 VGG16-4 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='633µs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='265µs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='700µs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='897µs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='162µs Model VGG16-5 PCA VAE CVAE Time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='562µs 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='412µs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='703µs 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='192µs Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Test inference time of machine learning-based CVS quality assessment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Inference Time In real-time monitoring, assessment should be performed quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The input for the proposed method was updated for every heartbeat in the EIT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Assuming a subject with a constant 80bpm, the CVS input is updated every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Roughly, the assessment should be faster than approximately 10´2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Table 3 shows the inference time for the test data, calculated by taking the average over the entire test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The ML models provided a test outcome with inference times between 100µs (10´4s) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1µs (10´7s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This confirms that the proposed method meets the speed requirements for real-time monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Conclusion and Discussion We developed a novel automated SQI method using two machine learning techniques, the discriminative model and manifold learning, to detect abnormal CVS caused by motion- induced artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We discussed how body movement influences the transconductance data and how the resulting CVS is degraded by movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Numerous experiments support the idea that the proposed method can successfuly filter motion-induced unrealistic variations in CVS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' To the best of our knowledge, this is the first attempt to assess CVS quality to enhance the clinical capability of an EIT-based cardiopulmonary monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' From a practical point of view, the proposed method can alert clinicians about CVS corruption to minimize misinformation about patient safety and facilitate adequate management of patients and medical resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The proposed method can be combined with a software system for existing EIT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The use of only healthy subject data in the training process did not fully consider possible influence of the subject’s illness on CVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' SQI performance might be degraded in patients with illnesses such as arrhythmias, in which irregular deformation may occur 15 in CVS due to premature ventricular contraction and lead to be classified as low signal quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' However, when ill patient data are available and appended in the training process, a slightly modified SQI can detect the illness and motion by adding another label class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Meanwhile, arrhythmia can be easily detected using ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A further collection of CVS data could be a strategy for enhancing model generalization or stability toward being equipped with an actual monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In discriminative models, even with additional data collection, generalization or stability might not be meaningfully improved because the class imbalance problem remains or increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In contrast, the manifold-learning models can accurately infer common features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', data manifolds) as the total number of normal CVS data grows regardless of class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In addition, it can be extended into a semi-supervised or unsupervised learning framework [2, 46], which reduces the requirement for labeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Thus, manifold-learning models might be favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Data Availability The data that support the findings of this study are available from the corresponding author, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Lee, upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Acknowledgements This work was supported by the Ministry of Trade, Industry and Energy (MOTIE) in Korea through the Industrial Strategic Technology Development Program under Grant 20006024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Hyun was supported by Samsung Science & Technology Foundation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' SRFC-IT1902-09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We are deeply grateful to BiLab (Pangyo, South Korea) for their help and collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Motion-induced Effect on Trans-conductance In the 16 channel EIT system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' the voltage data tV j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='k t uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='k in (1) are governed by the following complete electrode model [44]: At time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' the electric potential distribution 16 (uj t) and electric potential on an electrode (U j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='k t ) satisfy $ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ % ∇ ¨ pγt∇uj tq “ 0 in Ω Ă R3 γt∇uj t ¨ n “ 0 on BΩz Ť16 i Ek ˆ Ek γt∇uj t ¨ n “ 0 for k P Iztj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' j ` 1u uj t ` zkpγt∇uj t ¨ nq “ U j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='k t on Ek for k P I ˆ Ej γt∇uj t ¨ nds “ ´ ˆ Ej`1 γt∇uj t ¨ nds “ I (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1) where γt is a conductivity distribution in a human chest Ω at t, n is an unit normal vector outward BΩ, ds is a surface element, and zk is a skin-electrode contact impedance on Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The amount of electric current I, which is injected to the domain Ω, can be scaled and, thus, assumed to be I “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the case that the human chest Ω is time-varying owing to motions, Reynolds transport theorem yields the following approximation [39]: 9V j,k t « 9V j,k,normal t ` 9V j,k,motion t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2) where 9V j,k,normal t “ ´ ˆ Ω 9γtprq∇uj tprq ¨ ∇uk t prqdr (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3) 9V j,k,motion t “ ´ ˆ BΩ vnpr, tqγtprq∇uj tprq ¨ ∇uk t prqds (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4) Here, vn is an outward-normal directional velocity of BΩ and r P Ω is a position vector in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The term 9V j,k,normal t and 9V j,k,motion t can be viewed as voltage data acquirable in normal EIT measurement and motion-induced inference, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' A similar relation to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2) for trans-conductance can be derived as follows: Let us define a trans-conductance-related value gj,k t by gj,k t “ I RpV j,k t q (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='5) By differentiating gj,k t with respect to t, we obtain 9gj,k t “ ´IRp 9V j,k t q ´ RpV j,k t q ¯2 « ´IpRp 9V j,k,normal t q ` Rp 9V j,k,motion t qq ´ RpV j,k t q ¯2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6) The approximation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6) can be expressed as 9gj,k t « 9gj,k,normal t ` 9gj,k,motion t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='7) where 9gj,k,normal t “ ´IRp 9V j,k,normal t q pRpV j,k t qq2 and 9gj,k,motion t “ ´IRp 9V j,k,motion t q pRpV j,k t qq2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='8) 17 We note that, in the case of vn “ 0 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', EIT measurement is not affected by motions), the relation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='6) becomes 9gj,k t “ 9gj,k,normal t by the reason of V j,k,motion t “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In the form of trans-conductance vector, the following approximation holds: 9gt « 9gnormal t ` 9gmotion t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9) where 9gnormal t “ ” 9g1,3,normal t , ¨ ¨ ¨ , 9g16,14,normal t ı and 9gmotion t “ ” 9g1,3,motion t , ¨ ¨ ¨ , 9g16,14,motion t ı (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='10) If 9gnormal t satisfies the relation (5), we consequently obtain 9gt « 9gair t ` 9gblood t ` 9gmotion t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='11) Here, we note that 9gmotion t becomes more significant as motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', |vn| in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4)) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Machine Learning Models Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Discriminative Models Logistic Regression (LR) A LR model fLR consists of linear transformation and sigmoid as follows: fLRpXq “ σpwTX ` bq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1) where w P R150 and b P R are learnable weight and bias, and σ is a sigmoid function given by σpxq “ p1 ` exppxqq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Multilayer Perceptron (MLP) A MLP model fMLP has a hierarchical structure with nonlinearity compared to LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Each layer consists of linear transformation and nonlinear activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In our MLP models, ReLU is used in all layers except the last to avoid gradient vanishing [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Table B1 shows the architectures of the MLPs used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Convolutional Neural Network (CNN) A CNN model fCNN consists of two paths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 1) feature extraction and 2) classification paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In this study, the feature extraction path is based on VGG16 [45], as shown in Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The resultant feature map is flattened and then forwarded to the classification path, which is a MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The feature extraction path is a series of two convolutional and maxpooling (or flatten) layers, whose depth is associated with receptive field (RF) size of a unit in the last convolutional layer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' According to the length of this series, VGG16-3, -4, and -5 are defined, where 3, 4, and 5 represent the iteration number of the layers in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Here, RFs are given by 32, 68, and 140, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' 18 (a) MLP1 (MLP2) Layer Input Dim Output Dim Activation Linear 150 (150) 150 (150) ReLU Linear 150 (150) 300 (150) ReLU Linear 300 (150) 300 (100) ReLU Linear 300 (100) 150 (50) ReLU Linear 150 (50) 150 (25) ReLU Linear 150 (25) 150 (10) ReLU Linear 150 (10) 1 (1) Sigmoid (b) VGG16-5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [1] Feature extraction and [2] Classification networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Input Dim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Output Dim ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9ˆ64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3ˆ64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Flatten ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9ˆ64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='576ˆ1 [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='576ˆ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='576ˆ1 ReLU Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='576ˆ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='1ˆ1 Sigmoid Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Network architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' MLP and VGG16-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Manifold-learning Models This subsection explains structures of an encoder E and a decoder D in (23), which were used for the manifold-learning approach described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' The dimension of the latent vector z was constantly set as 10 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Principal Component Analysis (PCA) PCA learns principal vectors tvi P R150u10 i“1 in the following sense: For i “ 1, ¨ ¨ ¨ , 10, vi “ argmax }v}“1 }Xiv}2 2 and Xi “ Xi´1 ´ vi´1vT i´1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='2) where X1 :“ rX p1q pos, X p2q pos, ¨ ¨ ¨ , X pNposq pos sT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For ease of explanation, X1 is assumed to be zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' An encoder Epca and a decoder Dpca are given by EpcapXq “ z :“ ” xX, v1y, ¨ ¨ ¨ , xX, v10y ı and Dpcapzq “ 10 ÿ j“1 zivi (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='3) where zi is i-th component of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Variational Auto-encoder (VAE) Table B2 shows encoder-decoder models for VAE, whose network architecture is based on either MLP or CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' In VAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' z is given by the following sampling procedure: z “ µ ` σ d znoise and znoise „ Np0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Iq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' where µ and σ are substantial outputs generated by a neural network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' d is the element-wise product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='(a) VAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Input Dim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Output Dim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='10ˆ2 Sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='10ˆ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='10 Decoder ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='125 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='150ˆ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='150ˆ1 Table B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' VAE network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' and Np0, Iq is the normal distribution of mean 0 and covariance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Here, 0 is the zero vector and I is the identity matrix of 10 ˆ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For VAE training, the following term is added to the loss function (23): KLpNpµ, Σq}Np0, Iqq “ 1 2 10 ÿ i“1 pµ2 i ` σ2 i ´ log σi ´ 1q (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4) where KL is Kullback-Leibler divergence and Σ is a 10ˆ10 diagonal matrix whose pi, iq entry is σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This term enables VAE to learn dense and smooth latent space embedding in or near Np0, Iq [30,47,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' β-Variational Auto-encoder (β-VAE) β-VAE differs with VAE in terms of loss function while sharing a model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' For some β P R, β ˆ KL is added to the loss (23) instead of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='4) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=', VAE is the case of β “ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' This simple weighting is known to be advantageous on disentangled representation learning of underlying factors [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' We determined an optimal β as the empirical best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Table B3 showed SQI performance variation about β in the dataset where the scale and size normalization using linear interpolation were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Adler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Guardo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Berthiaume, “Impedance imaging of lung ventilation: Do we need to account for chest expansion?”' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content='9471 Table B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' β-VAE performance comparison about varying β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' An, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} +page_content=' Cho, “Variational autoencoder based anomaly detection using reconstruction probability,” Special Lecture on IE, 2(1), 1-18, 2015.' 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semi-supervised learning,” Synthesis lectures on artificial intelligence and machine learning, 3(1), 1-130, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfgfxi/content/2301.01469v1.pdf'} diff --git a/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/2301.13250v1.pdf.txt b/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/2301.13250v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6fdca55e69c87e7b356f179a50e48df36878a66 --- /dev/null +++ b/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/2301.13250v1.pdf.txt @@ -0,0 +1,2346 @@ +.Draft version February 1, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Molecular gas and star formation in nearby starburst galaxy mergers +Hao He,1 Connor Bottrell,2 Christine Wilson,1 Jorge Moreno,3 Blakesley Burkhart,4, 5 +Christopher C. Hayward,5 Lars Hernquist,6 and Angela Twum3 +1McMaster University +1280 Main St W, Hamilton, ON L8S 4L8, CAN +2Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, University of Tokyo +Kashiwa, Chiba 277-8583, Japan +3Department of Physics and Astronomy, Pomona College, +Claremont, CA 91711, USA +4Department of Physics and Astronomy, Rutgers University, +136 Frelinghuysen Rd., Piscataway, NJ 08854, USA +5Center for Computational Astrophysics, Flatiron Institute, +162 Fifth Avenue, New York, NY 10010, USA +6Center for Astrophysics, Harvard & Smithsonian, +60 Garden Street, Cambridge, MA 02138, USA +(Received February 1, 2023; Revised xxx; Accepted xxx) +Submitted to ApJ Letter +ABSTRACT +We employ the Feedback In Realistic Environments (FIRE-2) physics model to study how the prop- +erties of giant molecular clouds (GMCs) evolve during galaxy mergers. We conduct a pixel-by-pixel +analysis of molecular gas properties in both the simulated control galaxies and galaxy major mergers. +The simulated GMC-pixels in the control galaxies follow a similar trend in a diagram of velocity disper- +sion (σv) versus gas surface density (Σmol) to the one observed in local spiral galaxies in the Physics at +High Angular resolution in Nearby GalaxieS (PHANGS) survey. For GMC-pixels in simulated mergers, +we see a significant increase of factor of 5 – 10 in both Σmol and σv, which puts these pixels above the +trend of PHANGS galaxies in the σv vs Σmol diagram. This deviation may indicate that GMCs in the +simulated mergers are much less gravitationally bound compared with simulated control galaxies with +virial parameter (αvir) reaching 10 – 100. Furthermore, we find that the increase in αvir happens at +the same time as the increase in global star formation rate (SFR), which suggests stellar feedback is +responsible for dispersing the gas. We also find that the gas depletion time is significantly lower for +high αvir GMCs during a starburst event. This is in contrast to the simple physical picture that low +αvir GMCs are easier to collapse and form stars on shorter depletion times. This might suggest that +some other physical mechanisms besides self-gravity are helping the GMCs in starbursting mergers +collapse and form stars. +Keywords: ISM: clouds, ISM: kinematics and dynamics, ISM: structure, galaxies: interactions, galaxies: +starburst, galaxies: star formation +1. INTRODUCTION +Corresponding author: Hao He +heh15@mcmaster.ca +Despite the diversity of galaxy morphology and envi- +ronment, giant molecular clouds (GMCs) are the sites of +star formation across cosmic time (Krumholz et al. 2019; +Chevance et al. 2020). As one of the most promising +star formation model, the turbulence model (Krumholz +& McKee 2005; Hennebelle & Chabrier 2011) suggest a +arXiv:2301.13250v1 [astro-ph.GA] 30 Jan 2023 + +2 +He et al. +relatively uniform star formation efficiency per freefall +time (ϵff) for individual GMCs. They predict that the +observed scatter in ϵff could be account for by the diver- +sity in GMC properties (e.g.virial parameter αvir and +Mach number). However, Lee et al. (2016) show that +the observed scatter is larger than these early theoret- +ical predictions expected and updated models suggest +that cloud evolution, in addition to initial conditions +such as Mach number and αvir, should be accounted for +(see Burkhart 2018; Mocz & Burkhart 2018; Burkhart +& Mocz 2019). Furthermore, Grudi´c et al. (2018) show +in their simulation that GMCs in starburst galaxies can +have different ϵff in normal spiral galaxies. Hence, to un- +derstand the links between GMCs and star formation in +galaxies, it is essential to study various GMC properties +in a broad range of environments. +However, modeling of GMCs starting from the scales +of galaxies and cosmological zoom-ins is complicated by +challenges in capturing the structure of the coldest and +densest gas, which is heavily affected by various numeri- +cal choices, such as resolution (e.g. Bournaud et al. 2008; +Teyssier et al. 2010) and the treatment of feedback (Fall +et al. 2010; Murray et al. 2010; Dale et al. 2014; My- +ers et al. 2014; Raskutti et al. 2016; Kim et al. 2017; +Grudi´c et al. 2018; Smith et al. 2021). Most resolved +GMC simulations focus on the evolution of individual +GMCs (e.g. Burkhart et al. 2015; Howard et al. 2018; +Li et al. 2019; Decataldo et al. 2020; Burkhart et al. +2020) and ignore the wider environment. Only a hand- +ful of galaxy simulations have the ability to model GMC +populations inside Milky-Way-like galaxies (Jeffreson & +Kruijssen 2018; Benincasa et al. 2020) and mergers (Re- +naud et al. 2019a; Li et al. 2022). +High-resolution CO observations have successfully +characterized GMCs in the Milky Way (e.g. Rice et al. +2016; Rico-Villas et al. 2020; Miville-Deschˆenes et al. +2017; Colombo et al. 2019; Lada & Dame 2020) and +nearby galaxies (e.g. Donovan Meyer et al. 2013; Hughes +et al. 2013; Colombo et al. 2014; Leroy et al. 2016; +Schruba et al. 2019). In particular, the recently com- +pleted PHANGS-ALMA survey (Leroy et al. 2021) has +expanded these observations across a complete sample +of nearby spiral galaxies, providing direct measurements +of molecular gas surface density Σmol, velocity disper- +sion σv and size of GMCs, which are key quantities for +determining the physical state of GMCs (Larson 1981). +Observations show that the correlation between σ2 +v/R +and Σmol is nearly linear (e.g., Heyer & Dame 2015; Sun +et al. 2018, 2020), which is consistent with the theoreti- +cal prediction that most clouds follows the Larson’s sec- +ond law (Larson 1981), which indicates a constant ratio +between clouds’ kinetic energy and gravitational poten- +tial energy. This universal correlation provides us with +a starting point to study how other galactic environ- +mental factors (e.g., external pressure, stellar potential) +influence the dynamical state of GMCs. +Unlike studies targeting isolated galaxies, GMCs in +starburst galaxy mergers are less well studied. On the +observational side, the scarcity of nearby mergers means +that we have only a handful of systems with GMC res- +olution data (Wei et al. 2012; Ueda et al. 2012; Whit- +more et al. 2014; Elmegreen et al. 2017; Brunetti et al. +2020; Brunetti 2022; S´anchez-Garc´ıa et al. 2022; Bel- +locchi et al. 2022). These studies show that GMCs in +mergers have significantly higher gas surface densities +and are less gravitationally bound compared to GMCs +in normal spirals. However, it is difficult to draw sta- +tistically robust conclusions on how GMC properties +evolve across various merging stages based on these lim- +ited number of local galaxy mergers. On the simulation +front, only a handful of studies (e.g., Teyssier et al. 2010; +Renaud et al. 2014; Fensch et al. 2017) have the ability +to probe the cold gas at ∼pc scale starting from cosmo- +logical scales. Using a comprehensive library of idealized +galaxy merger simulations based on the FIRE-2 physics +model, Moreno et al. (2019) show that SFR enhance- +ment is accompanied by an increase in the cold dense +gas reservoir. +This simulation suite thus provides us +with the ideal tool to properly examine GMC evolution +along the entire merging sequence. +This paper explores how GMC properties evolve dur- +ing the starburst merging event using the FIRE-2 merger +suite from Moreno et al. (2019) and performs compar- +isons with observations to test the simulation model. +In Section 2, we describe this simulation suite and the +observational data used for comparison. Section 3 com- +pares the σv −Σmol relation between control simulated +galaxies and normal spirals in the PHANGS-ALMA +sample. Section 4 examines the σv −Σmol relation for +mergers in both observations and simulations. In Sec- +tion 5, we discuss and interpret various aspects of the +comparison between observations and simulations. +2. DATA PROCESSING +2.1. Simulated data +2.1.1. The FIRE-2 Model +We use the FIRE-2 model (Hopkins et al. 2018), which +employs the hydrodynamic code GIZMO (Hopkins 2015, +2017). +Compared with the previous version, FIRE-2 +adopts the updated meshless finite-mass (MFM) mag- +netohydrodynamics (MHD) solver, which is designed to +capture the advantages of both grid-based and particle- +based methods. We refer the reader to Hopkins (2015) +and Hopkins et al. (2018) for details. The model includes + +GMCs in Galaxy Mergers +3 +treatment of radiative heating and cooling from free- +free, photo-ionization/recombination, Compton, photo- +electric, dust-collisional, cosmic ray, molecular, metal +line, and fine-structure processes. Star formation occurs +in gas that is self-gravitating (3D αvir < 1 at the reso- +lution scale), self-shielded, and denser than 1000 cm−3 +(see Appendix C of Hopkins et al. 2018). Stellar feed- +back mechanisms include (i) mass, metal, energy, and +momentum flux from supernovae type Ia & II; (ii) con- +tinuous stellar mass-loss through OB/AGB winds; (iii) +photoionization and photoelectric heating; and (iv) radi- +ation pressure. Each stellar particle is treated as a single +stellar population. +Mass, age, metallicity, luminosity, +energy, mass-loss rate, and stellar feedback event rate +for each stellar particle are calculated using the STAR- +BURST99 stellar population synthesis model (Leitherer +et al. 1999). The model does not account for feedback +generated via accretion of gas onto a supermassive black +hole (SMBH). +2.1.2. Our FIRE-2 galaxy suite +Moreno et al. (2019) present a suite of idealized galaxy +merger simulations (Initial conditions are manually set +instead of from cosmological simulations; see also Bot- +trell et al. 2019; Moreno et al. 2021; McElroy et al. 2022) +covering a range of orbital parameters and mass ratios +between 4 disc galaxies (G1, G2, G3 and G4, in or- +der of increasing total stellar mass of (0.21, 1.24, 2.97 +and 5.5×1010 M⊙), along with separate runs for each +disk galaxy in isolation (the control runs). Their orbit +settings contain 3 orbital spin directions, 3 impact pa- +rameters and 3 impact velocities (see Fig. 3 in Moreno +et al. 2019). For these simulations, the highest gas den- +sity and spatial resolution are 5.8 × 105 cm−3 and 1.1 +pc, respectively. The gravitational softening lengths are +10 pc for the dark matter and stellar components and 1 +pc for the gaseous component. The mass resolution for +a gas particle is 1.4 ×104 M⊙. The time resolution of a +typical snapshot is 5 Myr (See further details in Moreno +et al. 2019). +Table 1. Orbital Parameter of ‘e1’ and ‘e2’ orbit +e1 +e2 +Apo. Dist. (kpc)a +60 +120 +Peri. Dist. (kpc)a +15.5 +9.3 +a First apocentric distance between the +centers of two galaxies. +b Second pericentric distance between +the centers of two galaxies. +For our analysis, we focus on the simulation run of +isolated G2 and G3 galaxies along with one of G2&G3 +merger suites. The detailed information of G2 and G3 +galaxies is in Moreno et al. (2019, Table 2). The G2&G3 +merger suites have a mass ratio of 1:2.5 and hence are +similar to major mergers such as the Antennae and NGC +3256 for which we have observational data. In addition, +G2 and G3 have stellar masses within the range of the +PHANGS sample (1010–1011 M⊙; Leroy et al. 2021). +We choose the ‘e’ orbit (Robertson et al. 2006, roughly +prograde), which is expected to maximally enhance the +star formation rate. In most of our analyses, we focus +on the ‘e2’ orbit since this is the fiducial run in Moreno +et al. (2019). +We use the ‘e1’ orbit as a comparison +in some cases as it has smaller impact parameter and is +more similar to the orbit of the Antennae merger (Privon +et al. 2013), for which we have GMC observational data. +The pericentric distance of ‘e1’ and ‘e2’ orbit is listed in +Table 1. +2.1.3. Molecular gas +We follow the scheme in Moreno et al. (2019) to sepa- +rate the ISM of our simulated galaxy mergers into 4 com- +ponents based on temperature and density: hot, warm, +cool, and cold-dense gas, which roughly correspond to +the hot, ionized, atomic, and molecular gas in observa- +tions (see Table 4 in Moreno et al. 2019). +The com- +ponents that are most important for this work are the +cool (temperatures below 8000 K and densities above +0.1 cm−3) and the cold-dense gas (temperatures below +300 K and densities above 10 cm−3), which corresponds +to H I and H2. +This choice captures HI and H2 gas +reasonably well (Orr et al. 2018). Orr et al. (2018) also +demonstrate that using this threshold to separate H2 +and HI yields reasonable agreement with the observed +Kennicutt-Schmidt law (Kennicutt 1998; Kennicutt & +Evans 2012). In the following, we refer to total gas as +the sum of the gas in the cool and cold-dense phases +(simulations) or in the atomic and molecular phases (ob- +servations). +We adopt the same definition of molecular gas as in +Moreno et al. (2019) (temperature below 300 K and den- +sity above 10 cm−3). Guszejnov et al. (2017) demon- +strate that the model successfully reproduces the GMC +mass function in the Milky Way (Rice et al. 2016) and +the size-linewidth relation (e.g., the Larson scaling re- +lationship, Larson 1981) in our Galaxy (Heyer et al. +2009; Heyer & Dame 2015) and in nearby galaxies (Bo- +latto et al. 2008; Fukui et al. 2008; Muraoka et al. 2009; +Roman-Duval et al. 2010; Colombo et al. 2014; Tosaki +et al. 2017). Given the density cut of 10 cm−3 and mass +resolution of 1.4 ×104 M⊙, the lower limit of our spa- +tial resolution ( 3� +M/ρ, where M is the mass resolution +and ρ is the mass volume density) is ∼ 40 pc, which + +4 +He et al. +is smaller than the typical scale of observed GMCs (40 +– 100 pc, Rosolowsky et al. 2021). In addition, GMC +mass function peaks at 105 – 106 M⊙ in Milky Way (Rice +et al. 2016), which is significantly larger than our mass +resolution. Therefore, we would generally expect more +than 1 gas particle is included for molecular gas in each +GMC-scale pixel. +For generating different components of the ISM, the +simulations start with a homogeneous ISM with a tem- +perature of 104 K and solar metallicity. The multi-phase +ISM then emerges quickly as a result of cooling and feed- +back from star formation. The initial gas mass for the +simulation is set to match the median HI mass from the +xCOLDGASS survey (Catinella et al. 2018). +2.1.4. Data cubes +We first convert the FIRE-2 molecular gas data into +mass-weighted position-position-velocity (p-p-v) data +cubes to match the format of the CO data from radio +observations (McMullin et al. 2007). We adopt the cube +construction method created for Bottrell et al. (2022) +and Bottrell & Hani (2022) and then adapted to the +FIRE-2 merger suite by McElroy et al. (2022). Kine- +matic cubes are produced along four lines-of-sight (la- +beled as ‘v0’, ‘v1’, ‘v2’, ‘v3’), defined by the vertices of a +tetrahedron centered at the primary galaxy (G3 in this +work). For the isolated galaxy simulations, we generate +p-p-v data cubes at different inclination angles (10 – 80 +degrees). We adopt a pixel size of 100 pc and velocity +resolution of 2 km s−1, which is similar to PHANGS +choice (Sun et al. 2020). The field of view (FOV) for +the data cube is set to be 25 kpc. +Then we create zeroth-moment maps of the gas surface +density Σmol and second-moment maps of the velocity +dispersion σv. We do not set any thresholds on these +moment maps since we argue that every gas particle in +the simulated cube should be treated as a real signal, +rather than observational noise. However, in later anal- +yses, when we display σv versus Σmol for the simulated +data, we select pixels with Σmol greater than 1 M⊙ pc−2, +which approximates the lower limit of the molecular gas +detection threshold in the observational data (Sun et al. +2018). +We also exclude pixels detected in fewer than +two velocity channels in the simulated cube to exclude +inaccurate measurements of σv. +To characterize clouds, we use a pixel-based analy- +sis (Leroy et al. 2016), which treats each pixel as an +individual GMC, rather than identifying each individ- +ual cloud from the data cube. This approach has been +widely applied to GMC analyses for PHANGS galax- +ies (Sun et al. 2018, 2020). +Compared to the tradi- +tional cube-based approach, this new method requires +minimal assumptions and can be easily applied to many +datasets in a uniform way, while still giving us the essen- +tial GMC properties (e.g., molecular gas surface density +Σmol, gas velocity dispersion σv). On the other hand, +the pixel-based method has a major disadvantage of not +able to decompose different cloud components along the +same line of sight. +Several observational studies (e.g. +Brunetti & Wilson 2022; Sun et al. 2022) have compared +this new approach with the traditional approach and +found good agreement on cloud properties between two +methods for both normal spiral galaxies and starburst +mergers, especially for clouds in galaxy disks. +These +comparisons show pixel-based analysis should be valid +for capturing individual cloud properties, especially for +galaxy disks which generally have single-layer of GMCs +(see Section 5.1 for detailed discussion about the pro- +jection effect). In this work, we adopt this approach to +match the method in Brunetti et al. (2020) and Brunetti +(2022). We also note that since we treat each pixel as +a GMC, these GMCs do not necessarily represent inde- +pendent ISM structures. In fact, given the mass resolu- +tion of 1.4 ×104 M⊙, we can barely resolve the internal +structure of most massive GMCs of ∼ 106 M⊙ (100 ele- +ments). We refer to them as GMCs in this paper to be +consistent with similar observational analyses (e.g. Sun +et al. 2018, 2020). +2.2. Star Formation Rate Maps +To further explore how the GMC properties at 100 pc +scale affect the star formation, we also make SFR maps +with the same resolution of 100 pc for the simulated +mergers at different times. We create these maps using +a method similar to the one used to create the gas cubes. +We include all the stellar particles with age younger than +10 Myr and create p-p-v data cubes for these stellar +particles. The mass-weighted cubes are integrated along +the velocity axis to produce 2D maps of stellar mass +formed within the last 10 Myr. These surface-density +mass maps are subsequently divided by 10 Myr to obtain +the average star-formation rates over the last 10 Myr. +2.3. Observational Data +We use several sets of observations for comparison +with our simulations. +2.3.1. Spiral galaxies: PHANGS data +For isolated galaxies, we mainly use the PHANGS +data from Sun et al. (2020) with resolution of 90 pc, +which is comparable to our pixel size choice of 100 pc. +Sun et al. (2020) apply the pixel-based method for sta- +tistical analyses of GMC properties for 70 galaxies in +the PHANGS sample. We also include GMC data for + +GMCs in Galaxy Mergers +5 +M31 from Sun et al. (2018) at resolution of 120 pc. M31 +is identified as a green-valley galaxy, similar to our own +Milky Way, and hence has a lower total gas fraction +than normal spiral galaxies (Mutch et al. 2011). Both +M31 and the Milky Way seem to be in a transition from +blue spiral galaxies to quenched galaxies via depletion of +their cold gas (Bland-Hawthorn & Gerhard 2016). M31 +has stellar mass of 1011 M⊙ (Sick et al. 2015), H2 mass +of 3.6 × 108 M⊙ and HI mass of 4.8 × 109 M⊙ (Nieten +et al. 2006). +2.3.2. Galaxy mergers: the Antennae and NGC 3256 +Table 2. Information about the observed mergers in this +work +Antennae +NGC 3256 +#References +M⋆ (1010 M⊙)a +4.5 +11.4 +(1); (2) +Mmol (1010 M⊙)b +1.2 +0.8 +(3); this work +SFR (M⊙yr−1) +8.5 +50 +(1); (4) +Sep. (kpc)c +7.3 +1.1 +(5); (4) +tnow (Myr)d +40 +· · · +(6) +mass ratiof +1:1 +· · · +(6) +Peri. Sep (kpc)g +10.4 +· · · +(6) +Notes: a. Stellar mass. b. Molecular gas mass. c. Current +separation between two nuclei. d. Current time since the sec- +ond passage. e. Mass ratio of the two progenitor galaxies. g. +Pericentric distance of two nuclei from the simulation model. +References: (1) Seill´e et al. (2022) (2) Howell et al. (2010) +(3) Wilson et al. (2000) (4) Sakamoto et al. (2014) (5) Zhang +et al. (2001) (6) Karl et al. (2010) +We use the CO 2-1 data for NGC 3256 (Brunetti et al. +2020) and the Antennae (Brunetti (2022, +Brunetti et +al. in prep)) at resolutions of 90 and 80 pc, respectively. +The GMC measurements use the same pixel-based ap- +proach as in Sun et al. (2018, 2020). Both NGC 3256 and +the Antennae are identified as late-stage major mergers +that have been through their second perigalactic pas- +sage (Privon et al. 2013). NGC 3256 has stellar mass +of 1.1 × 1011 M⊙, total molecular gas of 8 × 1019 M⊙ +(calculated based on CO 2-1 map in Brunetti & Wil- +son (2022), assuming αCO of 1.1 M⊙ (K km s−1 pc2)−1 +and CO 2-1/1-0 ratio of 0.8) and SFR of 50 M⊙yr−1 +(Sakamoto et al. 2014). In contrast, the Antennae has a +stellar mass of 4.5 × 1010 M⊙ and SFR of 8.5 M⊙yr−1 +(Seill´e et al. 2022). +NGC 3256 currently has a more +intense starburst, perhaps because it is at different evo- +lutionary stage in the merging process. +The detailed +information is in Table 2. +To convert the CO 2-1 emission to molecular gas mass +requires the assumption of a CO-to-H2 conversion factor +(αCO). The exact value of αCO has large uncertainties +and varies significantly among different types of galax- +ies, especially for starburst galaxies. Downes & Solomon +(1998) find that for starburst U/LIRGs, the αCO value +is generally 4 times smaller than that in our Milky Way. +The major method for direct measurement of αCO is +through large velocity gradient (LVG) radiative trans- +fer modeling of multiple CO and its isotope lines. For +αCO in the Antennae, various LVG modeling (e.g. Zhu +et al. 2003; Schirm et al. 2014) suggests that the An- +tennae has αCO close to the Milky Way value of 4.3 +M⊙ (K km s−1 pc2)−1. This is also supported by the +galaxy simulation that specifically matches the Anten- +nae (Renaud et al. 2019a). For NGC 3256, we do not +have a direct measurement of αCO. We therefore adopt +the treatment from Sargent et al. (2014) to determine +the αCO for an individual galaxy as +αCO = (1 − fSB) × αCO,MS + fSB × αCO,SB, +(1) +where αCO,MS and αCO,SB are the conversion fac- +tors for the Milky Way (4.3 M⊙ (K km s−1 pc2)−1) +and U/LIRGs (1.1 M⊙ (K km s−1 pc2)−1, including he- +lium), and fSB is the probability for a galaxy to be a +starburst galaxy, which is determined by its deviation +from the star-forming main sequence. +We adopt the +star-forming main sequence relation from Catinella et al. +(2018), +log sSFRMS = −0.344(log M⋆ − 9) − 9.822, +(2) +where sSFR = SFR / M⋆ +is the specific star forma- +tion rate. +NGC 3256 has an sSFR/sSFRMS ratio of +15 (Brunetti et al. 2020), which suggests NGC 3256 +should have αCO close to the U/LIRG value of 1.1 +M⊙ (K km s−1 pc2)−1. Therefore, in the following anal- +yses, we will adopt αCO of 4.3 M⊙ (K km s−1 pc2)−1 for +the Antennae and 1.1 M⊙ (K km s−1 pc2)−1 for NGC +3256. +3. CONTROL (ISOLATED) GALAXIES +To test if the simulation successfully reproduces ob- +served GMCs, Figure 1 shows the well-known correla- +tion between σv and Σmol for isolated simulated galaxies +and PHANGS-ALMA spiral galaxies. We show σv ver- +sus Σmol contours for G2 and G3 galaxies at an inclina- +tion angle of 30 degrees, compared with that of observed +galaxies. +The two simulated galaxies exhibit similar +properties (black and dark red solid contours) and gener- +ally lie on the trend followed by the PHANGS galaxies. +We also plot a red dashed line indicating GMCs with +constant virial parameter αvir of 3.1. For the pixel-based + +6 +He et al. +100 +101 +102 +103 +104 +mol (M pc +2) +10 +1 +100 +101 +102 +v (km s +1) +t: 0.625 Gyr +FIRE G2 +FIRE G3 +PHANGS disks +PHANGS barred galaxy centers +PHANGS unbarred galaxy centers +M31 +Figure 1. Velocity dispersion versus gas surface density for +the G2 (black solid contour) and G3 (brown solid contour) +simulated galaxies at 0.625 Gyr with inclination angle of 30 +degrees compared to the PHANGS galaxy sample. The con- +tours are mass-weighted and set to include 20%, 50% and +80% of the data. The density contours of PHANGS galaxies +(Sun et al. 2020) show the distribution of measurements in +galaxy disks (blue shaded contours), the centers of barred +galaxies (salmon shaded contours) and the centers of un- +barred galaxies (brown dashed contours) with a resolution +of 90 pc. The red dashed line marks the position of the me- +dian values of αvir for PHANGS galaxies of 3.1 (Sun et al. +2020). We also show the data for M31 (green solid contour) +at 120 pc resolution from Sun et al. (2018). We can see that +the FIRE-2 spiral galaxies follow the same σv- Σmol relation +as the PHANGS galaxies. +analysis, αvir is calculated as (Sun et al. 2018) +αvir = 9 ln 2 +2πG +σ2 +v +ΣmolR += 5.77 +� +σv +km s−1 +�2 �Σmol +M⊙ +�−1 � R +40pc +�−1 +, +(3) +where R is the GMC radius. In Sun et al. (2018), R is set +to be the radius of the beam in the image, as each beam +is treated as an independent GMC. We can see both +our simulated galaxies and observed PHANGS galaxies +follow the trend of the constant αvir, which yields the +relation of σ2 +v ∝ Σmol that suggests the simulations re- +produce GMCs similar to the observations. +However, +we can see that the two galaxies lie at the low surface- +density end of the PHANGS distribution and thus their +gas properties are more similar to those of M31 than a +typical PHANGS galaxy. Indeed, the molecular and to- +tal gas properties of the simulated galaxies are similar +to those of M 31, perhaps due to the choice of initial gas +mass in the simulations (see Appendix B). +4. MERGING GALAXIES +4.1. GMC linewidth and surface density +We performed a similar σv versus Σmol analysis for +our suite of galaxy merger simulations. +Since we are +particularly interested in how the starburst activity in- +fluences GMC properties, we focus on the period right +before and after the second passage where we can see +the largest contrast in SFR. In Fig. 2 we show some ex- +ample snapshots of σv versus Σmol for different merger +stages during the second passage, along with Σmol and +αvir maps at each snapshot. +Note that the datacube +is centered on the primary galaxy G3. At the time of +first snapshot (2.54 Gyr), right before the start of the +second perigalactic passage, the simulated mergers still +have Σmol and σv that are similar the isolated galaxies. +Then the molecular gas quickly transitions to a more +turbulent state with much higher σv after the second +passage along with a dramatic increase in global SFR, +as shown in the snapshot for 2.66 Gyr (middle panel of +Fig. 2). The merger at this time still shows two sep- +arate nuclei in the zeroth moment map; this is similar +to our observed mergers, the Antennae and NGC 3256. +At this time, the σv versus Σmol contours for the simu- +lated merger lie above the trend seen for the PHANGS +galaxies, similar to NGC 3256, but in contrast to the An- +tennae, which still lies along the trend of the PHANGS +galaxies. The larger deviation above the PHANGS trend +implies higher αvir. We note that different αCO choices +will affect the position of the contours. If we choose the +ULIRG αCO instead of the Milky Way value, the An- +tennae would have αvir similar to that of NGC 3256 and +our G2&G3 merger. The uncertainty in the correct αCO +value to use makes it difficult to interpret the data for +the Antennae in this context. +The bottom panel of Fig. 2 shows the snapshot at +2.87 Gyr, which marks the post-merger stage after the +final coalescence of two nuclei (defined here as the time +at which the two central supermassive black holes are +at a distance of 500 pc for the last time). This is the +time when both Σmol and σv reach their highest values. +We can see that most of the molecular gas is concen- +trated in the central 1 kpc region, with Σmol reaching +1000 M⊙ pc−2. σv reaches 200 km s−1, which is even +higher than the σv observed in NGC 3256, which is in +an earlier merging stage when the two nuclei have not +yet coalesced. +To better quantify the variation of Σmol and σv during +the second passage, we plot the 16th, 50th and 84th per- +centile of the mass-weighted values for all pixels of each +snapshot during the second passage in Fig. 3. We also +normalize both the median Σmol and σv to the median + +GMCs in Galaxy Mergers +7 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Time (Gyr) +100 +101 +SFR (M yr +1 +100 +101 +102 +103 +10 +1 +100 +101 +102 +v (km s +1) +t: 2.54 Gyr +SFR: 1.65 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +100 +101 +4 +2 +0 +2 +4 +vir +100 +101 +102 +100 +101 +102 +103 +10 +1 +100 +101 +102 +v (km s +1) +t: 2.66 Gyr +SFR: 6.04 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +101 +102 +4 +2 +0 +2 +4 +vir +100 +101 +102 +100 +101 +102 +103 +mol (M pc +2) +10 +1 +100 +101 +102 +v (km s +1) +t: 2.87 Gyr +SFR: 28.22 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +kpc +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +101 +102 +4 +2 +0 +2 +4 +kpc +vir +100 +101 +102 +Figure 2. (Top) SFR history for the G2&G3 merger with ‘e2’ orbit with viewing angle of ‘v0’. The 3 solid black vertical lines +indicate the time for each snapshot displayed below. The two dashed lines indicate the times at the start of second merging +and the final coalesce of two nuclei. (Bottom) Three snapshots. For each snapshot, the left panel shows the σv versus Σmol +mass-weighted contour with the same setting as Fig. 1. We also show the density contours for the PHANGS galaxies (filled +blue region), NGC 3256 (blue contours) and the Antennae (orange shaded region). For NGC 3256, Σmol is calculated using +the ULIRG αCO of 1.1 M⊙ (K km s−1 pc2)−1. For the Antennae, the gas surface density is calculated using the Milky Way +αCO of 4.3 M⊙ (K km s−1 pc2)−1. The red dashed line indicate the line of constant αvir of 3.1. The right two panels show the +Σmol and αvir maps of inner 5 kpc regions where we have most of our detected pixels. The interactive version of the animation +is available at https://htmlpreview.github.io/?https://github.com/heh15/merger animations/blob/main/G2G3 e2 v0.html. We +can see that the properties of the GMCs right before the second passage still resemble those of normal spiral galaxies, while +GMCs after the second passage lie above the PHANGS trend in the σv vs Σmol plot and show significantly higher αvir. + +8 +He et al. +2.6 +2.8 +3.0 +3.2 +3.4 +101 +102 +Σmol(t) (M⊙ pc−2) +2.6 +2.8 +3.0 +3.2 +3.4 +100 +101 +Σmol(t) / Σmol,0 +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +101 +102 +σv (km s−1) +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +100 +101 +σv(t) / σv,0 +Figure 3. The Σmol and σv variation across the second passage and final coalescence of the G2&G3 merger at ‘e2’ orbit with +viewing angle of ‘v0’. The two dashed vertical lines indicate the times when the simulated merger begin the second passage and +experience final coalescence. The three solid vertical lines correspond to the 3 snapshots shown in Fig, 2. The horizontal dashed +lines indicate the median value of the isolated G3 galaxy at time of 0.625 Gyr (Fig. 1) as a baseline for comparison. (Upper +left) Σmol vs time. Blue lines shows the mass weighted median Σmol of the entire merger while the orange filled area indicates +Σmol range between 16th and 84th percentile. The two dashed lines indicate the time for the start of the second passage and +the final coalesce of the two nuclei. (Upper right) The ratio between median Σmol at given time and the median value Σmol,0 for +the isolated G3 galaxy at 0.625 Gyr. (Lower left) The mass-weighted median σv versus time. (Lower right) The ratio between +the median σv and the value σv,0 for isolated G3 galaxies at 0.625 Gyr. We can see both Σmol and σv increase dramatically +during the second passage when the extreme starburst happens. +values of the isolated G3 galaxy at 0.625 Gyr (Fig. 1) +to show how the merging event affects the GMC prop- +erties during the second passage. Both Σmol and σv in- +crease significantly during the merger, with a maximum +increase of a factor of 10. The increase in σv and Σmol +is roughly of the same order. Eq. 3 shows that a con- +stant αvir requires σ2 +v ∝ Σmol. These results imply that +our simulated merger will have higher αvir compared to +PHANGS galaxies. +4.2. The virial parameters of GMCs +During the second passage, we see that the σv vs +Σmol distribution for our simulated merger lies above +the trend observed for the PHANGS galaxies. A higher +σv for a given Σmol means the GMCs in these mergers +are more turbulent and less gravitationally bound than +in normal spiral galaxies. +We adopt the same approach as in observations to +calculate αvir for pixel-based GMC pixels using Eq. +3. +Since the simulation data do not have a telescope +“beam” and each pixel in this analysis is treated as +an independent GMC, we set R to be half the size of +each pixel (50 pc). With constant R, αvir depends only +on σv and Σmol. Higher σv at a similar Σmol thus im- +plies that αvir values for GMCs in simulated mergers +are higher than the values for PHANGS or simulated +isolated galaxies. Higher values for αvir are also found +for NGC 3256 (Brunetti et al. 2020; Brunetti & Wilson +2022) and the Antennae (Brunetti 2022). +Fig. 4 shows αvir as a function of time during the pe- +riod near the second pericentric passage for the merger +simulations with “e2” and “e1” orbits and viewed from +“v0” angle. αvir stays low before the second passage and +suddenly rises after the passage along with a sudden in- +crease in SFR. The peak of median αvir can reach ∼100. +After the second passage, αvir gradually dies down as +the SFR also decreases. During the entire merging pro- +cess, we generally see a good correspondence between +the SFR and αvir peaks, which suggests that the αvir +value is either regulated by feedback from star forma- +tion or that both SFR and αvir increase together as a +result of the merger. +αvir for our fiducial ‘e2’ orbit is generally higher than +that of the ‘e1’ orbit and stays at higher values for a +significantly longer time. +The ‘e2’ orbit has a higher +impact parameter than the ‘e1’ orbit (Section 2.1.2). +Therefore, we would expect more gravitational poten- +tial energy transferred to the kinetic energy of individual +GMCs, potentially making these GMCs less gravitation- +ally bound. The αvir values for the ‘e1’ orbit are more + +GMCs in Galaxy Mergers +9 +100 +101 +SFR (M⊙ yr−1) +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +101 +102 +103 +αvir +e2 orbit +PHANGS +NGC 3256 +Antennae +101 +SFR (M⊙ yr−1) +1.2 +1.4 +1.6 +1.8 +2.0 +Time (Gyr) +101 +102 +103 +αvir +e1 orbit +PHANGS +NGC 3256 +Antennae +Figure 4. αvir versus time for the G2&G3 mergers in (left) the e2 orbit and (right) the e1 orbit viewed from ‘v0’ angle during +the final coalescence. (Left) The red line is the mass-weighted median for αvir from the simulation. The orange shaded region +includes data within the 16th and 84th quantile of αvir values. The dashed lines correspond to the start of the second passage +and the final coalescence of the two nuclei. The three solid lines correspond to the merger times shown in Fig. 2. The horizontal +dashed line indicates the median αvir for the isolated G3 galaxy at 0.625 Gyr (Fig . 1) as a baseline for comparison. The upper +panel shows SFR versus time for the second coalescence and the right panel shows the 16th, 50th and 84th quantile of αvir for +PHANGS, NGC 3256 and the Antennae from the observations. In calculating αvir, we use the U/LIRG αCO for NGC 3256 and +the Milky Way value for PHANGS and the Antennae. (Right) Same plot for G2&G3 merger in the ‘e1’ orbit during the final +coalescence. The 3 solid lines correspond to 3 snapshots in Fig. C1. The ‘e1’ orbit has a smaller impact parameter than the +‘e2’ orbit. We can see the global αvir increases dramatically right after the second passage as SFR rises. The peak SFR also +roughly corresponds with the peak αvir, which suggests the high αvir might be caused by the feedback from the starburst. +similar to the αvir of NGC 3256 and the Antennae and +the ‘e1’ orbit is more similar to the orbit of the Anten- +nae. We note that both the Antennae and NGC 3256 are +at the very start of their second passages (Privon et al. +2013; Renaud et al. 2019a). +At this stage, there are +significant variations in αvir, which makes it difficult to +pick the exact snapshot that matches the observation. +If we use the U/LIRG αCO instead of the Milky Way +value, αvir for the Antennae would be similar to that of +NGC 3256. We will discuss our αCO choices further in +Section 5.2. +4.3. Molecular Gas in the central 1 kpc region +From the moment 0 maps in Fig. 2, we can see that +most molecular gas is concentrated in the center during +the post-merger phase after 2.83 Gyr. This is consis- +tent with the traditional scenario that the central star- +burst activity is caused by the inflow of molecular gas +due to the loss of angular momentum (Hernquist 1989; +Barnes & Hernquist 1991; Mihos & Hernquist 1994, +1996; Barnes & Hernquist 1996; Moreno et al. 2015). +To quantify how much of the molecular gas is concen- +trated in the center, Fig. 5 shows the molecular gas mass +within the central 1 kpc, and the ratio between this value +and total molecular gas mass. The fraction of molecular +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +0% +20% +40% +60% +80% +100% +Mmol, central / Mmol, total +Figure 5. +The ratio between molecular gas mass within +the central 1 kpc radius circle of the G3 galaxy and total +molecular gas inside our FOV of 25 kpc. During the second +coalescence between 2.7 Gyr and 3.2 Gyr, more than 50% of +molecular gas is concentrated within the central 1 kpc region, +which indicates the Σmol increase we see in the simulated +merger during the second passage is probably due to this gas +concentration. +gas concentrated in the center reaches as high as 80% for +a significant period of time (∼ 500 Myr) around the final +coalescence. On the other hand, Moreno et al. (2019) +shows that the total molecular gas mass decreases dur- +ing the second passage. Therefore, the overall high Σmol + +10 +He et al. +values of GMCs across our simulated merger compared +to isolated galaxies are mostly due to the central gas +concentration. +Fig. 6 shows the σv versus Σmol distribution for pix- +els in the central kpc region of the G2&G3 merger at +2.87 Gyr (red aperture in Fig. +2), along with pix- +els in the center of PHANGS galaxies, the Antennae +and NGC 3256. +We can see the pixels in the center +of the G2&G3 merger have a larger deviation from the +PHANGS trend than NGC 3256, which indicates that +the G2&G3 merger has GMCs with larger αvir in the +center. We also show the mass weighted median αvir for +the entire and central region of G2&G3 merger as a func- +tion of time (Fig. 6 right). αvir in the center is generally +higher than for the entire region, which indicates that +GMCs in the center are more perturbed and less grav- +itationally bound. At the time right after the second +passage, we see dramatic fluctuations of αvir for both +the center and the entire galaxy, which is probably due +to the complex and constantly varying gas morphology +during this period. Moreover, we might see two GMCs +that are far apart in 3D space but lie along the same +line of sight, which cause large measured αvir value, but +in a short time they no longer lie along the same line of +sight, which causes a sudden drop of αvir. At the post- +merger phase, αvir values are more stable. However, we +see that αvir of the disk region gradually settles down +while the central αvir keeps increasing. This might indi- +cate that GMCs in the central region take more time to +settle down to their normal states, which may be due to +the starburst activity in the center. We also see high αvir +for the center at the very start (2.54 Gyr), which prob- +ably means GMCs in the center at this time have not +recovered from the starburst event that occured during +the first peri-galactic passage. +4.4. Correlation between the central SFR and GMC +Properties +The driving mechanism behind the SFR enhancement +in mergers is of great interest to the study of star forma- +tion and galaxy evolution. One approach to tackle this +problem is to decompose the SFR into the following 2 +terms +SFR = Mmol +tdep +, +(4) +where tdep is the depletion time, defined as the time +for star formation to consume the available molecular +gas. +This approach makes it clearer that the rise in +SFR could be either due to a larger amount of molecular +gas “fuel driven”) or shorter depletion time (“efficiency +driven”). The simulations (e.g. Moreno et al. 2021) and +observations (e.g. Thorp et al. 2022) indicate that both +terms contribute to the SFR enhancement at kpc scales. +Moreover, many studies of the Kennicutt-Schmidt rela- +tion in U/LIRGs at kpc scales show that these starburst +mergers have relatively short tdep of ∼ 108 yr compared +to normal spiral galaxies of ∼ 109 yr (e.g. Daddi et al. +2010), which confirms the role of efficiency driving in +mergers. With our simulations being able to probe the +molecular gas at GMC scales, we can explore how tdep +is correlated with GMC populations in different regions. +For this analysis, we focus on the molecular gas and +star formation in the central 1 kpc region since most gas +is concentrated here during the second passage (see Sec- +tion 4.3). We measure the mass-weighted median αvir in +this central region as a metric for GMC dynamical state +in the center. Fig. 7 shows Σmol and ΣSFR color-coded +by αvir for the central region as a function of time. We +calculate the average Σmol and ΣSFR by dividing the to- +tal Mmol or SFR in the central region by the aperture +size. We show the data points within the period of 2.54 – +2.61 Gyr (before the second passage) and 2.73 – 3.47 Gyr +(after the second passage) for comparison. We exclude +the data points between the start of the second passage +(2.62 Gyr) and the time when the central/total gas frac- +tion starts to reach 50% (2.73 Gyr) because data points +from this period show a large deviation from the major +trend in ΣSFR vs Σmol diagram. The large deviation is +probably because the limited amount of molecular gas +is highly perturbed in the central region. Gas is either +quickly consumed without being replenished in time, or +just concentrated and has not formed stars yet, which +causes the large scatter in the ΣSFR vs Σmol relation. On +the other hand, before and after this period, the central +region is in a relatively stable state when the molecu- +lar gas is constantly replenished to fuel star formation +activity. +In the left panel of Fig. 7, we can see that tdep be- +comes shorter as Σmol and ΣSFR increase. The points at +the lower left end of the Σmol correspond to the times be- +fore the second passage, which also have relatively low +αvir. In contrast, the αvir after the second passage is +significantly higher. We also note that tdep even before +the second passage (∼ 108 yr) is quite shorter than that +of normal spiral galaxies (109 yr). The difference could +be due to different dynamical timescales of simulated +and observed galaxies. +At this time, we can see αvir +for the central region is already ∼ 10 which indicates +the molecular gas in the central region has already been +disturbed. +There is no significant correlation between tdep and +αvir, with Spearman coefficient of -0.08 for all data +points and of 0.18 for data after the second passage, +which is against our expectation that low αvir gas form + +GMCs in Galaxy Mergers +11 +101 +102 +103 +104 +mol (M pc +2) +100 +101 +102 +v (km s +1) +t: 2.87 Gyr +FIRE G2&G3 +FIRE G2&G3 Center +PHANGS galaxy center +Antennae nucleus +NGC 3256 nucleus +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +101 +102 +vir +G2&G3 +G2&G3 center +Figure 6. +(Left) The σv versus Σmol contour for the entire (dashed contour) and central 1 kpc region (solid contour) of the +G2&G3 merger at 2.87 Gyr viewed from the ‘v0’ angle. We also show contours for the centers of PHANGS galaxies (brown +dashed contours), the Antennae (orange shaded contours) and NGC 3256 (blue contours). We can see the central region in our +simulated merger generally has the highest σv and αvir. (Right) The mass weighted median αvir for molecular gas in the entire +(blue) and central (orange) region of G2&G3 merger viewed from ‘v0’ angle. We see that αvir for the entire disk gradually +settles back to the original low value, while that for the central region keeps a high value until the end of the simulation. +101 +mol (M pc +2) +10 +1 +100 +SFR (M yr +1 kpc +2) +tdep = 107 yr +tdep = 108 yr +101 +102 +vir +101 +102 +vir +107 +108 +tdep (yr) +rs all: -0.08 +rs after: 0.18 +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +101 +mol (M yr +1 pc +2) +107 +108 +tdep (yr) +rs all: -0.47 +rs after: -0.32 +2.6 +2.8 +3.0 +3.2 +3.4 +Time (Gyr) +Figure 7. (Left) SFR surface density ΣSFR versus Σmol color coded by the mass-weighted median αvir for the central 1 kpc +region of the the G2&G3 merger during the second passage of the ‘e2’ orbit viewed from ‘v0’ orientation. We include simulated +data points within 2.54 – 2.61 Gyr (before the second passage; trangle) and 2.73 – 3.47 Gyr (after the second passage; circle). +Both ΣSFR and Σmol are calculated as total central SFR or Mmol within 1 kpc radius divided by the aperture size, while αvir is the +mass weighted median of pixels inside the aperture. The two dashed line indicate constant depletion times (tdep = Σmol/ΣSFR) +of 107 and 108 years. (Middle) tdep versus αvir for the central 1 kpc. (Right) tdep versus Σmol for the central region. The label +“rs all” shows the Spearman coefficient between tdep and αvir and Σmol for all data points while “rs after” shows the Spearman +coefficient only for data after the second passage. We can see there is no significant correlation between αvir and tdep, which is +against our expectation that low αvir clouds will consume molecular gas at faster rate. +stars more quickly. We can also clearly see a distinc- +tion between αvir before and after the second passage. +The αvir before the second passage is relatively small +and corresponds to larger tdep while the αvir after the +second passage is significantly larger but corresponds to +shorter tdep. This again is inconsistent with our expecta- +tion that low αvir GMCs should form stars more easily. +Other physical mechanisms rather than self-gravity of +individual GMCs may be needed to help the molecular +gas to collapse (see detailed discussion in Section 5.1). +On the other hand, we see an anti-correlation between +tdep and Σmol. +This relation is similar to the global +Kennicutt-Schmidt relation where the gas rich U/LIRGs +have the shorter tdep (Daddi et al. 2010). One expla- + +12 +He et al. +nation for this trend is that the fraction of dense gas +(traced by HCN) that are actually forming stars (i.e. +traces the self-gravitating gas fraction) is increasing as +Σmol increases (e.g. +Gao & Solomon 2004; Bemis & +Wilson 2023). If we assume Σmol is proportional to the +mean volume density of molecular gas in the central re- +gion, we would expect larger fraction of molecular gas +above the dense gas threshold (n > 104 cm−3) in FIRE- +2 simulation (Hopkins 2015), which leads to faster star +formation and shorter tdep. +5. DISCUSSION +5.1. How can high αvir gas form stars in simulated +mergers? +As shown in Section 4.2, αvir generally stays above 10 +during the second passage for the G2&G3 merger. If we +assume star formation occurs in individual GMCs and is +driven by the collapse of the clouds due to self-gravity , +we would expect star forming GMCs to have αvir below +1. The combination of high αvir values and starburst +activity is inconsistent with this expectation, unless the +velocity dispersion is being driven to higher values by +infall motion. Furthermore, we find no correlation be- +tween αvir and tdep (Section 4.4), which suggests low αvir +values do not strongly affect the depletion time in our +simulations. However, we need to note that our mea- +surement of αvir from pixel-based method might not re- +flect the real αvir of individual GMC components, espe- +cially for the post-second-passage phase when molecular +gas is concentrated in the center. Although observations +(Brunetti & Wilson 2022; Sun et al. 2020) show that +cloud properties extracted from a pixel-based approach +is generally consistent with the traditional cloud-based +approach, they also show the pixel-based approach gives +higher σv and αvir for molecular gas in galaxy centers. +This is likely due to the superimposition of different +GMC components along the same line of sight in gas- +concentrated galaxy centers. Sun et al. (2022) find that +αvir from pixel-based approach is ∼ 3 times higher than +the cloud-based approach for galaxy centers. If we as- +sume the same degree of overestimate in our simulation +data for the merger center, we would expect the real αvir +to be ∼ 10 during the second passage, still significantly +higher than the critical value of 1 when clouds reach the +self-collapsing criterion. We also note that even the ob- +servational cloud-based approach by extracting different +GMC components from p-p-v data cube might still suf- +fer from the projection effects. Beaumont et al. (2013) +find that αvir from p-p-p and p-p-v cubes have a factor +of 2 difference for substructures in their cloud simulation +due to a mismatch of substructures from these two data +cubes. Therefore, one of our next steps is to perform +cloud-finding algorithm (Burkhart et al. 2013) on both +p-p-p and p-p-v simulation data cubes to fully under- +stand how GMCs evolve during the merging events. +A possible explanation for large αvir is that GMCs +that satisfy the self-collapsing criterion have already +formed stars and become unbound or destroyed due to +the stellar feedback. +However, if this is the case, we +would expect αvir to fluctuate around the critical value +of 1. Furthermore, according to Benincasa et al. (2020), +GMCs with high αvir (>10) have significantly shorter +lifetimes (∼2 Myr) than GMCs with low αvir (∼1; life- +time of ∼10 Myr). If we assume all GMCs are of the +same population but at different evolutionary stages, we +would expect GMCs to stay at low αvir state for a longer +time and hence we should be more likely to catch these +low αvir GMCs in our simulation snapshots. Instead, +we see αvir constantly higher than 10 during the star- +burst activity (Fig. 4), which is inconsistent with this +scenario. +It is perhaps likely that the explanation is that these +GMCs are experiencing compression from the large-scale +gravitational potential. This compression could add ad- +ditional potential energy to balance the kinetic energy. +Furthermore, they can trigger inflow of gas into GMCs +and bring radial velocity (Vr) component into our σv +measurement. Ganguly et al. (2022) find in their simu- +lation that vr could be an important factor to produce +high measured αvir clouds. For GMCs in normal spi- +ral galaxies and galaxy pairs (e.g., M 51), Meidt et al. +(2018) show that the large-scale stellar potential could +be responsible for holding individual GMCs in energy +equipartition state. Compared to galaxies in their study, +the starburst mergers in our study undergo more dra- +matic morphological changes, which could generate com- +plicated gravitational tidal fields. Renaud et al. (2009) +show in their simulation that major mergers can produce +fully compressive tidal fields that concentrate molecular +gas and trigger starburst activities. These compressive +tidal fields are believed to be responsible for creating the +off-nuclei gas concentration region in the ULIRG, Arp +220 (Downes & Solomon 1998). In our next step to test +this scenario, we will need to calculate tidal deformation +timescale (as in Ganguly et al. 2022) for each individual +GMC and compare it with GMC free-fall and crossing +timescales to see how important the external tidal field +is compared to GMC self-gravity. +Another possible explanation is that molecular gas +is smoothly distributed rather than clumped into in- +dividual GMCs during the starburst activities. If this +is the case, the star formation is regulated by the en- +tire molecular disk rather than individual GMC compo- +nents (Krumholz et al. 2018). Wilson et al. (2019) pro- + +GMCs in Galaxy Mergers +13 +pose that the star formation in U/LIRGs is regulated by +the hydrodynamic pressure of the molecular disk with a +constant scale height. In observation, one way to test +the smoothness of gas distribution is by comparing av- +erage gas surface density at different observing resolu- +tions (Leroy et al. 2017). Brunetti et al. (2020) show +that molecular gas in the LIRG, NGC 3256, is smoothly +distributed based on this method. +For our simulated +merger, gas might be smoothly distributed during the +second passage when most gas is concentrated in the +center (e.g. at 2.87 Gyr, Fig. 2. We could test this sce- +nario by changing the pixel size in our p-p-v cubes and +compare the average gas surface densities in the central +region at different pixel resolutions. +5.2. Comparison with observations +As shown in Section 4, our simulated merger gener- +ally has lower Σmol and higher σv and αvir compared to +the two observed mergers, the Antennae and NGC 3256. +We note that this simulation is not set to match the ex- +act condition of the observed mergers, so some discrep- +ancy between observations and simulations would be ex- +pected. +From the observational side, the biggest un- +certainty that comes into the measurement is the value +of αCO. As mentioned in Section 4.1, if we adopt the +ULIRG αCO instead of the Milky Way value for the +Antennae, we would find the Antennae to have similar +Σmol and αvir as NGC 3256. In contrast, if we assume +an even smaller αCO for NGC 3256, that might bring +the contours of the observations further away from the +PHANGS trend and hence more similar to the simu- +lation contours. However, various LVG modelings (Pa- +padopoulos et al. 2012; Harrington et al. 2021) show that +local U/LIRGs and high-z starburst galaxies generally +have αCO above 0.8 M⊙ (K km s−1 pc2)−1. In fact, a +recent study by Dunne et al. (2022) concludes that these +starburst galaxies might actually have αCO equal to the +Milky Way value by cross-correlating the CO luminos- +ity with dust and CI luminosity. Therefore, a factor of 3 +discrepancy in αvir between simulated mergers and NGC +3256 is probably real rather than due to measurement +uncertainties. +For the comparison between observations and simula- +tions, we also note that the two observed mergers are +both in an early stage after the second passage since we +can still identify two separate nuclei. In this stage, αvir +is quite time-sensitive and it is difficult to match the +exact same stage between the simulated and observed +galaxies. Therefore, it is possible that both NGC 3256 +and Antennae are caught at a specific merger stage with +a lower αvir (although in the case of NGC 3256, still +enhanced relative to PHANGS galaxies). In compari- +son, αvir in the simulations is relatively stable in the +post-merger stage. This stability suggests that a com- +parison between simulations and observations of post- +merger galaxies could be a useful next step. Moreover, +post-mergers have a rather simple morphology, which +simplifies the task of making quantitative comparisons. +It would also be interesting to compare the simula- +tion results with starburst galaxies at high redshift. Re- +cent works (e.g. Dessauges-Zavadsky et al. 2019; Meˇstri´c +et al. 2022) show that we can probe GMC-scale star- +forming clumps in gravitationally lensed objects at high +redshift. +These star-forming clumps generally show a +similar αvir to GMCs of normal spiral galaxies in our lo- +cal Universe despite using different αCO, and therefore +lower than what we see in the simulations. However, +these high-z targets likely live in a completely different +environment than our idealized mergers. +Specifically, +high-z galaxies tend to have a much higher gas fraction, +and thus can form self-gravitating clumps with low αvir +more easily (Fensch & Bournaud 2021). +5.3. Comparison with other simulations +In this work, we use the non-cosmological simulations +from Moreno et al. (2019) to compare GMC properties in +mergers and normal spiral galaxies. Two major advan- +tages of this simulation suite are that it has a resolution +of 1.1 pc (which is much smaller than typical GMC sizes) +and it can model the ISM down to low temperatures (∼ +10 K), both of which allow us to match the molecular +gas in simulations with CO observations. Various cos- +mological simulations show that mergers are responsible +for enhancing gas fractions and triggering starburst ac- +tivity (Scudder et al. 2015; Knapen et al. 2015; Patton +et al. 2013; Martin et al. 2017; Rodr´ıguez Montero et al. +2019; Patton et al. 2020, e.g.,). However, these simu- +lations can only model gas with temperatures down to +104 K and hence are incapable of capturing the turbu- +lent multi-phase structure of the ISM. An alternative +approach is to compare observations with cosmological +zoom-in simulations, which allows for higher resolution, +more realistic feedback star formation thresholds, and +more realistic modeling of the multi-phase ISM. Vari- +ous authors have explored GMC properties, mostly in +Milky-Way-like galaxies (e.g. Guedes et al. 2011; Cev- +erino et al. 2014; Sawala et al. 2014; Benincasa et al. +2020; Orr et al. 2021), and they generally reproduce the +GMC mass function in our Milky Way. However, only a +handful of work (e.g. Rey et al. 2022) has been done for +GMCs in mergers. Also, the Milky Way is identified as a +green-valley galaxy (Mutch et al. 2011) with lower SFR +than typical spiral galaxies in the local universe. There- +fore, due to the lack of zoom-in cosmological simulations + +14 +He et al. +on local mergers, we have adopted idealized simulations +for this study. +Furthermore, idealized simulations al- +low us to compare GMCs of control galaxies with those +of mergers to directly study the impact of the merging +event. +Several idealized simulations have been performed to +study molecular gas and GMC properties in mergers. +Karl et al. (2013) perform a merger simulation closely +matched to the Antennae and find a great match on +CO distributions between simulation and observations, +which suggests insufficient stellar feedback efficiencies in +the Antennae. Li et al. (2022) perform a study of GMCs +and young massive star clusters (YMCs) in Antennae- +like mergers. They find that GMC mass functions for +mergers have similar power-law slopes to normal spi- +rals during the second coalescence but with much higher +mass values. Narayanan et al. (2011) compare the αCO +in mergers and normal spiral galaxies and find that the +low αCO in mergers is mostly due to the high temper- +ature and αvir of GMCs in the merger. They predict +there is a transition stage with αCO between U/LIRG +and Milky Way values and that αvir is tightly anti- +correlated with αCO. In contrast, Renaud et al. (2019b) +show that αCO values drop quickly during each coales- +cence between two galaxies. We find similar behavior for +αvir during the second coalescence, which might imply +a similar drop in αCO (Narayanan et al. 2011). +6. CONCLUSIONS +We summarize our main conclusions below: +• Our pixel-by-pixel analysis shows that the FIRE- +2 simulation by Moreno et al. (2019) successfully +reproduces the σv vs Σmol relation for GMC-scale +pixels measured for galaxies in the PHANGS sur- +vey. +• The simulated mergers show a significant increase +in both Σmol and σv for GMC-pixels by a factor of +5 – 10 during the second passage when SFR peaks, +which brings these pixels above PHANGS-trend in +the σv vs Σmol diagram. This may indicate GMCs +in these mergers are less gravitationally bound. +We quantify this deviation by the virial param- +eter αvir and find that our simulated mergers have +αvir of 10 − 100, which is even higher than the +observed αvir in NGC 3256. +However, this dis- +crepancy could be partly due to the high impact +parameter in the initial set-up of the simulated +mergers. Furthermore, we see a good correspon- +dence between the increase in SFR and αvir, which +suggest either the starburst feedback is responsi- +ble for dispersing the gas or the correlation is in +response to gas compression. +• Our simulated mergers show a clear gas concentra- +tion in the center during the second passage, with +up to 80% of molecular gas in the central 1 kpc +region. Therefore, the GMC-pixels in the central +region tend to have the highest Σmol. We also find +these pixels tend to have the highest σv and αvir, +which could be caused by the starburst feedback +and the inflow of gas. +• We explore if αvir at GMC scales is responsible +for the varying depletion time (tdep) in observed +mergers. While we do not find a significant corre- +lation between tdep and αvir, we see a clear distinc- +tion before (small αvir, long tdep) and after (large +αvir, short tdep) the second passage. This could +be due to projection effects (multiple GMCs along +the same line of sight) during the second passage +when most of the molecular gas is concentrated in +the central 1 kpc region. The next step is to run a +cloud-identification algorithm on the data to dis- +entangle this factor. We also suspect there might +be some other mechanism, such as the stellar po- +tential and inflow of gas, that helps the GMCs in +starburst mergers to collapse and form stars. We +also find that tdep has a significant anti-correlation +with Σmol for the central region. This may be due +to higher Σmol leading to a higher fraction of dense +gas, which shortens tdep. +In the future, we would like to expand our compar- +ison to more observed and simulated mergers. +From +the observational side, we need larger samples of galaxy +mergers spanning different evolutionary stages in order +to understand how GMCs evolve throughout the merg- +ing. In addition, it is easier to compare the observations +with simulations in the post-merger stage since the mor- +phology is simpler and easier to quantify. The ALMA +archive contains ∼ 40 U/LIRGs with GMC resolution +CO 2-1 observations that can be used to build a more +complete sample of GMCs in mergers at different stages. +From the simulation side, it would be helpful to have +simulations that better match the observed galaxies. +The Antennae has been widely studied and matched by +non-cosmological simulations (e.g. Renaud et al. 2019a; +Li et al. 2022) but NGC 3256 is less well studied. Besides +comparing with these non-cosmological simulations, we +could also compare observation with cosmological sim- +ulations, such as FIREBox (Feldmann et al. 2022), that +include local mergers. + +GMCs in Galaxy Mergers +15 +We thank Dr. +Jiayi Sun for his help to access to +PHANGS data and insightful discussions about the com- +parison between simulation and observation. We thank +Dr. Nathan Brunetti for access to CO 2-1 image and +GMC catalogs of the observed mergers in his paper. +This work was carried out as part of the FIRE collab- +oration. The research of C.D.W. is supported by grants +from the Natural Sciences and Engineering Research +Council of Canada and the Canada Research Chairs +program. The research of H.H. is partially supported +by the New Technologies for Canadian Observatories, +an NSERC-CREATE training program. The computa- +tions in this paper were run on the Odyssey cluster sup- +ported by the FAS Division of Science, Research Com- +puting Group at Harvard University. Support for JM is +provided by the Hirsch Foundation, by the NSF (AST +Award Number 1516374), and by the Harvard Insti- +tute for Theory and Computation, through their Visit- +ing Scholars Program. B.B. acknowledges support from +NSF grant AST-2009679. B.B. is grateful for the gener- +ous support of the David and Lucile Packard Foundation +and Alfred P. Sloan Foundation. The Flatiron Institute +is supported by the Simons Foundation. +This +paper +makes +use +of +the +following +ALMA +data: +ADS/JAO.ALMA +#2015.1.00714.S +and +ADS/JAO.ALMA #2018.1.00272.S. ALMA is a part- +nership of ESO (representing its member states), +NSF (USA), and NINS (Japan), together with NRC +(Canada), MOST and ASIAA (Taiwan), and KASI +(Republic of Korea), in cooperation with the Republic +of Chile. The Joint ALMA Observatory is operated by +ESO, AUI/ NRAO, and NAOJ. The National Radio +Astronomy Observatory is a facility of the National Sci- +ence Foundation operated under cooperative agreement +by Associated Universities, Inc. +Facilities: ALMA +Software: +astropy (Collaboration et al. 2013), +Spectral-Cube (Ginsburg et al. 2019) +REFERENCES +Barnes, J. E., & Hernquist, L. 1996, The Astrophysical +Journal, 471, 115, doi: 10.1086/177957 +Barnes, J. E., & Hernquist, L. E. 1991, The Astrophysical +Journal, 370, L65, doi: 10.1086/185978 +Beaumont, C. 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For resolved spherical clouds, the ob- +served surface density should always be the same despite +different viewing angles. +100 +101 +102 +103 +104 +mol (M pc +2) +10 +1 +100 +101 +102 +v (km s +1) +FIRE G3, 30 deg +FIRE G3, 60 deg +FIRE G3, 80 deg +PHANGS disks +PHANGS barred galaxy centers +PHANGS unbarred galaxy centers +M31 +Figure A1. Similar to Fig. 1 but with the simulated G3 +galaxy viewed at different inclination angles (30, 60 and 80 +degrees). +For the simulated mergers, the molecular gas structure +is more complicated than a single layer of gas disk. In +this case, we might pick a specific angle where multiple +clouds happen to lie along the same line of sight, which +gives us large σv values. To test if this is the actual case, +we examine a snapshot at 2.87 when we reach maximal +Σmol and σv from a different angle (’v1’). As shown in +Fig. A2, we see similar gas distribution contour in σv vs +Σmol contour. This along with the velocity spectrum in +Fig. 6 suggests that the large σv and αvir we measured +is intrinsic properties of individual GMCs. +B. GLOBAL GAS FRACTION +On global scales, we compare the molecular gas mass +and total gas mass (including HI) of the FIRE-2 galax- +100 +101 +102 +103 +mol (M pc +2) +10 +1 +100 +101 +102 +v (km s +1) +2.87 Gyr +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +kpc +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +101 +102 +4 +2 +0 +2 +4 +kpc +vir +100 +101 +102 +Figure A2. The snapshot of FIRE-2 merger at a time of +2.87 Gyr with viewing angles of ’v1’, which is roughly per- +pendicular (with an angle of 109 degree) to the ’v0’ angle. +(Upper) the σv vs Σmol distribution of GMCs. (Lower) The +Σmol map of the snapshot. We can see that viewing from dif- +ferent angles still give us high σv measurement, which also +suggests that σv we measure is not the velocity dispersion +among GMCs along the line of sight. (lower) The Σmol and +σv for this snapshot from ’v1’ angle. +ies with observed values for normal spiral galaxies. We +compare to both the PHANGS galaxies (Leroy et al. +2021) as well as to the global gas properties from the +xCOLDGASS survey, to confirm that the PHANGS +galaxies are representative of star forming main se- +quence galaxies in our local universe. For xCOLDGASS, +the molecular gas mass is extracted from Saintonge et al. +(2017) and the total gas mass is from Catinella et al. +(2018). +In interpreting the lower Σmol values seen in Fig. +1, +one possibility is that there may not be as much gas +available to form high surface density clouds in the two +simulated galaxies compared to the PHANGS galax- +ies. Fig. B1 compares the global molecular gas masses, +Mmol, and molecular gas fractions, fmol = Mmol / M⋆, +for the FIRE-2 mergers with those of the PHANGS + +GMCs in Galaxy Mergers +21 +1010 +1011 +M⋆ (M⊙) +108 +109 +1010 +Mmol (M⊙) +PHANGS median +xCOLDGASS median +FIRE G3 +FIRE G2 +M31 +PHANGS +1010 +1011 +M⋆ (M⊙) +10−2 +10−1 +Mmol / M⋆ +1010 +1011 +M⋆ (M⊙) +109 +1010 +Mgas (M⊙) +1010 +1011 +M⋆ (M⊙) +10−1 +100 +Mgas / M⋆ +Figure B1. (Upper Left) Mmol versus M⋆ for PHANGS galaxies (salmon dots; Leroy et al. 2021), M 31 (green filled circle; +Nieten et al. 2006) and the G2 (red points) and G3 (blue points) simulated galaxies at different times in their evolution. Note +that the G2 and G3 simulated galaxies lie significantly below the star-forming main sequence defined by the xCOLDGASS +sample. (Upper Right) fmol versus M⋆ for the same galaxies. The molecular gas fractions of G2 and G3 are significantly lower +than most of the PHANGS spiral galaxies. (Lower left) total gas versus M⋆. (Lower right) total gas fraction versus M⋆. These +comparisons suggest that the low Σmol measured for the simulated galaxies might be due to the low total and molecular gas +fraction in the initial set-up. +galaxies from Sun et al. (2020). We also show the me- +dian value of Mmol and fgas in each M⋆ bin for the +PHANGS galaxies, as well as the weighted median of +M⋆ and fmol for galaxies in xCOLDGASS sample (Sain- +tonge et al. 2017). +The two median values are quite +close to each other for galaxies with M⋆ of 109.5 – 1011 +M⊙, although the PHANGS galaxies seem to deviate +somewhat from the xCOLDGASS sample in the high- +est and lowest mass bins. In contrast, the G2 and G3 +galaxies both have fmol ∼ 3 times lower than typical +PHANGS or xCOLDGASS galaxies of the same stellar +mass. Therefore, the small global fmol may be respon- +sible for producing the low Σmol values seen in the sim- +ulated galaxies. +The low values of fmol could be produced either by the +initial set-up of the simulations or by physical mecha- +nisms in the simulation that lead to inefficient conversion +of gas into the cold phase. We can distinguish between +these two options by calculating the total gas fraction +fgas including both HI and H2. The lower panel of Fig. +B1 shows the median of Mgas and fgas for the PHANGS +galaxies and xGASS-CO samples (Catinella et al. 2018) +compared to the two simulated galaxies. The values of +fgas for both simulated galaxies are still ∼ 3 times lower +than those of typical spiral galaxies with similar M⋆. +Therefore, it seems most likely that the low cold gas frac- +tion, fmol, is produced by a low total (cold+warm+hot) +gas mass in the initial set-up of the simulations. +C. SNAPSHOTS FOR ‘E1’ ORBIT +Here we show the SFR history and 3 example snap- +shots for G2&G3 ‘e1’ orbit in Fig. C1. + +22 +He et al. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time (Gyr) +100 +101 +SFR (M yr +1 +100 +101 +102 +103 +10 +1 +100 +101 +102 +v (km s +1) +t: 1.22 Gyr +SFR: 4.81 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +100 +101 +4 +2 +0 +2 +4 +vir +100 +101 +102 +100 +101 +102 +103 +10 +1 +100 +101 +102 +v (km s +1) +t: 1.46 Gyr +SFR: 23.59 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +101 +102 +4 +2 +0 +2 +4 +vir +100 +101 +102 +100 +101 +102 +103 +mol (M pc +2) +10 +1 +100 +101 +102 +v (km s +1) +t: 1.56 Gyr +SFR: 25.37 M yr +1 +FIRE G2&G3 +PHANGS galaxies +Antennae, +CO = 4.3 +NGC 3256, +CO = 1.1 +M31 +4 +2 +0 +2 +4 +kpc +4 +2 +0 +2 +4 +kpc +mol +(M pc +2) +101 +102 +4 +2 +0 +2 +4 +kpc +vir +100 +101 +102 +Figure C1. Similar plot as Figure 2 but with SFR history and 3 snapshots from ‘e1’ orbit. The interactive version of the +animation is available at https://htmlpreview.github.io/?https://github.com/heh15/merger animations/blob/main/G2G3 e1 +v0.html + diff --git a/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/load_file.txt b/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18d77a6e431dae03f8978865c31e88a27b476e4d --- /dev/null +++ b/8NFQT4oBgHgl3EQfHzW5/content/tmp_files/load_file.txt @@ -0,0 +1,1881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf,len=1880 +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='Draft version February 1, 2023 Typeset using LATEX twocolumn style in AASTeX63 Molecular gas and star formation in nearby starburst galaxy mergers Hao He,1 Connor Bottrell,2 Christine Wilson,1 Jorge Moreno,3 Blakesley Burkhart,4, 5 Christopher C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Hayward,5 Lars Hernquist,6 and Angela Twum3 1McMaster University 1280 Main St W, Hamilton, ON L8S 4L8, CAN 2Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, University of Tokyo Kashiwa, Chiba 277-8583, Japan 3Department of Physics and Astronomy, Pomona College, Claremont, CA 91711, USA 4Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', Piscataway, NJ 08854, USA 5Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA 6Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA (Received February 1, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Revised xxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Accepted xxx) Submitted to ApJ Letter ABSTRACT We employ the Feedback In Realistic Environments (FIRE-2) physics model to study how the prop- erties of giant molecular clouds (GMCs) evolve during galaxy mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We conduct a pixel-by-pixel analysis of molecular gas properties in both the simulated control galaxies and galaxy major mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The simulated GMC-pixels in the control galaxies follow a similar trend in a diagram of velocity disper- sion (σv) versus gas surface density (Σmol) to the one observed in local spiral galaxies in the Physics at High Angular resolution in Nearby GalaxieS (PHANGS) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For GMC-pixels in simulated mergers, we see a significant increase of factor of 5 – 10 in both Σmol and σv, which puts these pixels above the trend of PHANGS galaxies in the σv vs Σmol diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This deviation may indicate that GMCs in the simulated mergers are much less gravitationally bound compared with simulated control galaxies with virial parameter (αvir) reaching 10 – 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, we find that the increase in αvir happens at the same time as the increase in global star formation rate (SFR), which suggests stellar feedback is responsible for dispersing the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also find that the gas depletion time is significantly lower for high αvir GMCs during a starburst event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is in contrast to the simple physical picture that low αvir GMCs are easier to collapse and form stars on shorter depletion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This might suggest that some other physical mechanisms besides self-gravity are helping the GMCs in starbursting mergers collapse and form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Keywords: ISM: clouds, ISM: kinematics and dynamics, ISM: structure, galaxies: interactions, galaxies: starburst, galaxies: star formation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' INTRODUCTION Corresponding author: Hao He heh15@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='ca Despite the diversity of galaxy morphology and envi- ronment, giant molecular clouds (GMCs) are the sites of star formation across cosmic time (Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Chevance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' As one of the most promising star formation model, the turbulence model (Krumholz & McKee 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Hennebelle & Chabrier 2011) suggest a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='13250v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='GA] 30 Jan 2023 2 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' relatively uniform star formation efficiency per freefall time (ϵff) for individual GMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' They predict that the observed scatter in ϵff could be account for by the diver- sity in GMC properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='virial parameter αvir and Mach number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2016) show that the observed scatter is larger than these early theoret- ical predictions expected and updated models suggest that cloud evolution, in addition to initial conditions such as Mach number and αvir, should be accounted for (see Burkhart 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Mocz & Burkhart 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Burkhart & Mocz 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, Grudi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018) show in their simulation that GMCs in starburst galaxies can have different ϵff in normal spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Hence, to un- derstand the links between GMCs and star formation in galaxies, it is essential to study various GMC properties in a broad range of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, modeling of GMCs starting from the scales of galaxies and cosmological zoom-ins is complicated by challenges in capturing the structure of the coldest and densest gas, which is heavily affected by various numeri- cal choices, such as resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010) and the treatment of feedback (Fall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' My- ers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Raskutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Grudi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Most resolved GMC simulations focus on the evolution of individual GMCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Burkhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Decataldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Burkhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020) and ignore the wider environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Only a hand- ful of galaxy simulations have the ability to model GMC populations inside Milky-Way-like galaxies (Jeffreson & Kruijssen 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Benincasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020) and mergers (Re- naud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' High-resolution CO observations have successfully characterized GMCs in the Milky Way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Rice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Rico-Villas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Miville-Deschˆenes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Lada & Dame 2020) and nearby galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Donovan Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Schruba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In particular, the recently com- pleted PHANGS-ALMA survey (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021) has expanded these observations across a complete sample of nearby spiral galaxies, providing direct measurements of molecular gas surface density Σmol, velocity disper- sion σv and size of GMCs, which are key quantities for determining the physical state of GMCs (Larson 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Observations show that the correlation between σ2 v/R and Σmol is nearly linear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', Heyer & Dame 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018, 2020), which is consistent with the theoreti- cal prediction that most clouds follows the Larson’s sec- ond law (Larson 1981), which indicates a constant ratio between clouds’ kinetic energy and gravitational poten- tial energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This universal correlation provides us with a starting point to study how other galactic environ- mental factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', external pressure, stellar potential) influence the dynamical state of GMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Unlike studies targeting isolated galaxies, GMCs in starburst galaxy mergers are less well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the observational side, the scarcity of nearby mergers means that we have only a handful of systems with GMC res- olution data (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Whit- more et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Elmegreen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Brunetti 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' S´anchez-Garc´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Bel- locchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These studies show that GMCs in mergers have significantly higher gas surface densities and are less gravitationally bound compared to GMCs in normal spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, it is difficult to draw sta- tistically robust conclusions on how GMC properties evolve across various merging stages based on these lim- ited number of local galaxy mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the simulation front, only a handful of studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fensch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017) have the ability to probe the cold gas at ∼pc scale starting from cosmo- logical scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Using a comprehensive library of idealized galaxy merger simulations based on the FIRE-2 physics model, Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) show that SFR enhance- ment is accompanied by an increase in the cold dense gas reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This simulation suite thus provides us with the ideal tool to properly examine GMC evolution along the entire merging sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This paper explores how GMC properties evolve dur- ing the starburst merging event using the FIRE-2 merger suite from Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) and performs compar- isons with observations to test the simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In Section 2, we describe this simulation suite and the observational data used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Section 3 com- pares the σv −Σmol relation between control simulated galaxies and normal spirals in the PHANGS-ALMA sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Section 4 examines the σv −Σmol relation for mergers in both observations and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In Sec- tion 5, we discuss and interpret various aspects of the comparison between observations and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' DATA PROCESSING 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Simulated data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The FIRE-2 Model We use the FIRE-2 model (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018), which employs the hydrodynamic code GIZMO (Hopkins 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Compared with the previous version, FIRE-2 adopts the updated meshless finite-mass (MFM) mag- netohydrodynamics (MHD) solver, which is designed to capture the advantages of both grid-based and particle- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We refer the reader to Hopkins (2015) and Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The model includes GMCs in Galaxy Mergers 3 treatment of radiative heating and cooling from free- free, photo-ionization/recombination, Compton, photo- electric, dust-collisional, cosmic ray, molecular, metal line, and fine-structure processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Star formation occurs in gas that is self-gravitating (3D αvir < 1 at the reso- lution scale), self-shielded, and denser than 1000 cm−3 (see Appendix C of Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Stellar feed- back mechanisms include (i) mass, metal, energy, and momentum flux from supernovae type Ia & II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (ii) con- tinuous stellar mass-loss through OB/AGB winds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (iii) photoionization and photoelectric heating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' and (iv) radi- ation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Each stellar particle is treated as a single stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Mass, age, metallicity, luminosity, energy, mass-loss rate, and stellar feedback event rate for each stellar particle are calculated using the STAR- BURST99 stellar population synthesis model (Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The model does not account for feedback generated via accretion of gas onto a supermassive black hole (SMBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Our FIRE-2 galaxy suite Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) present a suite of idealized galaxy merger simulations (Initial conditions are manually set instead of from cosmological simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' see also Bot- trell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' McElroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) covering a range of orbital parameters and mass ratios between 4 disc galaxies (G1, G2, G3 and G4, in or- der of increasing total stellar mass of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='21, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='24, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='97 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5×1010 M⊙), along with separate runs for each disk galaxy in isolation (the control runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Their orbit settings contain 3 orbital spin directions, 3 impact pa- rameters and 3 impact velocities (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 3 in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For these simulations, the highest gas den- sity and spatial resolution are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 × 105 cm−3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The gravitational softening lengths are 10 pc for the dark matter and stellar components and 1 pc for the gaseous component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The mass resolution for a gas particle is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 ×104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The time resolution of a typical snapshot is 5 Myr (See further details in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Orbital Parameter of ‘e1’ and ‘e2’ orbit e1 e2 Apo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (kpc)a 60 120 Peri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (kpc)a 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 a First apocentric distance between the centers of two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' b Second pericentric distance between the centers of two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For our analysis, we focus on the simulation run of isolated G2 and G3 galaxies along with one of G2&G3 merger suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The detailed information of G2 and G3 galaxies is in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The G2&G3 merger suites have a mass ratio of 1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 and hence are similar to major mergers such as the Antennae and NGC 3256 for which we have observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In addition, G2 and G3 have stellar masses within the range of the PHANGS sample (1010–1011 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We choose the ‘e’ orbit (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2006, roughly prograde), which is expected to maximally enhance the star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In most of our analyses, we focus on the ‘e2’ orbit since this is the fiducial run in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We use the ‘e1’ orbit as a comparison in some cases as it has smaller impact parameter and is more similar to the orbit of the Antennae merger (Privon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013), for which we have GMC observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The pericentric distance of ‘e1’ and ‘e2’ orbit is listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Molecular gas We follow the scheme in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) to sepa- rate the ISM of our simulated galaxy mergers into 4 com- ponents based on temperature and density: hot, warm, cool, and cold-dense gas, which roughly correspond to the hot, ionized, atomic, and molecular gas in observa- tions (see Table 4 in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The com- ponents that are most important for this work are the cool (temperatures below 8000 K and densities above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 cm−3) and the cold-dense gas (temperatures below 300 K and densities above 10 cm−3), which corresponds to H I and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This choice captures HI and H2 gas reasonably well (Orr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Orr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018) also demonstrate that using this threshold to separate H2 and HI yields reasonable agreement with the observed Kennicutt-Schmidt law (Kennicutt 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Kennicutt & Evans 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In the following, we refer to total gas as the sum of the gas in the cool and cold-dense phases (simulations) or in the atomic and molecular phases (ob- servations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We adopt the same definition of molecular gas as in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) (temperature below 300 K and den- sity above 10 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Guszejnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2017) demon- strate that the model successfully reproduces the GMC mass function in the Milky Way (Rice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016) and the size-linewidth relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', the Larson scaling re- lationship, Larson 1981) in our Galaxy (Heyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Heyer & Dame 2015) and in nearby galaxies (Bo- latto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fukui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Muraoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Roman-Duval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Tosaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Given the density cut of 10 cm−3 and mass resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 ×104 M⊙, the lower limit of our spa- tial resolution ( 3� M/ρ, where M is the mass resolution and ρ is the mass volume density) is ∼ 40 pc, which 4 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' is smaller than the typical scale of observed GMCs (40 – 100 pc, Rosolowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In addition, GMC mass function peaks at 105 – 106 M⊙ in Milky Way (Rice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016), which is significantly larger than our mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, we would generally expect more than 1 gas particle is included for molecular gas in each GMC-scale pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For generating different components of the ISM, the simulations start with a homogeneous ISM with a tem- perature of 104 K and solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The multi-phase ISM then emerges quickly as a result of cooling and feed- back from star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The initial gas mass for the simulation is set to match the median HI mass from the xCOLDGASS survey (Catinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Data cubes We first convert the FIRE-2 molecular gas data into mass-weighted position-position-velocity (p-p-v) data cubes to match the format of the CO data from radio observations (McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We adopt the cube construction method created for Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) and Bottrell & Hani (2022) and then adapted to the FIRE-2 merger suite by McElroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Kine- matic cubes are produced along four lines-of-sight (la- beled as ‘v0’, ‘v1’, ‘v2’, ‘v3’), defined by the vertices of a tetrahedron centered at the primary galaxy (G3 in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the isolated galaxy simulations, we generate p-p-v data cubes at different inclination angles (10 – 80 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We adopt a pixel size of 100 pc and velocity resolution of 2 km s−1, which is similar to PHANGS choice (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The field of view (FOV) for the data cube is set to be 25 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Then we create zeroth-moment maps of the gas surface density Σmol and second-moment maps of the velocity dispersion σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We do not set any thresholds on these moment maps since we argue that every gas particle in the simulated cube should be treated as a real signal, rather than observational noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, in later anal- yses, when we display σv versus Σmol for the simulated data, we select pixels with Σmol greater than 1 M⊙ pc−2, which approximates the lower limit of the molecular gas detection threshold in the observational data (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also exclude pixels detected in fewer than two velocity channels in the simulated cube to exclude inaccurate measurements of σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' To characterize clouds, we use a pixel-based analy- sis (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2016), which treats each pixel as an individual GMC, rather than identifying each individ- ual cloud from the data cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This approach has been widely applied to GMC analyses for PHANGS galax- ies (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Compared to the tradi- tional cube-based approach, this new method requires minimal assumptions and can be easily applied to many datasets in a uniform way, while still giving us the essen- tial GMC properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', molecular gas surface density Σmol, gas velocity dispersion σv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the other hand, the pixel-based method has a major disadvantage of not able to decompose different cloud components along the same line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Several observational studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Brunetti & Wilson 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) have compared this new approach with the traditional approach and found good agreement on cloud properties between two methods for both normal spiral galaxies and starburst mergers, especially for clouds in galaxy disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These comparisons show pixel-based analysis should be valid for capturing individual cloud properties, especially for galaxy disks which generally have single-layer of GMCs (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 for detailed discussion about the pro- jection effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In this work, we adopt this approach to match the method in Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020) and Brunetti (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also note that since we treat each pixel as a GMC, these GMCs do not necessarily represent inde- pendent ISM structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In fact, given the mass resolu- tion of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 ×104 M⊙, we can barely resolve the internal structure of most massive GMCs of ∼ 106 M⊙ (100 ele- ments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We refer to them as GMCs in this paper to be consistent with similar observational analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Star Formation Rate Maps To further explore how the GMC properties at 100 pc scale affect the star formation, we also make SFR maps with the same resolution of 100 pc for the simulated mergers at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We create these maps using a method similar to the one used to create the gas cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We include all the stellar particles with age younger than 10 Myr and create p-p-v data cubes for these stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The mass-weighted cubes are integrated along the velocity axis to produce 2D maps of stellar mass formed within the last 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These surface-density mass maps are subsequently divided by 10 Myr to obtain the average star-formation rates over the last 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Observational Data We use several sets of observations for comparison with our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Spiral galaxies: PHANGS data For isolated galaxies, we mainly use the PHANGS data from Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020) with resolution of 90 pc, which is comparable to our pixel size choice of 100 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020) apply the pixel-based method for sta- tistical analyses of GMC properties for 70 galaxies in the PHANGS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also include GMC data for GMCs in Galaxy Mergers 5 M31 from Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018) at resolution of 120 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' M31 is identified as a green-valley galaxy, similar to our own Milky Way, and hence has a lower total gas fraction than normal spiral galaxies (Mutch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Both M31 and the Milky Way seem to be in a transition from blue spiral galaxies to quenched galaxies via depletion of their cold gas (Bland-Hawthorn & Gerhard 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' M31 has stellar mass of 1011 M⊙ (Sick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2015), H2 mass of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 × 108 M⊙ and HI mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 × 109 M⊙ (Nieten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Galaxy mergers: the Antennae and NGC 3256 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Information about the observed mergers in this work Antennae NGC 3256 #References M⋆ (1010 M⊙)a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2) Mmol (1010 M⊙)b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' this work SFR (M⊙yr−1) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 50 (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (4) Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (kpc)c 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (4) tnow (Myr)d 40 · · (6) mass ratiof 1:1 · · (6) Peri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sep (kpc)g 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 · · (6) Notes: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Molecular gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Current separation between two nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Current time since the sec- ond passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Mass ratio of the two progenitor galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Pericentric distance of two nuclei from the simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' References: (1) Seill´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) (2) Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2010) (3) Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2000) (4) Sakamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2014) (5) Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2001) (6) Karl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2010) We use the CO 2-1 data for NGC 3256 (Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020) and the Antennae (Brunetti (2022, Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' in prep)) at resolutions of 90 and 80 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The GMC measurements use the same pixel-based ap- proach as in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Both NGC 3256 and the Antennae are identified as late-stage major mergers that have been through their second perigalactic pas- sage (Privon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' NGC 3256 has stellar mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 × 1011 M⊙, total molecular gas of 8 × 1019 M⊙ (calculated based on CO 2-1 map in Brunetti & Wil- son (2022), assuming αCO of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M⊙ (K km s−1 pc2)−1 and CO 2-1/1-0 ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8) and SFR of 50 M⊙yr−1 (Sakamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In contrast, the Antennae has a stellar mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 × 1010 M⊙ and SFR of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 M⊙yr−1 (Seill´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' NGC 3256 currently has a more intense starburst, perhaps because it is at different evo- lutionary stage in the merging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The detailed information is in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' To convert the CO 2-1 emission to molecular gas mass requires the assumption of a CO-to-H2 conversion factor (αCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The exact value of αCO has large uncertainties and varies significantly among different types of galax- ies, especially for starburst galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Downes & Solomon (1998) find that for starburst U/LIRGs, the αCO value is generally 4 times smaller than that in our Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The major method for direct measurement of αCO is through large velocity gradient (LVG) radiative trans- fer modeling of multiple CO and its isotope lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For αCO in the Antennae, various LVG modeling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Schirm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014) suggests that the An- tennae has αCO close to the Milky Way value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 M⊙ (K km s−1 pc2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is also supported by the galaxy simulation that specifically matches the Anten- nae (Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For NGC 3256, we do not have a direct measurement of αCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We therefore adopt the treatment from Sargent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2014) to determine the αCO for an individual galaxy as αCO = (1 − fSB) × αCO,MS + fSB × αCO,SB, (1) where αCO,MS and αCO,SB are the conversion fac- tors for the Milky Way (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 M⊙ (K km s−1 pc2)−1) and U/LIRGs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M⊙ (K km s−1 pc2)−1, including he- lium), and fSB is the probability for a galaxy to be a starburst galaxy, which is determined by its deviation from the star-forming main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We adopt the star-forming main sequence relation from Catinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018), log sSFRMS = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='344(log M⋆ − 9) − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='822, (2) where sSFR = SFR / M⋆ is the specific star forma- tion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' NGC 3256 has an sSFR/sSFRMS ratio of 15 (Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020), which suggests NGC 3256 should have αCO close to the U/LIRG value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M⊙ (K km s−1 pc2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, in the following anal- yses, we will adopt αCO of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 M⊙ (K km s−1 pc2)−1 for the Antennae and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M⊙ (K km s−1 pc2)−1 for NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' CONTROL (ISOLATED) GALAXIES To test if the simulation successfully reproduces ob- served GMCs, Figure 1 shows the well-known correla- tion between σv and Σmol for isolated simulated galaxies and PHANGS-ALMA spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We show σv ver- sus Σmol contours for G2 and G3 galaxies at an inclina- tion angle of 30 degrees, compared with that of observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two simulated galaxies exhibit similar properties (black and dark red solid contours) and gener- ally lie on the trend followed by the PHANGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also plot a red dashed line indicating GMCs with constant virial parameter αvir of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the pixel-based 6 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 100 101 102 103 104 mol (M pc 2) 10 1 100 101 102 v (km s 1) t: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr FIRE G2 FIRE G3 PHANGS disks PHANGS barred galaxy centers PHANGS unbarred galaxy centers M31 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Velocity dispersion versus gas surface density for the G2 (black solid contour) and G3 (brown solid contour) simulated galaxies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr with inclination angle of 30 degrees compared to the PHANGS galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The con- tours are mass-weighted and set to include 20%, 50% and 80% of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The density contours of PHANGS galaxies (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020) show the distribution of measurements in galaxy disks (blue shaded contours), the centers of barred galaxies (salmon shaded contours) and the centers of un- barred galaxies (brown dashed contours) with a resolution of 90 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The red dashed line marks the position of the me- dian values of αvir for PHANGS galaxies of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also show the data for M31 (green solid contour) at 120 pc resolution from Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see that the FIRE-2 spiral galaxies follow the same σv- Σmol relation as the PHANGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' analysis, αvir is calculated as (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018) αvir = 9 ln 2 2πG σ2 v ΣmolR = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='77 � σv km s−1 �2 �Σmol M⊙ �−1 � R 40pc �−1 , (3) where R is the GMC radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018), R is set to be the radius of the beam in the image, as each beam is treated as an independent GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see both our simulated galaxies and observed PHANGS galaxies follow the trend of the constant αvir, which yields the relation of σ2 v ∝ Σmol that suggests the simulations re- produce GMCs similar to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, we can see that the two galaxies lie at the low surface- density end of the PHANGS distribution and thus their gas properties are more similar to those of M31 than a typical PHANGS galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Indeed, the molecular and to- tal gas properties of the simulated galaxies are similar to those of M 31, perhaps due to the choice of initial gas mass in the simulations (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' MERGING GALAXIES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' GMC linewidth and surface density We performed a similar σv versus Σmol analysis for our suite of galaxy merger simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Since we are particularly interested in how the starburst activity in- fluences GMC properties, we focus on the period right before and after the second passage where we can see the largest contrast in SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2 we show some ex- ample snapshots of σv versus Σmol for different merger stages during the second passage, along with Σmol and αvir maps at each snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Note that the datacube is centered on the primary galaxy G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At the time of first snapshot (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='54 Gyr), right before the start of the second perigalactic passage, the simulated mergers still have Σmol and σv that are similar the isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Then the molecular gas quickly transitions to a more turbulent state with much higher σv after the second passage along with a dramatic increase in global SFR, as shown in the snapshot for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='66 Gyr (middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The merger at this time still shows two sep- arate nuclei in the zeroth moment map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' this is similar to our observed mergers, the Antennae and NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At this time, the σv versus Σmol contours for the simu- lated merger lie above the trend seen for the PHANGS galaxies, similar to NGC 3256, but in contrast to the An- tennae, which still lies along the trend of the PHANGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The larger deviation above the PHANGS trend implies higher αvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We note that different αCO choices will affect the position of the contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we choose the ULIRG αCO instead of the Milky Way value, the An- tennae would have αvir similar to that of NGC 3256 and our G2&G3 merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The uncertainty in the correct αCO value to use makes it difficult to interpret the data for the Antennae in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2 shows the snapshot at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr, which marks the post-merger stage after the final coalescence of two nuclei (defined here as the time at which the two central supermassive black holes are at a distance of 500 pc for the last time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is the time when both Σmol and σv reach their highest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see that most of the molecular gas is concen- trated in the central 1 kpc region, with Σmol reaching 1000 M⊙ pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' σv reaches 200 km s−1, which is even higher than the σv observed in NGC 3256, which is in an earlier merging stage when the two nuclei have not yet coalesced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' To better quantify the variation of Σmol and σv during the second passage, we plot the 16th, 50th and 84th per- centile of the mass-weighted values for all pixels of each snapshot during the second passage in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also normalize both the median Σmol and σv to the median GMCs in Galaxy Mergers 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 Time (Gyr) 100 101 SFR (M yr 1 100 101 102 103 10 1 100 101 102 v (km s 1) t: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='54 Gyr SFR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='65 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 4 2 0 2 4 kpc mol (M pc 2) 100 101 4 2 0 2 4 vir 100 101 102 100 101 102 103 10 1 100 101 102 v (km s 1) t: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='66 Gyr SFR: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='04 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 4 2 0 2 4 kpc mol (M pc 2) 101 102 4 2 0 2 4 vir 100 101 102 100 101 102 103 mol (M pc 2) 10 1 100 101 102 v (km s 1) t: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr SFR: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='22 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 kpc 4 2 0 2 4 kpc mol (M pc 2) 101 102 4 2 0 2 4 kpc vir 100 101 102 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Top) SFR history for the G2&G3 merger with ‘e2’ orbit with viewing angle of ‘v0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The 3 solid black vertical lines indicate the time for each snapshot displayed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two dashed lines indicate the times at the start of second merging and the final coalesce of two nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Bottom) Three snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For each snapshot, the left panel shows the σv versus Σmol mass-weighted contour with the same setting as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also show the density contours for the PHANGS galaxies (filled blue region), NGC 3256 (blue contours) and the Antennae (orange shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For NGC 3256, Σmol is calculated using the ULIRG αCO of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M⊙ (K km s−1 pc2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the Antennae, the gas surface density is calculated using the Milky Way αCO of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 M⊙ (K km s−1 pc2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The red dashed line indicate the line of constant αvir of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The right two panels show the Σmol and αvir maps of inner 5 kpc regions where we have most of our detected pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The interactive version of the animation is available at https://htmlpreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='io/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='com/heh15/merger animations/blob/main/G2G3 e2 v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see that the properties of the GMCs right before the second passage still resemble those of normal spiral galaxies, while GMCs after the second passage lie above the PHANGS trend in the σv vs Σmol plot and show significantly higher αvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 8 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 101 102 Σmol(t) (M⊙ pc−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 100 101 Σmol(t) / Σmol,0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 101 102 σv (km s−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 100 101 σv(t) / σv,0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The Σmol and σv variation across the second passage and final coalescence of the G2&G3 merger at ‘e2’ orbit with viewing angle of ‘v0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two dashed vertical lines indicate the times when the simulated merger begin the second passage and experience final coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The three solid vertical lines correspond to the 3 snapshots shown in Fig, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The horizontal dashed lines indicate the median value of the isolated G3 galaxy at time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1) as a baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Upper left) Σmol vs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Blue lines shows the mass weighted median Σmol of the entire merger while the orange filled area indicates Σmol range between 16th and 84th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two dashed lines indicate the time for the start of the second passage and the final coalesce of the two nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Upper right) The ratio between median Σmol at given time and the median value Σmol,0 for the isolated G3 galaxy at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Lower left) The mass-weighted median σv versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Lower right) The ratio between the median σv and the value σv,0 for isolated G3 galaxies at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see both Σmol and σv increase dramatically during the second passage when the extreme starburst happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' values of the isolated G3 galaxy at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1) to show how the merging event affects the GMC prop- erties during the second passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Both Σmol and σv in- crease significantly during the merger, with a maximum increase of a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The increase in σv and Σmol is roughly of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 3 shows that a con- stant αvir requires σ2 v ∝ Σmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These results imply that our simulated merger will have higher αvir compared to PHANGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The virial parameters of GMCs During the second passage, we see that the σv vs Σmol distribution for our simulated merger lies above the trend observed for the PHANGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' A higher σv for a given Σmol means the GMCs in these mergers are more turbulent and less gravitationally bound than in normal spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We adopt the same approach as in observations to calculate αvir for pixel-based GMC pixels using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Since the simulation data do not have a telescope “beam” and each pixel in this analysis is treated as an independent GMC, we set R to be half the size of each pixel (50 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' With constant R, αvir depends only on σv and Σmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Higher σv at a similar Σmol thus im- plies that αvir values for GMCs in simulated mergers are higher than the values for PHANGS or simulated isolated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Higher values for αvir are also found for NGC 3256 (Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Brunetti & Wilson 2022) and the Antennae (Brunetti 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4 shows αvir as a function of time during the pe- riod near the second pericentric passage for the merger simulations with “e2” and “e1” orbits and viewed from “v0” angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' αvir stays low before the second passage and suddenly rises after the passage along with a sudden in- crease in SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The peak of median αvir can reach ∼100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' After the second passage, αvir gradually dies down as the SFR also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' During the entire merging pro- cess, we generally see a good correspondence between the SFR and αvir peaks, which suggests that the αvir value is either regulated by feedback from star forma- tion or that both SFR and αvir increase together as a result of the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' αvir for our fiducial ‘e2’ orbit is generally higher than that of the ‘e1’ orbit and stays at higher values for a significantly longer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The ‘e2’ orbit has a higher impact parameter than the ‘e1’ orbit (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, we would expect more gravitational poten- tial energy transferred to the kinetic energy of individual GMCs, potentially making these GMCs less gravitation- ally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The αvir values for the ‘e1’ orbit are more GMCs in Galaxy Mergers 9 100 101 SFR (M⊙ yr−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 101 102 103 αvir e2 orbit PHANGS NGC 3256 Antennae 101 SFR (M⊙ yr−1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 Time (Gyr) 101 102 103 αvir e1 orbit PHANGS NGC 3256 Antennae Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' αvir versus time for the G2&G3 mergers in (left) the e2 orbit and (right) the e1 orbit viewed from ‘v0’ angle during the final coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Left) The red line is the mass-weighted median for αvir from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The orange shaded region includes data within the 16th and 84th quantile of αvir values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The dashed lines correspond to the start of the second passage and the final coalescence of the two nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The three solid lines correspond to the merger times shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The horizontal dashed line indicates the median αvir for the isolated G3 galaxy at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='625 Gyr (Fig .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1) as a baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The upper panel shows SFR versus time for the second coalescence and the right panel shows the 16th, 50th and 84th quantile of αvir for PHANGS, NGC 3256 and the Antennae from the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In calculating αvir, we use the U/LIRG αCO for NGC 3256 and the Milky Way value for PHANGS and the Antennae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Right) Same plot for G2&G3 merger in the ‘e1’ orbit during the final coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The 3 solid lines correspond to 3 snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The ‘e1’ orbit has a smaller impact parameter than the ‘e2’ orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see the global αvir increases dramatically right after the second passage as SFR rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The peak SFR also roughly corresponds with the peak αvir, which suggests the high αvir might be caused by the feedback from the starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' similar to the αvir of NGC 3256 and the Antennae and the ‘e1’ orbit is more similar to the orbit of the Anten- nae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We note that both the Antennae and NGC 3256 are at the very start of their second passages (Privon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At this stage, there are significant variations in αvir, which makes it difficult to pick the exact snapshot that matches the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we use the U/LIRG αCO instead of the Milky Way value, αvir for the Antennae would be similar to that of NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We will discuss our αCO choices further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Molecular Gas in the central 1 kpc region From the moment 0 maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2, we can see that most molecular gas is concentrated in the center during the post-merger phase after 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='83 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is consis- tent with the traditional scenario that the central star- burst activity is caused by the inflow of molecular gas due to the loss of angular momentum (Hernquist 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Barnes & Hernquist 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Mihos & Hernquist 1994, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Barnes & Hernquist 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' To quantify how much of the molecular gas is concen- trated in the center, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 5 shows the molecular gas mass within the central 1 kpc, and the ratio between this value and total molecular gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The fraction of molecular 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 0% 20% 40% 60% 80% 100% Mmol, central / Mmol, total Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The ratio between molecular gas mass within the central 1 kpc radius circle of the G3 galaxy and total molecular gas inside our FOV of 25 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' During the second coalescence between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='7 Gyr and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 Gyr, more than 50% of molecular gas is concentrated within the central 1 kpc region, which indicates the Σmol increase we see in the simulated merger during the second passage is probably due to this gas concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' gas concentrated in the center reaches as high as 80% for a significant period of time (∼ 500 Myr) around the final coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the other hand, Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) shows that the total molecular gas mass decreases dur- ing the second passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, the overall high Σmol 10 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' values of GMCs across our simulated merger compared to isolated galaxies are mostly due to the central gas concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 6 shows the σv versus Σmol distribution for pix- els in the central kpc region of the G2&G3 merger at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr (red aperture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2), along with pix- els in the center of PHANGS galaxies, the Antennae and NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see the pixels in the center of the G2&G3 merger have a larger deviation from the PHANGS trend than NGC 3256, which indicates that the G2&G3 merger has GMCs with larger αvir in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also show the mass weighted median αvir for the entire and central region of G2&G3 merger as a func- tion of time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 6 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' αvir in the center is generally higher than for the entire region, which indicates that GMCs in the center are more perturbed and less grav- itationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At the time right after the second passage, we see dramatic fluctuations of αvir for both the center and the entire galaxy, which is probably due to the complex and constantly varying gas morphology during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreover, we might see two GMCs that are far apart in 3D space but lie along the same line of sight, which cause large measured αvir value, but in a short time they no longer lie along the same line of sight, which causes a sudden drop of αvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At the post- merger phase, αvir values are more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, we see that αvir of the disk region gradually settles down while the central αvir keeps increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This might indi- cate that GMCs in the central region take more time to settle down to their normal states, which may be due to the starburst activity in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also see high αvir for the center at the very start (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='54 Gyr), which prob- ably means GMCs in the center at this time have not recovered from the starburst event that occured during the first peri-galactic passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Correlation between the central SFR and GMC Properties The driving mechanism behind the SFR enhancement in mergers is of great interest to the study of star forma- tion and galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' One approach to tackle this problem is to decompose the SFR into the following 2 terms SFR = Mmol tdep , (4) where tdep is the depletion time, defined as the time for star formation to consume the available molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This approach makes it clearer that the rise in SFR could be either due to a larger amount of molecular gas “fuel driven”) or shorter depletion time (“efficiency driven”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021) and observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Thorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) indicate that both terms contribute to the SFR enhancement at kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreover, many studies of the Kennicutt-Schmidt rela- tion in U/LIRGs at kpc scales show that these starburst mergers have relatively short tdep of ∼ 108 yr compared to normal spiral galaxies of ∼ 109 yr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010), which confirms the role of efficiency driving in mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' With our simulations being able to probe the molecular gas at GMC scales, we can explore how tdep is correlated with GMC populations in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For this analysis, we focus on the molecular gas and star formation in the central 1 kpc region since most gas is concentrated here during the second passage (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We measure the mass-weighted median αvir in this central region as a metric for GMC dynamical state in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 7 shows Σmol and ΣSFR color-coded by αvir for the central region as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We calculate the average Σmol and ΣSFR by dividing the to- tal Mmol or SFR in the central region by the aperture size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We show the data points within the period of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='54 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='61 Gyr (before the second passage) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='73 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='47 Gyr (after the second passage) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We exclude the data points between the start of the second passage (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='62 Gyr) and the time when the central/total gas frac- tion starts to reach 50% (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='73 Gyr) because data points from this period show a large deviation from the major trend in ΣSFR vs Σmol diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The large deviation is probably because the limited amount of molecular gas is highly perturbed in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Gas is either quickly consumed without being replenished in time, or just concentrated and has not formed stars yet, which causes the large scatter in the ΣSFR vs Σmol relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the other hand, before and after this period, the central region is in a relatively stable state when the molecu- lar gas is constantly replenished to fuel star formation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 7, we can see that tdep be- comes shorter as Σmol and ΣSFR increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The points at the lower left end of the Σmol correspond to the times be- fore the second passage, which also have relatively low αvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In contrast, the αvir after the second passage is significantly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also note that tdep even before the second passage (∼ 108 yr) is quite shorter than that of normal spiral galaxies (109 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The difference could be due to different dynamical timescales of simulated and observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' At this time, we can see αvir for the central region is already ∼ 10 which indicates the molecular gas in the central region has already been disturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' There is no significant correlation between tdep and αvir, with Spearman coefficient of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='08 for all data points and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='18 for data after the second passage, which is against our expectation that low αvir gas form GMCs in Galaxy Mergers 11 101 102 103 104 mol (M pc 2) 100 101 102 v (km s 1) t: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr FIRE G2&G3 FIRE G2&G3 Center PHANGS galaxy center Antennae nucleus NGC 3256 nucleus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 101 102 vir G2&G3 G2&G3 center Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Left) The σv versus Σmol contour for the entire (dashed contour) and central 1 kpc region (solid contour) of the G2&G3 merger at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr viewed from the ‘v0’ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also show contours for the centers of PHANGS galaxies (brown dashed contours), the Antennae (orange shaded contours) and NGC 3256 (blue contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see the central region in our simulated merger generally has the highest σv and αvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Right) The mass weighted median αvir for molecular gas in the entire (blue) and central (orange) region of G2&G3 merger viewed from ‘v0’ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We see that αvir for the entire disk gradually settles back to the original low value, while that for the central region keeps a high value until the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 101 mol (M pc 2) 10 1 100 SFR (M yr 1 kpc 2) tdep = 107 yr tdep = 108 yr 101 102 vir 101 102 vir 107 108 tdep (yr) rs all: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='08 rs after: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) 101 mol (M yr 1 pc 2) 107 108 tdep (yr) rs all: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='47 rs after: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4 Time (Gyr) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Left) SFR surface density ΣSFR versus Σmol color coded by the mass-weighted median αvir for the central 1 kpc region of the the G2&G3 merger during the second passage of the ‘e2’ orbit viewed from ‘v0’ orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We include simulated data points within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='54 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='61 Gyr (before the second passage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' trangle) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='73 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='47 Gyr (after the second passage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Both ΣSFR and Σmol are calculated as total central SFR or Mmol within 1 kpc radius divided by the aperture size, while αvir is the mass weighted median of pixels inside the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two dashed line indicate constant depletion times (tdep = Σmol/ΣSFR) of 107 and 108 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Middle) tdep versus αvir for the central 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Right) tdep versus Σmol for the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The label “rs all” shows the Spearman coefficient between tdep and αvir and Σmol for all data points while “rs after” shows the Spearman coefficient only for data after the second passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see there is no significant correlation between αvir and tdep, which is against our expectation that low αvir clouds will consume molecular gas at faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' stars more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can also clearly see a distinc- tion between αvir before and after the second passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The αvir before the second passage is relatively small and corresponds to larger tdep while the αvir after the second passage is significantly larger but corresponds to shorter tdep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This again is inconsistent with our expecta- tion that low αvir GMCs should form stars more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Other physical mechanisms rather than self-gravity of individual GMCs may be needed to help the molecular gas to collapse (see detailed discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' On the other hand, we see an anti-correlation between tdep and Σmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This relation is similar to the global Kennicutt-Schmidt relation where the gas rich U/LIRGs have the shorter tdep (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' One expla- 12 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' nation for this trend is that the fraction of dense gas (traced by HCN) that are actually forming stars (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' traces the self-gravitating gas fraction) is increasing as Σmol increases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Gao & Solomon 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Bemis & Wilson 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we assume Σmol is proportional to the mean volume density of molecular gas in the central re- gion, we would expect larger fraction of molecular gas above the dense gas threshold (n > 104 cm−3) in FIRE- 2 simulation (Hopkins 2015), which leads to faster star formation and shorter tdep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' How can high αvir gas form stars in simulated mergers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' As shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2, αvir generally stays above 10 during the second passage for the G2&G3 merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we assume star formation occurs in individual GMCs and is driven by the collapse of the clouds due to self-gravity , we would expect star forming GMCs to have αvir below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The combination of high αvir values and starburst activity is inconsistent with this expectation, unless the velocity dispersion is being driven to higher values by infall motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, we find no correlation be- tween αvir and tdep (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='4), which suggests low αvir values do not strongly affect the depletion time in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, we need to note that our mea- surement of αvir from pixel-based method might not re- flect the real αvir of individual GMC components, espe- cially for the post-second-passage phase when molecular gas is concentrated in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Although observations (Brunetti & Wilson 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020) show that cloud properties extracted from a pixel-based approach is generally consistent with the traditional cloud-based approach, they also show the pixel-based approach gives higher σv and αvir for molecular gas in galaxy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is likely due to the superimposition of different GMC components along the same line of sight in gas- concentrated galaxy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) find that αvir from pixel-based approach is ∼ 3 times higher than the cloud-based approach for galaxy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we as- sume the same degree of overestimate in our simulation data for the merger center, we would expect the real αvir to be ∼ 10 during the second passage, still significantly higher than the critical value of 1 when clouds reach the self-collapsing criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also note that even the ob- servational cloud-based approach by extracting different GMC components from p-p-v data cube might still suf- fer from the projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Beaumont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2013) find that αvir from p-p-p and p-p-v cubes have a factor of 2 difference for substructures in their cloud simulation due to a mismatch of substructures from these two data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, one of our next steps is to perform cloud-finding algorithm (Burkhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013) on both p-p-p and p-p-v simulation data cubes to fully under- stand how GMCs evolve during the merging events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' A possible explanation for large αvir is that GMCs that satisfy the self-collapsing criterion have already formed stars and become unbound or destroyed due to the stellar feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, if this is the case, we would expect αvir to fluctuate around the critical value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, according to Benincasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020), GMCs with high αvir (>10) have significantly shorter lifetimes (∼2 Myr) than GMCs with low αvir (∼1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' life- time of ∼10 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If we assume all GMCs are of the same population but at different evolutionary stages, we would expect GMCs to stay at low αvir state for a longer time and hence we should be more likely to catch these low αvir GMCs in our simulation snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Instead, we see αvir constantly higher than 10 during the star- burst activity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 4), which is inconsistent with this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' It is perhaps likely that the explanation is that these GMCs are experiencing compression from the large-scale gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This compression could add ad- ditional potential energy to balance the kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, they can trigger inflow of gas into GMCs and bring radial velocity (Vr) component into our σv measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Ganguly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) find in their simu- lation that vr could be an important factor to produce high measured αvir clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For GMCs in normal spi- ral galaxies and galaxy pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', M 51), Meidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018) show that the large-scale stellar potential could be responsible for holding individual GMCs in energy equipartition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Compared to galaxies in their study, the starburst mergers in our study undergo more dra- matic morphological changes, which could generate com- plicated gravitational tidal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2009) show in their simulation that major mergers can produce fully compressive tidal fields that concentrate molecular gas and trigger starburst activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These compressive tidal fields are believed to be responsible for creating the off-nuclei gas concentration region in the ULIRG, Arp 220 (Downes & Solomon 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In our next step to test this scenario, we will need to calculate tidal deformation timescale (as in Ganguly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) for each individual GMC and compare it with GMC free-fall and crossing timescales to see how important the external tidal field is compared to GMC self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Another possible explanation is that molecular gas is smoothly distributed rather than clumped into in- dividual GMCs during the starburst activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' If this is the case, the star formation is regulated by the en- tire molecular disk rather than individual GMC compo- nents (Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) pro- GMCs in Galaxy Mergers 13 pose that the star formation in U/LIRGs is regulated by the hydrodynamic pressure of the molecular disk with a constant scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In observation, one way to test the smoothness of gas distribution is by comparing av- erage gas surface density at different observing resolu- tions (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Brunetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020) show that molecular gas in the LIRG, NGC 3256, is smoothly distributed based on this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For our simulated merger, gas might be smoothly distributed during the second passage when most gas is concentrated in the center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We could test this sce- nario by changing the pixel size in our p-p-v cubes and compare the average gas surface densities in the central region at different pixel resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Comparison with observations As shown in Section 4, our simulated merger gener- ally has lower Σmol and higher σv and αvir compared to the two observed mergers, the Antennae and NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We note that this simulation is not set to match the ex- act condition of the observed mergers, so some discrep- ancy between observations and simulations would be ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' From the observational side, the biggest un- certainty that comes into the measurement is the value of αCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' As mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1, if we adopt the ULIRG αCO instead of the Milky Way value for the Antennae, we would find the Antennae to have similar Σmol and αvir as NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In contrast, if we assume an even smaller αCO for NGC 3256, that might bring the contours of the observations further away from the PHANGS trend and hence more similar to the simu- lation contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, various LVG modelings (Pa- padopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021) show that local U/LIRGs and high-z starburst galaxies generally have αCO above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='8 M⊙ (K km s−1 pc2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In fact, a recent study by Dunne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) concludes that these starburst galaxies might actually have αCO equal to the Milky Way value by cross-correlating the CO luminos- ity with dust and CI luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, a factor of 3 discrepancy in αvir between simulated mergers and NGC 3256 is probably real rather than due to measurement uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the comparison between observations and simula- tions, we also note that the two observed mergers are both in an early stage after the second passage since we can still identify two separate nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In this stage, αvir is quite time-sensitive and it is difficult to match the exact same stage between the simulated and observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, it is possible that both NGC 3256 and Antennae are caught at a specific merger stage with a lower αvir (although in the case of NGC 3256, still enhanced relative to PHANGS galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In compari- son, αvir in the simulations is relatively stable in the post-merger stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This stability suggests that a com- parison between simulations and observations of post- merger galaxies could be a useful next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Moreover, post-mergers have a rather simple morphology, which simplifies the task of making quantitative comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' It would also be interesting to compare the simula- tion results with starburst galaxies at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Re- cent works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Dessauges-Zavadsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Meˇstri´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) show that we can probe GMC-scale star- forming clumps in gravitationally lensed objects at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These star-forming clumps generally show a similar αvir to GMCs of normal spiral galaxies in our lo- cal Universe despite using different αCO, and therefore lower than what we see in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, these high-z targets likely live in a completely different environment than our idealized mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Specifically, high-z galaxies tend to have a much higher gas fraction, and thus can form self-gravitating clumps with low αvir more easily (Fensch & Bournaud 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Comparison with other simulations In this work, we use the non-cosmological simulations from Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) to compare GMC properties in mergers and normal spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Two major advan- tages of this simulation suite are that it has a resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 pc (which is much smaller than typical GMC sizes) and it can model the ISM down to low temperatures (∼ 10 K), both of which allow us to match the molecular gas in simulations with CO observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Various cos- mological simulations show that mergers are responsible for enhancing gas fractions and triggering starburst ac- tivity (Scudder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Knapen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Rodr´ıguez Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=',).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, these simu- lations can only model gas with temperatures down to 104 K and hence are incapable of capturing the turbu- lent multi-phase structure of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' An alternative approach is to compare observations with cosmological zoom-in simulations, which allows for higher resolution, more realistic feedback star formation thresholds, and more realistic modeling of the multi-phase ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Vari- ous authors have explored GMC properties, mostly in Milky-Way-like galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Guedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Cev- erino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sawala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Benincasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Orr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021), and they generally reproduce the GMC mass function in our Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, only a handful of work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) has been done for GMCs in mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Also, the Milky Way is identified as a green-valley galaxy (Mutch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2011) with lower SFR than typical spiral galaxies in the local universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' There- fore, due to the lack of zoom-in cosmological simulations 14 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' on local mergers, we have adopted idealized simulations for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, idealized simulations al- low us to compare GMCs of control galaxies with those of mergers to directly study the impact of the merging event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Several idealized simulations have been performed to study molecular gas and GMC properties in mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Karl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2013) perform a merger simulation closely matched to the Antennae and find a great match on CO distributions between simulation and observations, which suggests insufficient stellar feedback efficiencies in the Antennae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2022) perform a study of GMCs and young massive star clusters (YMCs) in Antennae- like mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' They find that GMC mass functions for mergers have similar power-law slopes to normal spi- rals during the second coalescence but with much higher mass values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2011) compare the αCO in mergers and normal spiral galaxies and find that the low αCO in mergers is mostly due to the high temper- ature and αvir of GMCs in the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' They predict there is a transition stage with αCO between U/LIRG and Milky Way values and that αvir is tightly anti- correlated with αCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In contrast, Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019b) show that αCO values drop quickly during each coales- cence between two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We find similar behavior for αvir during the second coalescence, which might imply a similar drop in αCO (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' CONCLUSIONS We summarize our main conclusions below: Our pixel-by-pixel analysis shows that the FIRE- 2 simulation by Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2019) successfully reproduces the σv vs Σmol relation for GMC-scale pixels measured for galaxies in the PHANGS sur- vey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The simulated mergers show a significant increase in both Σmol and σv for GMC-pixels by a factor of 5 – 10 during the second passage when SFR peaks, which brings these pixels above PHANGS-trend in the σv vs Σmol diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This may indicate GMCs in these mergers are less gravitationally bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We quantify this deviation by the virial param- eter αvir and find that our simulated mergers have αvir of 10 − 100, which is even higher than the observed αvir in NGC 3256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' However, this dis- crepancy could be partly due to the high impact parameter in the initial set-up of the simulated mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Furthermore, we see a good correspon- dence between the increase in SFR and αvir, which suggest either the starburst feedback is responsi- ble for dispersing the gas or the correlation is in response to gas compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Our simulated mergers show a clear gas concentra- tion in the center during the second passage, with up to 80% of molecular gas in the central 1 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, the GMC-pixels in the central region tend to have the highest Σmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also find these pixels tend to have the highest σv and αvir, which could be caused by the starburst feedback and the inflow of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We explore if αvir at GMC scales is responsible for the varying depletion time (tdep) in observed mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' While we do not find a significant corre- lation between tdep and αvir, we see a clear distinc- tion before (small αvir, long tdep) and after (large αvir, short tdep) the second passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This could be due to projection effects (multiple GMCs along the same line of sight) during the second passage when most of the molecular gas is concentrated in the central 1 kpc region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The next step is to run a cloud-identification algorithm on the data to dis- entangle this factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also suspect there might be some other mechanism, such as the stellar po- tential and inflow of gas, that helps the GMCs in starburst mergers to collapse and form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also find that tdep has a significant anti-correlation with Σmol for the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This may be due to higher Σmol leading to a higher fraction of dense gas, which shortens tdep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In the future, we would like to expand our compar- ison to more observed and simulated mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' From the observational side, we need larger samples of galaxy mergers spanning different evolutionary stages in order to understand how GMCs evolve throughout the merg- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In addition, it is easier to compare the observations with simulations in the post-merger stage since the mor- phology is simpler and easier to quantify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The ALMA archive contains ∼ 40 U/LIRGs with GMC resolution CO 2-1 observations that can be used to build a more complete sample of GMCs in mergers at different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' From the simulation side, it would be helpful to have simulations that better match the observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The Antennae has been widely studied and matched by non-cosmological simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Renaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022) but NGC 3256 is less well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Besides comparing with these non-cosmological simulations, we could also compare observation with cosmological sim- ulations, such as FIREBox (Feldmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2022), that include local mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' GMCs in Galaxy Mergers 15 We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Jiayi Sun for his help to access to PHANGS data and insightful discussions about the com- parison between simulation and observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Nathan Brunetti for access to CO 2-1 image and GMC catalogs of the observed mergers in his paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This work was carried out as part of the FIRE collab- oration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The research of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' is supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The research of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' is partially supported by the New Technologies for Canadian Observatories, an NSERC-CREATE training program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The computa- tions in this paper were run on the Odyssey cluster sup- ported by the FAS Division of Science, Research Com- puting Group at Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Support for JM is provided by the Hirsch Foundation, by the NSF (AST Award Number 1516374), and by the Harvard Insti- tute for Theory and Computation, through their Visit- ing Scholars Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' acknowledges support from NSF grant AST-2009679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' is grateful for the gener- ous support of the David and Lucile Packard Foundation and Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Sloan Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The Flatiron Institute is supported by the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This paper makes use of the following ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='ALMA #2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='00714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='S and ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='ALMA #2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='00272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' ALMA is a part- nership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/ NRAO, and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Sci- ence Foundation operated under cooperative agreement by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Facilities: ALMA Software: astropy (Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2013), Spectral-Cube (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2019) REFERENCES Barnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2000, The Astrophysical Journal, 542, 120, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1086/309504 Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', Fall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', & Whitmore, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2001, Observatory, 10 Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', Seaquist, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=', & Kuno, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2003, The Astrophysical Journal, 588, 243, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1086/368353 20 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' INFLUENCE FROM DIFFERENT VIEWING ANGLES One important factor that might influence Σmol mea- sured from the simulations is the inclination angle at which the galaxy is viewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the simulated control galaxies, we pick the inclination angle of 30 degrees in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' By increasing inclination angle we might see significant increase in Σmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' A1 shows the data for the G3 galaxy viewed with inclination angles of 30, 60 and 80 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We see little increase in Σmol even for an inclination of 80 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This is consistent with our expectation that individual clouds are resolved in the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For resolved spherical clouds, the ob- served surface density should always be the same despite different viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 100 101 102 103 104 mol (M pc 2) 10 1 100 101 102 v (km s 1) FIRE G3, 30 deg FIRE G3, 60 deg FIRE G3, 80 deg PHANGS disks PHANGS barred galaxy centers PHANGS unbarred galaxy centers M31 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1 but with the simulated G3 galaxy viewed at different inclination angles (30, 60 and 80 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For the simulated mergers, the molecular gas structure is more complicated than a single layer of gas disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In this case, we might pick a specific angle where multiple clouds happen to lie along the same line of sight, which gives us large σv values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' To test if this is the actual case, we examine a snapshot at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 when we reach maximal Σmol and σv from a different angle (’v1’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' A2, we see similar gas distribution contour in σv vs Σmol contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' This along with the velocity spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 6 suggests that the large σv and αvir we measured is intrinsic properties of individual GMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' GLOBAL GAS FRACTION On global scales, we compare the molecular gas mass and total gas mass (including HI) of the FIRE-2 galax- 100 101 102 103 mol (M pc 2) 10 1 100 101 102 v (km s 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 kpc 4 2 0 2 4 kpc mol (M pc 2) 101 102 4 2 0 2 4 kpc vir 100 101 102 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The snapshot of FIRE-2 merger at a time of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='87 Gyr with viewing angles of ’v1’, which is roughly per- pendicular (with an angle of 109 degree) to the ’v0’ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Upper) the σv vs Σmol distribution of GMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Lower) The Σmol map of the snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can see that viewing from dif- ferent angles still give us high σv measurement, which also suggests that σv we measure is not the velocity dispersion among GMCs along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (lower) The Σmol and σv for this snapshot from ’v1’ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' ies with observed values for normal spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We compare to both the PHANGS galaxies (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021) as well as to the global gas properties from the xCOLDGASS survey, to confirm that the PHANGS galaxies are representative of star forming main se- quence galaxies in our local universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' For xCOLDGASS, the molecular gas mass is extracted from Saintonge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2017) and the total gas mass is from Catinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In interpreting the lower Σmol values seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 1, one possibility is that there may not be as much gas available to form high surface density clouds in the two simulated galaxies compared to the PHANGS galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' B1 compares the global molecular gas masses, Mmol, and molecular gas fractions, fmol = Mmol / M⋆, for the FIRE-2 mergers with those of the PHANGS GMCs in Galaxy Mergers 21 1010 1011 M⋆ (M⊙) 108 109 1010 Mmol (M⊙) PHANGS median xCOLDGASS median FIRE G3 FIRE G2 M31 PHANGS 1010 1011 M⋆ (M⊙) 10−2 10−1 Mmol / M⋆ 1010 1011 M⋆ (M⊙) 109 1010 Mgas (M⊙) 1010 1011 M⋆ (M⊙) 10−1 100 Mgas / M⋆ Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Upper Left) Mmol versus M⋆ for PHANGS galaxies (salmon dots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2021), M 31 (green filled circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Nieten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2006) and the G2 (red points) and G3 (blue points) simulated galaxies at different times in their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Note that the G2 and G3 simulated galaxies lie significantly below the star-forming main sequence defined by the xCOLDGASS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Upper Right) fmol versus M⋆ for the same galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The molecular gas fractions of G2 and G3 are significantly lower than most of the PHANGS spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Lower left) total gas versus M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (Lower right) total gas fraction versus M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' These comparisons suggest that the low Σmol measured for the simulated galaxies might be due to the low total and molecular gas fraction in the initial set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' galaxies from Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We also show the me- dian value of Mmol and fgas in each M⋆ bin for the PHANGS galaxies, as well as the weighted median of M⋆ and fmol for galaxies in xCOLDGASS sample (Sain- tonge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The two median values are quite close to each other for galaxies with M⋆ of 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 – 1011 M⊙, although the PHANGS galaxies seem to deviate somewhat from the xCOLDGASS sample in the high- est and lowest mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' In contrast, the G2 and G3 galaxies both have fmol ∼ 3 times lower than typical PHANGS or xCOLDGASS galaxies of the same stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, the small global fmol may be respon- sible for producing the low Σmol values seen in the sim- ulated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The low values of fmol could be produced either by the initial set-up of the simulations or by physical mecha- nisms in the simulation that lead to inefficient conversion of gas into the cold phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' We can distinguish between these two options by calculating the total gas fraction fgas including both HI and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' B1 shows the median of Mgas and fgas for the PHANGS galaxies and xGASS-CO samples (Catinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 2018) compared to the two simulated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The values of fgas for both simulated galaxies are still ∼ 3 times lower than those of typical spiral galaxies with similar M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Therefore, it seems most likely that the low cold gas frac- tion, fmol, is produced by a low total (cold+warm+hot) gas mass in the initial set-up of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' SNAPSHOTS FOR ‘E1’ ORBIT Here we show the SFR history and 3 example snap- shots for G2&G3 ‘e1’ orbit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 22 He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='0 Time (Gyr) 100 101 SFR (M yr 1 100 101 102 103 10 1 100 101 102 v (km s 1) t: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='22 Gyr SFR: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='81 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 4 2 0 2 4 kpc mol (M pc 2) 100 101 4 2 0 2 4 vir 100 101 102 100 101 102 103 10 1 100 101 102 v (km s 1) t: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='46 Gyr SFR: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='59 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 4 2 0 2 4 kpc mol (M pc 2) 101 102 4 2 0 2 4 vir 100 101 102 100 101 102 103 mol (M pc 2) 10 1 100 101 102 v (km s 1) t: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='56 Gyr SFR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='37 M yr 1 FIRE G2&G3 PHANGS galaxies Antennae, CO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='3 NGC 3256, CO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content='1 M31 4 2 0 2 4 kpc 4 2 0 2 4 kpc mol (M pc 2) 101 102 4 2 0 2 4 kpc vir 100 101 102 Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' Similar plot as Figure 2 but with SFR history and 3 snapshots from ‘e1’ orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFQT4oBgHgl3EQfHzW5/content/2301.13250v1.pdf'} +page_content=' The interactive version of the animation is available at https://htmlpreview.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..e30d34d4d067fc125cab3e722214749f3863d51c --- /dev/null +++ b/AdAzT4oBgHgl3EQf_v_f/content/tmp_files/2301.01954v1.pdf.txt @@ -0,0 +1,3702 @@ + +1 + + + +Corrupted by Algorithms? How AI-generated and Human-written Advice Shape +(Dis)honesty +by +Margarita Leib*, Nils Köbis*, Rainer Michael Rilke, Marloes Hagens, & Bernd Irlenbusch + + +* shared first-authorship +This research has been approved by the Ethics Commission of the Faculty of Management, Economics, and +Social Sciences of the University of Cologne under reference 200010BI. +Acknowledgements: We thank Clara Bersch, Yulia Litvinova, Ann-Kathrin Blanke, Toan Huynh, Anna Vogts +and Matteo Tinè for research assistance, and Iyad Rahwan, Jean-Francois Bonnefon, Aljaz Ule, Anne-Marie +Nussberger as well as the attendees of the Cognition, Values & Behaviour Research Group (Ludwig- +Maximilians Universität München / LMU Munich), Moral AI lab meeting (Max Planck Institute for Human +Development & Toulouse School of Economics), Applied Ethics & Morality Group (Prague University of +Business and Economics), Centre for Decision Research (University of Leeds), Department of Economics and +Management (University of Pisa), Decision Making and Economic Psychology Center (Ben-Gurion University), +Behavioral and Management Science group (Technion), Colloquium of the Department of Social Psychology +(Tilburg University) & Seminar at Department of Computer Science (Friedrich-Alexander University +Erlangen-Nuremberg) for their helpful comments. +Funding: The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) under Germany's Excellence Strategy–EXC 2126/1–390838866' ECONtribute: Markets and +Public Policy', the European Research Council (ERC-StG-637915), and the Chamber of Commerce and +Industry (IHK) Koblenz. + + +2 +Abstract +Artificial Intelligence (AI) increasingly becomes an indispensable advisor. New +ethical concerns arise if AI persuades people to behave dishonestly. In an experiment, we +study how AI advice (generated by a Natural-Language-Processing algorithm) affects +(dis)honesty, compare it to equivalent human advice, and test whether transparency about +advice source matters. We find that dishonesty-promoting advice increases dishonesty, +whereas honesty-promoting advice does not increase honesty. This is the case for both AI- +and human advice. Algorithmic transparency, a commonly proposed policy to mitigate AI +risks, does not affect behaviour. The findings mark the first steps towards managing AI +advice responsibly. + +Keywords: Artificial Intelligence, Machine Behaviour, Behavioural Ethics, Advice + + + +3 +Corrupted by Algorithms? How AI-generated and Human-written Advice Shape +(Dis)honesty +Artificial Intelligence (AI) shapes people's life on a daily basis (Rahwan et al., 2019). +It sets prices in online markets (Calvano et al., 2020), predicts crucial outcomes such as +healthcare costs (Obermeyer et al., 2019) and criminal sentences (Kleinberg et al., 2018), +and makes recommendations ranging from entertainment content and purchasing +decisions to romantic partners (Dellaert et al., 2020; Yeomans et al., 2019). Increasingly, AI +has become an indispensable advisor, thereby affecting people's behaviour (Fast & +Schroeder, 2020; Kim & Duhachek, 2020). As a case in point, Amazon's chief scientist, Rohit +Prasad, envisions that Alexa's role for its over 100 million users "keeps growing from more +of an assistant to an advisor" (Strong, 2020). Given AI's increasing role as an advisor, it is +crucial to examine whether people are persuaded to follow or break ethical rules based on +AI advice (Köbis et al., 2021). +Large companies like LinkedIn and Zillow are already implementing AI advisors, +thereby potentially shaping their employees' ethical behaviour. In such companies, natural +language processing (NLP) algorithms (e.g., provided by software such as Gong.io) analyse +employees' recorded sales calls and advise them on how to increase their sales. Without +supervision, such algorithms may detect that deceiving customers pays off and thus advise +salespeople to do so. Indeed, NLP algorithms can already autonomously detect deception as +a useful strategy in a negotiation task (Lewis et al., 2017). An ethical risk arises if people +follow such corruptive AI advice. Here we examine (i) whether people meaningfully alter +their (un)ethical behaviour following AI-generated advice and (ii) how such advice + + +4 +compares to human-written advice. Lastly, we test (iii) whether knowledge about the +advice source (AI vs human) matters. +Receiving advice on (un)ethical behaviour: Humans vs AI +Generally, people are reluctant to take advice from others ("egocentric advice +discounting", e.g., Yaniv & Kleinberger, 2000), especially when it is unsolicited (Bonaccio & +Dalal, 2006). However, when facing an ethical dilemma, advice has several compelling +benefits for the advised. Advice encouraging an ethical course of action may validate one's +moral preferences. It thereby might reduce negative feelings such as regret for not taking +the opportunity to maximise profits by lying. Advice encouraging an unethical course of +action may free people to violate ethical rules for profit without spoiling their moral self- +image (Cross et al., 2001). Indeed, taking advice can even provide a sense of shared +responsibility with the advisor (Harvey & Fischer, 1997). +Compared to receiving human advice, how would people react to advice from an AI? +Recent technological advances in the field of NLP reveal that AI text can already be +indistinguishable from human text, suggesting AI advice is as convincing as human advice. +For instance, GoogleDuplex, an AI-based call assistant, can book appointments while having +full-fledged conversations without the recipient even realising that an AI is on the line. +Further, AI can generate anything from poems (Köbis & Mossink, 2021) and Airbnb profiles +(Jakesch et al., 2019) to news articles (Kreps et al., 2021) on par with humans. It thus +stands to reason that when people are not informed about the sources of advice, they will +not recognise the advice source correctly and be affected by AI and human advice similarly. +Testing Algorithmic Transparency + + +5 +To make sure people know whom they interact with, governments, policymakers, +and researchers univocally call for algorithmic transparency (Jobin et al., 2019) — the +mandatory disclosure of AI presence (Diakopoulos, 2016). The recent Artificial Intelligence +Act released by the EU demands AI systems such as chatbots and call assistants to disclose +themselves as AI when interacting with humans (European Commission, 2021). Although it +is a popular policy recommendation, empirical evidence for its effectiveness in shaping +people's ethical behaviour is lacking. +How transparency about the advice source affects people's reaction to the advice is +not trivial. Prior work informs three competing possibilities. The first possibility is that +when informed about the source of advice, people follow human advice more than AI +advice. This account rests on the literature on algorithm aversion (Dietvorst et al., 2015). +People readily rely on AI in objective and technical domains (e.g., numeric estimation, data +analysis, and giving directions, Castelo et al., 2019; Logg et al., 2019). However, they are +reluctant to use AI for subjective decisions, especially with ethical implications (e.g., parole +sentences, trolley-type dilemmas, Bigman & Gray, 2018; Castelo et al., 2019; Laakasuo et al., +2021). Further, people follow perceived social norms when making (un)ethical decisions +(Bowles, 2016; Fehr, 2018; Gächter & Schulz, 2016; Gino et al., 2009; Köbis, Troost, et al., +2019). Compared to AI advice, human advice might be a stronger signal of social norms +because social norms regulate and emerge from human (not AI) behaviour. Consequently, +people should be more likely to follow human advice. Suppose people indeed prefer human +input in ethically charged settings and perceive human advice as a stronger cue for social +norms. In that case, we should expect that human advice sways people's (un)ethical +behaviour more than AI advice. + + +6 + +The second possibility is that when informed about the source of advice, people +follow advice from humans less than from AI. A closer look at the technical design of AI +advice systems would support this account. NLP algorithms are trained on a large corpus of +human-written texts (Radford et al., 2019). When people know that NLP algorithms draw +on large compiled human input, they might perceive AI advice as a better representation of +most people's beliefs and behaviours than the advice they receive from one human. If AI +advice is indeed a stronger cue for social norms than a single piece of human-written +advice, we should expect that AI advice sways people's (un)ethical behaviour more than +human advice. +The third possibility is that when people receive information about the source of +advice, they are affected equally by human and AI advice. Support for this account comes +from the observation that people already seek advice from AI agents. For instance, more +than 7 million people turn to Replika, the "AI companion who cares. Always here to listen +and talk. Always on your side" (replika.ai) for virtual companionship, socialising, and also +for advice (Murphy, 2019). Such AI advisors might also help justify questionable behaviour. +When tempted to break ethical rules for profit, people do so as long as they can justify their +actions (Barkan et al., 2015; Fischbacher & Föllmi-Heusi, 2013; Shalvi et al., 2015). +Receiving advice that encourages rule-breaking can serve as a welcomed justification, +possibly even when the advice stems from AI. Indeed, people deflect blame and share the +responsibility for harmful outcomes not only with other people (Bartling & Fischbacher, +2011; Bazerman & Gino, 2012; Tenbrunsel & Messick, 2004) but also with AI systems +(Hohenstein & Jung, 2020). If following AI and human advice is equally justifiable and leads + + +7 +to similar attribution of responsibility between the two, we should expect that human and +AI advice sway people's (un)ethical behaviour to the same extent. +The current study +The current study tests how advice type (honesty- vs dishonesty-promoting), advice +source (AI vs Human), and information about advice source (transparency vs opacity) +shape humans' (un)ethical behaviour. Prior work has examined people's stated preferences +about hypothetical scenarios describing AI advice (Bigman & Gray, 2018; Castelo et al., +2019; Kim & Duhachek, 2020; Logg et al., 2019). We supplement such work by adopting a +machine behaviour approach (Rahwan et al., 2019) and examine people's behavioural +reactions to actual AI-generated output. +To measure people's (un)ethical behaviour, we use the well-established incentivised +die-rolling task (Fischbacher & Föllmi-Heusi, 2013). In it, participants roll a die in private +and report the outcome, with higher outcomes corresponding to higher pay (see for similar +approaches, Abeler et al., 2019; Fehrler et al., 2020; Sutter, 2008). To generate real AI +advice, we employ the state-of-the-art algorithm GPT-J. We fine-tune the algorithm using +minimal training to produce relevant advice texts. Using this "few-shot" learning approach +mimics many real-world settings where language models (e.g., Alexa) are not optimised to +maximise users' profits or provide the most convincing advice but simply produce relevant +information based on prompts (Brown et al., 2020). +Method +We conducted a large-scale, pre-registered experiment. The design entailed two +parts. In part 1, we collected human-written advice and generated AI advice; in part 2, we +examined the effect of advice on (dis)honest behaviour (see Figure 1 for an overview). By + + +8 +using real AI text outputs, we avoid experimental deception and can gain insights into how +people react to actual AI advice. We report all measures, manipulations, and data +exclusions in the main text and Appendix. The IRB board of our department approved the +experiment, and all materials, pre-registrations, and data are available on the Open Science +Framework. +Part 1 - Advice-giving task +Human-written advice. First, we conducted an advice-giving task in which we +recruited advisors (N = 367, Mage = 30.28, SDage = 9.56, 45.50% females) via Prolific.co (pre- +registration https://osf.io/nbke2). The task took 10-15 minutes, and participants earned a +base pay of £1.27. Advisors learned that a separate group of participants (advisees) would +engage in a die-rolling task (Fischbacher & Föllmi-Heusi, 2013), in which they roll a die +privately and report the outcome (with higher outcomes corresponding to higher pay). +Advisors were asked to write advice for these advisees. +Advisors were randomly assigned to either write honesty-promoting or dishonesty- +promoting advice. We informed advisors in the Honesty-promoting advice treatment that if +an advisee read their advice and subsequently reported the actual die-roll outcome (i.e., +acted honestly), they would earn a bonus of £3. Advisors in the Dishonesty-promoting +advice treatment learned that if an advisee read their advice and subsequently reported the +highest possible outcome, 6, they would earn a bonus of £31. +Advisors had to follow pre-specified advice writing rules to ensure they produced +coherent advice texts that could be used to train GPT-J. Specifically, their advice had to (i) + +1If advisees follow the advice in the dishonesty-promoting treatments, they will lie in the majority of the cases (5 out +of 6 cases). Only when the actual die-roll outcome is 6, following the advice does not entail lying. + + +9 +entail at least 50 words, (ii) not use concrete numbers in numeric or written form2, (iii) be +in English and in their own words, (iv) be written in complete sentences, (v) be about the +advisee's die-roll outcome reporting decision, and (vi) not inform the advisee that the +advisor's payoff depended on their behaviour3. +To incentivise advisors to follow the advice writing rules, they stood to gain a bonus. +Namely, out of all advice texts, we randomly selected one, and if that text followed the +writing rules, the advisor earned a bonus of £10. Moreover, as incentivisation for writing +convincing texts, 1 per cent of advice texts (4 out of 400) were implemented. If advisees +acted according to the implemented advice, the respective advisor earned a bonus based on +the treatment they were in (Honesty- vs Dishonesty-promoting advice)4. +AI-generated advice. To generate AI advice (see Figure 1A), we employed GPT-J5, +an open-source NLP algorithm published by Eleuther AI (https://www.eleuther.ai/). GPT-J +is trained on a curated and diverse data set of 825 GiB texts to predict the next word in a +sequence of words and contains 6 billion parameters (Wang & Komatsuzaki, 2021). GPT-J +can be fine-tuned with extra training to produce a specific type of text. We fine-tuned GPT-J + +2 Advisors were not allowed to use concrete numbers to allow generating high-quality AI advice. GPT-J is trained to +predict the next word in a sentence (see ‘AI-generated advice‘ section). If advisors were allowed to concretely +mention numbers, training GPT-J on the human written advice could have resulted in random numbers appearing +out of context in the GPT-J output, reducing the quality of AI-generated advice. +3 Advisors were not allowed to mention their incentive structure to the advisees so that we could keep the prosocial +motivation for advisees who read AI and human advice constant (at zero). +4 Paying advisors required knowing whether participants, after reading the advice, reported the observed die-roll +honestly or not. To do so, we ran a modified version of the die-rolling task in which advisees received randomly +selected advice, saw a die-roll on the computer screen and were asked to report it. We implemented this procedure +for four randomly selected advice texts (1% of the advice) and four advisees. This non-private procedure provided +certainty about whether an advisee reported honestly or not and enabled us to pay advisors accordingly. Doing so +meant that our experimental setup was incentivised and did not entail experimental deception. In the main +experiment, the die-roll outcomes were private (see ‘Part 2 - Advice-taking task’). +5 As one can read in our pre-registration, we originally planned on deploying GPT-2 (see +https://openai.com/blog/better-language-models/) to generated AI advice. However, we opted to use GPT-J instead +because it is open source, which increases reproducibility and is more advanced as it is much larger and more potent +than GPT-2. + + +10 +with "few shot" learning by separately training it on the human-written honesty-promoting +and dishonesty-promoting advice from the advice-giving task. We only used advice texts +that adhered to the advice writing rules (as coded by a naive coder) for fine-tuning. More +details on the calibration of GPT-J are reported in the Appendix. + + +A)Part1-Advicegivingtask +Task +- Reads description of die rolling task +Advisor + Writes advice to advisee +个 +个 +Honesty promoting advice +Dishonesty promoting advice +(advisors earn 3 if advisee reports honestly) +(advisors +Human-written advice +Al-generated advice +trained on human-written advice +B) Part 2 -Advice taking task +Treatments +Baseline (No Advice) +Human-written vs. Al-generated +Honesty vs. Dishonesty promoting +Transparent vs.Opaque Information +Advisee's payoff +Reported outcome +Payoff +Advisee +1 +0.5 +Task +2 +1.0 +- Rolls a die +3 +1.5 +-Reports an outcome +4 +2.0 +5 +2.5 +6 +3.0 +11 +Figure 1. (A) Part 1 - advice-giving task (B) Part 2 - advice-taking tasks. (A) Participants were +incentivised to write honesty- or dishonesty-promoting advice texts, which were then used to +generate AI advice. (B) Another group of participants engaged in the die-rolling task. Advisees read +advice, then reported a die-roll outcome. In total, we administered nine treatments: Participants +read honesty or dishonesty-promoting advice that was human-written or AI-generated. +Participants were either informed about the source of advice (Transparency) or not (Opacity). As a +baseline, another group of participants did not read any advice. + + +Screening. After collecting human advice and generating AI advice, we employed +the same pre-specified screening procedure for both sources (see Figure 2). First, we +excluded texts that exceeded 100 words. Next, to ensure advisees read coherent and +relevant advice texts, we randomly selected 100 advice texts per cell. Two independent +coders, who were naive to the experimental treatments, coded each piece of advice on the +following criteria: (a) is the text coherent? (Y/N); (b) does the text contain clear advice? +(Y/N); (c) which type of behaviour does the advice encourage? +(honesty/dishonesty/unclear); (d) does the advice follow advice writing rules? (Y/N). +Further, we used the objective Grammarly and Readability scores as computational proxies +for the quality of the texts6. +Among the texts that passed the coding procedure7 and received a Grammarly score +equal or above 50, we randomly selected 20 advice texts per treatment (AI-generated vs +Human-written, by Honesty- vs Dishonesty-promoting), yielding a final sample of 80 advice +texts used in part 2 (see all advice texts in the Appendix). By applying the same screening + +6 We obtained Grammarly and Readability scores from grammarly.com. Grammarly score compares texts to all +other texts checked on the platform. A score of 80 indicates that a text scores better than 80% of all texts checked on +grammarly.com in terms of grammatical correctness. Readability score employs the Flesch-Reading-ease test and +represents how easy a text is to read. The score is calculated by the average sentence length and the average number +of syllables per word, with higher scores indicating easier readability. +7 Texts that passed the coding procedure (i) are coherent, (ii) contain clear advice, (iii) encourage honesty in the +honesty-promoting treatment and dishonesty in the dishonesty-promoting treatment, and (iv) follow the advice +writing rules. Moreover, the coding by both independent coders had to match each other in order for the text to pass. + + +12 +procedure for human and AI advice, we ensure that the advice texts fulfil minimal quality +criteria and are as comparable as possible. + + + + +Figure 2. Overview of the selection procedure of advice texts. H = Honesty-promoting advice; DH = +Dishonesty-promoting advice. + +Part 2 - Advice-taking task +The advice-taking task took about 8 minutes to complete, and participants earned a +fixed pay of £1.20. We pre-registered (https://osf.io/nqvf3) to collect a sample size that +would allow us to detect a small to medium effect size (200 participants per cell, 1,800 in + +Human-writtenadvice +Al-generatedadvice +Nμ = 186 +NpH = 181 +Screening +Independent coder assesses whether the text +follows writing rules +Nμ = 157 +NpH = 158 +Nμ = 152 +NpH =150 +Filter +50 ≥ number of words ≤ 100 +Nμ = 145 +NpH = 149 +Nμ = 140 +NH = 117 +Randomselection +100 texts per treatment and source +Nμ = 100 +NDH = 100 +Nμ = 100 +NpH = 100 +Coding +Two naive coders assess: (i) coherence, (ii) clear +advice, & (ii) writing rules. Agreement between +coders + with initial treatment is needed. +Nμ = 57 +NDH = 65 +Nμ = 79 +NpH = 79 +Grammarly score ≥ 50 +Randomselection +20 texts per treatment +Nμ = 20 +NpH = 20 +Nμ = 20 +NpH = 20 +13 +total) via Prolific.co to take part in the advice-taking task. Overall, 1,817 (Mage = 32.39; SDage += 11.68, 48.72% females) participants were included in the analyses. These participants +completed the task and self-report items and passed the comprehension and attention +checks (see below). Sensitivity analysis for a regression with 90% power and a significance +level of .05 revealed our sample was sufficient to detect small effect sizes (f 2 = .006 and +.010, see Appendix for details). +Participants read the instructions, received advice, and finally engaged in the die- +rolling task. Specifically, participants were asked to roll a die privately and report the +outcome. Participants' pay corresponded to their report: for reporting a '1' they earned +£0.5; for a '2' = £1, '3' = £1.5, '4' = £4, '5' = £2.5, '6' = £3. After reading the instructions and +before engaging in the die-rolling task, all participants learned that 10 per cent of +participants would be randomly selected and paid for the die-rolling task. +Assessing dishonesty by employing the die-rolling task is a common practice in +economics and psychology (see meta-analyses, Abeler et al., 2019; Gerlach et al., 2019; +Köbis, Verschuere, et al., 2019; Leib et al., 2021). Further, the task has good external +validity, as lying in the die-rolling task correlates with unethical behaviour outside the lab, +such as free-riding public transportation (Dai et al., 2018) and being absent from work +without reason (Hanna & Wang, 2017). Importantly, the die-rolling task pits two competing +motivations against each other: to be a moral, honest person vs maximising financial +profits, thus rendering advice particularly valuable. +After reading the instructions, participants had to answer three comprehension +questions correctly and pass an attention check. If participants answered a comprehension +question incorrectly, they could try again. If they answered the same question incorrectly + + +14 +twice, or if they answered the attention check incorrectly, they were disqualified from +participating in the study and were not paid. +Treatments. Before reporting the die-roll outcome, participants were randomly +assigned to one of nine different treatments. Participants either (i) read Honesty-promoting +or Dishonesty-promoting advice, (ii) that was either Human-written or AI-generated. +Further, (iii) participants were either informed about the advice source (Transparency +treatments) or not (Opacity treatments). Additionally, in a Baseline treatment, participants +did not receive any advice. Thus, the overall design was a 2 (Advice source: Human vs AI) +by 2 (Advice type: Honesty-promoting vs Dishonesty-promoting) by 2 (Information about +the advice source: Transparency vs Opacity) + 1 (Baseline, no advice) between-subject +design. +Participants who read AI-generated advice and were informed about it (in the +Transparency treatments) read: +"To help you with your decision, an artificially intelligent (AI) algorithm generated an +advice. +Information about AI-advice: +The algorithm is trained on large datasets of English text (> 820 GiB, which is more +than 500 million pages of text) and produces text by predicting the next word in a +sentence (similar to predictive text on smartphones). On top of that, it is also possible +to train the algorithm on additional specific pieces of text. +To generate multiple AI advice texts, the algorithm was trained on advice texts +collected from other participants in the study. These participants did not take part in +the die rolling task and were only instructed to write advice regarding the decision in + + +15 +the die rolling task. The advice you will read is one advice text that was generated by +the algorithm." +Participants who read human-written advice and were informed about it (in the +Transparency treatments) read: +"To help you with your decision, another participant wrote an advice. +Information about advice: +To collect multiple advice texts, another group of participants was asked to write +advice regarding the decision in the die rolling task. These participants did not take +part in the die rolling task and were only instructed to write advice regarding the +decision in the die rolling task. The advice you will read is advice written by one +participant. " +Participants who were in the Opacity treatments and thus not informed about the advice +source read: +"To help you with your decision, you will read an advice. +This advice has been written either by another participant or by an artificially intelligent (AI) +algorithm. There is a 50% chance the advice is written by a participant and a 50% chance it is +written by an algorithm." +In the Opacity treatments, this text was followed by the same two descriptions of how +advice text from each source was collected or generated in the Transparency treatments. In +the Opacity treatment, this information about AI advice generation and human advice +collection appeared in random order.8 + +8 To control for participants' beliefs about the potential advice sources, we opted to inform them that there is a 50-50 +chance that a human or AI wrote the advice. We believed that not providing any information about the advice source +would reasonably lead participants to assume the advice source is another human, as AI might not be a salient +source of advice for participants. + + +16 +A static Turing test. After completing the die-rolling task, participants in the +Opacity treatment engaged in an incentivised version of a static Turing Test (Köbis & +Mossink, 2021). In contrast to the classical Turing Test (Turing, 1950), participants did not +interact back and forth with the source of advice. Instead, they read the advice text and +indicated whether they thought a human or an AI had written it. Participants learned that +20 of them would be randomly selected, and if their guess in the static Turing test was +correct, they would earn an additional £1. +Potential mechanisms. Finally, to explore possible mechanisms, participants +completed a post-experimental survey. Participants indicated on a scale from 0 to 100 their +perceived (i) appropriateness (injunctive social norm), (ii) prevalence (descriptive social +norm), and (iii) justifiability of reporting a higher die-roll than the one observed. +Additionally, all participants, except those who did not receive any advice, rated how they +attribute responsibility between themselves and the advisor for the reported outcome in +the die-rolling task. The answer scale ranged from 0 (= I am fully responsible) over 50 (= +The advisor and I share responsibility equally) to 100 (= The advisor is fully responsible). +Participants further indicated (on a scale from 0 to 100) to what extent they feel guilty after +completing the task (see Appendix for results regarding guilt and wording of all items). +Finally, all participants indicated their age and gender. +Results +In all nine treatments, participants lied as the average die-roll outcomes +significantly exceeded the expected average if participants were honest (EV = 3.5), one- +sample t-test, ts > 3.43, ps < .001. +Is people's (un)ethical behaviour influenced by AI-generated advice? + + +17 + +Yes, when it comes to dishonesty-promoting advice; no, when it comes to honesty- +promoting advice. We first focus on the Opacity treatments, where participants are not +informed about the advice source. Here, linear regression analyses reveal that the average +die-roll reports following AI-generated Dishonesty-promoting advice (M = 4.59, SD = 1.37) +significantly exceed reports in the Baseline, no advice treatment (M = 3.98, SD = 1.55, b = +.609; p < .001; 95% CI = [.324, .894]). However, die-roll reports following AI-generated +Honesty-promoting advice (M = 4.00, SD = 1.62) do not significantly differ from reports in +the Baseline treatment (b = .019; p = .898; 95% CI = [-.275, .314]), see Figure 3 and Table 1 +(model 1). Further, die-roll reports in the AI-generated Dishonesty-promoting treatment +significantly exceed those in the AI-generated Honesty-promoting advice treatment (b = - +.590, p < .001; 95% CI = [-.881, -.299]). Thus, while dishonesty-promoting AI advice +successfully corrupts people, honesty-promoting AI advice fails to sway people toward +honesty. +How does AI-generated advice square compared to human-written advice? + +AI-generated advice affects behaviour similarly to human-written advice, for both +honesty-promoting and dishonesty-promoting advice. Focusing on the Opacity treatments, +the two-way interaction (advice source by advice type) is not significant (b = .069, p = .744; +95% CI = [-.349, .489]), see Figure 3 and Table 1 (model 2). Specifically, the average die-roll +reports do not differ between the AI-generated (M = 4.00, SD = 1.62) and Human-written +advice when advice was Honesty-promoting (M = 3.92, SD = 1.51, b = -.076, p = .631; 95% CI += [-.387, .235]). Similarly, average die-roll reports do not differ between the AI-generated +(M = 4.59, SD = 1.37) and Human-written advice when advice was Dishonesty-promoting (M += 4.58, SD = 1.53, b = -.006, p = .964; 95% CI = [-.289, .276]). + + +18 +In addition, the results of the static version of the Turing Test indicate that +individuals cannot distinguish AI-generated advice from human-written advice. +Specifically, in the Opacity treatments, 49.93 per cent (401 out of 803) of participants +guessed the source of advice correctly, which does not differ from chance levels (50%, +binomial test: p = .999; 95% CI = [.464, .534]). +Does transparency about the advice source matter? +No, informing participants about the algorithmic or human source of advice does +not change their behaviour. Linear regression analyses reveal that the three-way +interaction (advice type by source by information) is not significant (b = .100, p = .735; +95% CI = [-.482, .683]), Figure 3 and Table 1 (model 3). Both among the Opacity and +Transparency treatments, the two-way interactions (advice source by advice type) are not +significant (Transparency: b = .170, p = .409, 95% CI = [-.234, .575]; Opacity: b = .069, p = +.744, 95% CI = [-.349, .489]). +Overall, the popular policy recommendation of algorithmic transparency does not +alleviate the corrupting effect of AI advice. Namely, die-roll reports following AI-generated +Dishonesty-promoting advice under the Opacity treatment (M = 4.59, SD = 1.37) are on par +with reports following the same advice in the Transparency treatment (M = 4.61, SD = 1.40, +b = .020, p = .878; 95% CI = [-.244, .286]). Specifically, when participants are not informed +about the advice source, they boost their reports by 15.2% following AI-generated +Dishonesty-promoting advice, compared to the Baseline [(4.59-3.98)/3.98 = .152], which is +equivalent to the 15.2% increase when they are informed about the source of the advice +[(4.61-3.98)/3.98 = .158]. Bayesian analyses corroborate these conclusions (see Appendix). + + +19 +Overall, results align with the idea that people increasingly follow AI advice (e.g., Replika) +and use AI-generated advice to justify breaking ethical rules for profit. +Robustness of the obtained results. In our experimental design, advisors in the +Dishonesty-promoting treatment received £3 only if advisees reported the highest value, '6'. +Such an incentive scheme is comparable with the Honesty-promoting treatment in which +advisors earned £3 only if advisees reported honestly. In both cases, advisors earn money +for 1 out of 6 potential advisee's reports (i.e., when the advisee reports' 6' or honestly, +depending on the treatment) and do not earn money in the remaining 5 of the advisee's +reports. However, advisors' incentive scheme in the Dishonesty-promoting treatments may +have resulted in advice texts that predominantly focused on convincing participants to +report the outcome 6. To assess the robustness of our results, we (i) conducted additional +analyses and (ii) ran additional treatments. +Proportion of sixes. First, as an additional analysis, we examined whether the +proportion of 6's, as an alternative outcome variable, led to the same conclusions. We found +very similar results (see Figure 3, the white dots represent the proportion of 6's across all +treatments). Specifically, focusing on the Opacity treatments, linear regression analyses +reveal that the proportion of sixes following AI-generated Dishonesty-promoting advice +(32.44%) significantly exceeds the proportion of sixes in the Baseline, no advice treatment +(20.65%; b = .612; p = .005, 95% CI = [.182, 1.051]). However, the proportion of sixes +following AI-generated Honesty-promoting advice (21.93%) does not significantly differ +from the Baseline (b = .076; p = .751, 95% CI = [-.398, .551]). Further, the proportion of +sixes in the AI-generated Dishonesty-promoting treatment significantly exceeds that in AI- +generated Honesty-promoting advice treatment (b = -.535, p = .016, 95% CI = [-.979, -.101]). + + +20 +Further, focusing on the Opacity treatments, the two-way interaction (advice source +by advice type) is not significant (b = .444, p = .171, 95% CI = [-.191, 1.082]). The +proportion of sixes does not differ between the AI-generated (21.93%) and Human-written +treatments when the advice is Honesty-promoting (19.28%, b = -.162, p = .516, 95% CI = [- +.654, .327]). Similarly, the proportion of sixes does not differ between the AI-generated +(32.44%) and Human-written treatments when the advice is Dishonesty-promoting +(38.91%, b = .282, p = .173, 95% CI = [-.123, 690]). Lastly, the three-way interaction (advice +type by source by information) is also not significant (b = -.523, p = .257, 95% CI = [-1.430, +.382]). Both among the Opacity and Transparency treatments, the two-way interactions +(advice source by advice type) are not significant (Transparency: b = -.078, p = .810, 95% CI += [-.724, .566]; Opacity: b = .444, p = .171, 95% CI = [-.191, 1.082]). +Additional (Aligned) treatments. To assess the robustness of our results to the +advisor's incentive scheme, we ran four additional treatments (advice source: Human- +written vs AI-generated by information: Transparency vs Opacity). In these Aligned +treatments, advisees read advice written by advisors whose incentives were aligned with +those of the advisees. For these advisors (n = 207), if the advisee reported '1', both the +advisor and advisee earned £0.5 each; if the advisee reported '2', both the advisor and +advisee earned £1 each and so on. We again fine-tuned GPT-J on such human-written +advice texts. These treatments led to comparable results to the Dishonesty-promoting +treatment. In particular, the average die-roll outcomes in all four Aligned treatments were +significantly higher than in the Baseline treatment (p = .066 for the AI-generated, Opacity +treatment, and ps < .001 for the remaining three treatments, see Appendix for more details + + +21 +about these treatments and elaborated results). This consistency in results suggests that +our results are robust to such variation in the advisors' incentive scheme. + +Figure 3. Mean reported die-roll outcomes (in bars) and proportion of reported 6s (in white dots) +across advice type (honesty vs dishonesty-promoting), source (AI vs human), and information +treatments (opacity vs transparency). The dashed black line represents the expected mean if +participants were honest (EV = 3.5), and the dashed white line represents the expected proportion +of 6s if participants were honest (16.67%). Mean (SD) of die-roll reports are at the bottom of each +bar; ***p < .001; ns: p >.05. + + + +Honesty-promotingadvice +Dishonesty-promotingadvice +ns +100% +ns +ns +5 +80% +outcome +ns +sixes +4 +60 % +Proportion of s +3 +40% +2 +20% +3.99 +4.01 +4.60 +3.93 +4.59 +4.07 +4.62 +3.88 +4.59 +(1.56) +(1.63) +(1.37) +(1.52) +(1.54) +(1.48) +(1.40) +(1.58) +(1.36) +0 % +No +AI +Human +Al +Human +advice +generated +written +generated +writteh +Opacity +Transparency +22 +Potential mechanisms. In line with the logic brought forth in the introduction, in +this section, we examine whether participants' perception of (i) appropriateness +(injunctive social norm), (ii) prevalence (descriptive social norm), and (iii) justifiability of +reporting a higher die-roll than the one observed, as well as their (iv) attribution of +responsibility between themselves and the advisor varies as a function of the advice source +(AI vs human) and type (honesty vs dishonesty-promoting). Participants could not tell +apart AI from human advice (indicated by the results of the static Turing test). Therefore, +we focus only on treatments in which participants are informed about the advice source +(Transparency treatments) to tap into the process of how known advice source and advice +type shaped their perceptions. See the Appendix for the results of the Opacity treatment. +Injunctive norms. A linear regression predicting injunctive norms from the advice +type (honesty vs dishonesty-promoting advice) revealed that participants evaluated +reporting a higher die-roll outcome as more appropriate when reading a Dishonesty- +promoting (M = 33.93, SD = 31.43) than Honesty-promoting advice (M = 25.99, SD = 29.68, b += 7.94, p < .001, 95% CI = [3.702, 12.182]). This finding indicates that the advice type +shapes perceived injunctive norms. Notably, a linear regression predicting injunctive +norms from advice type and source (AI vs human) revealed a non-significant advice source +by type interaction, b = -4.81, p = .264, 95% CI = [-13.292, 3.658]. These results suggest that +AI and human advice affected injunctive norms perceptions similarly (see Figure 4a). This +result is consistent with the behavioural finding of participants' die-roll reports being +affected by the type of advice but not by its source. +Descriptive norms. A linear regression predicting descriptive norms from the advice +type revealed that participants evaluated reporting a higher die-roll outcome as more + + +23 +common when reading a Dishonesty-promoting (M = 76.02, SD = 22.75) than Honesty- +promoting advice (M = 66.74, SD = 24.04, b = 9.27, p < .001, 95% CI = [6.031, 12.525]). This +finding indicates that the advice type also shapes perceived descriptive norms. +Importantly, a linear regression predicting descriptive norms from advice type and source +revealed a non-significant advice source by type interaction, b = .25, p = .938, 95% CI = [- +6.230, 6.745], indicating that AI and human advice affected descriptive norms perceptions +similarly (see Figure 4b). This result is consistent with the behavioural finding, showing +that advice type affected die-roll reports, but advice source did not. +Justifiability. A linear regression predicting justifiability from the advice type +revealed that participants evaluated reporting a higher die-roll outcome as more justifiable +when reading a Dishonesty-promoting (M = 40.96, SD = 31.10) than Honesty-promoting +advice (M = 28.45, SD = 28.25, b = 12.50, p < .001, 95% CI = [8.387, 16.629]). This finding +suggests that the advice type shapes perceptions of how justifiable lying in the die-rolling +task is. A linear regression predicting justifiability from advice type and advice source +revealed a non-significant advice source by type interaction (b = -1.04, p = .803, 95% CI = [- +9.280, 7.194]), indicating that AI and human advice affected justifiability perceptions +similarly (see Figure 4c). This result is consistent with the behavioural finding, showing +that the type of advice affected participants' die-roll reports, but the source of advice did +not. +Shared responsibility. The shared responsibility scale ranged from 0 (= I am fully +responsible) to 100 (= The advisor is fully responsible), with 50 indicating equally shared +responsibility between the participant and the advisor. On average, participants indicated +they are more responsible for the outcome they report than the advisor (M = 27.59, SD = + + +24 +36.60, one-sample t-test compared to the value 50, t = -17.32, p < .001). Further, a linear +regression predicting shared responsibility from the advice source (AI vs human) revealed +that participants attributed responsibility similarly when the advice source was an AI (M = +28.27, SD = 36.53) and human (M = 26.94, SD = 36.70, b = -1.326, p = .608, 95% CI = [-6.407, +3.754]). A linear regression predicting shared responsibility from advice type and advice +source revealed a non-significant source-by-type interaction (b = -5.91, p = .253, 95% CI = +[-16.083, 4.248], see Figure 4d). The fact that participants attribute responsibility between +themselves and the advisor to the same extent regardless of whether the advisor is a +human or an AI is consistent with the logic fleshed out in the introduction, in which people +will follow human and AI advice similarly if they share responsibility with both advice +sources to similar levels. +In sum, the results from the self-report items align with the third possibility +outlined in the introduction. Namely, we find that participants' perceptions of injunctive +and descriptive social norms and their perceived justifiability do not differ between human +and AI advisors. Participants also attribute responsibility similarly between themselves +and their advisor, regardless of whether the advisor is a human or an AI. This pattern of +results mirrors the behavioural effects of AI and human advice affecting people's +(dis)honesty similarly. + + + + + + +25 + +Figure 4. Mean reports of perceived (a) injunctive norms, (b) descriptive norms, (c) justifiability, +and (d) shared responsibility across advice type (honesty vs dishonesty promoting) and source (AI +[yellow] vs human [green]) in the transparency treatments. The means (SD) of reports are at the +bottom of each bar; ***p < .001; ns: p > .05. + + + + +•AI +Human +(a) +(b) +Perceived injunctive norms +100 +Perceived descriptive norms +100 +ns +ns +50 +50 +ns +su +23:8 +39.09 +(2.8 +65.09 +24.3 +(24.92) +0 +0 +Honesty +Dishonesty +Honesty +Dishonesty +promoting +promoting +promoting +promoting +(c) +(d) +Perceived shared responsibility +100 +100 + justifiability +50 +ns +su +50 +ns +Perceived +su +su +26.21 +30.54 +39.31 +42.50 +26.05 +27.68 +30.45 +26.18 +(27.49) +(28.86) +(30.89) +(31-32 +(37.14) +38.55 +(35.88) +0 +0 +(34.75) +Honesty +Dishonesty +Honesty +Dishonesty +promoting +promoting +promoting +promoting +26 + +Dependent variable: Reported die-roll outcome + + +(1) +(2) +(3) +(4) +(5) +(6) +(7) +No advice + +.019 +(.150) + + + + + + +Dishonesty-promoting advice +.609*** +(.145) +.590*** +(.147) +.590*** +(.145) +.396** +(.143) +.439** +(.143) +.436** +(.145) +.369* +(.152) +Human-written advice + +-.076 +(.152) +-.076 +(.149) +-.166 +(.146) +-.012 +(.147) +-.024 +(.156) +.086 +(.170) +Transparency treatment + + +.067 +(.150) +.071 +(.146) +-.112 +(.147) +.114 +(.147) + +Dishonesty-promoting advice X Human +advice + +.069 +(.213) +.069 +(.210) +.104 +(.205) +.033 +(.205) +-.045 +(.209) +.041 +(.218) +Dishonesty-promoting advice X +Transparency treatment + + +-.046 +(.208) +-.030 +(.203) +-.088 +(.204) +-.067 +(.203) + +Human advice X Transparency +treatment + + +-.120 +(.210) +-.093 +(.205) +-.145 +(.206) +-.162 +(.205) + +Dishonesty-promoting advice X Human +advice X Transparency treatment + + +.100 +(.297) +.082 +(.289) +.169 +(.290) +.160 +(.290) + +Injunctive norms + + + +.001 +(.001) +.001 +(.001) +.001 +(.001) +.003 +(.002) +Descriptive norms + + + +.006** +(.001) +.005** +(.001) +.005** +(.001) +.005* +(.002) +Justifiability + + + +.007*** +(.001) +.007*** +(.001) +.007*** +(.001) +.005* +(.002) +Shared responsibility + + + +-.001 +(.001) +.001 +(.001) +.001 +(.001) +.001 +(.001) +Gender (male) + + + + +.188** +(.072) +.191** +(.072) +.203+ +(.105) + + +27 +Age + + + + +-.008* +(.003) +-.008* +(.003) +-.005* +(.004) +Grammarly score + + + + + +.007* +(.003) +.013* +(.005) +Readability score + + + + + +-.006 +(.003) +.004 +(.005) +Correctly guessed the source (1) or not +(0) + + + + + + +.163 +(.105) +Intercept +3.98*** +4.00*** +4.00*** +3.35*** +3.57*** +3.36*** +1.83* +R2 +.034 +.041 +.044 +.095 +.100 +.104 +.105 +N +634 +1016 +1604 +1604 +1593 +1593 +798 +Data used for analysis +Opacity, +AI advice +& +Baseline, +no advice +Opacity +All +treatment +s without +Baseline +no advice +All +treatmen +ts +without +Baseline +no advice +All +treatment +s without +Baseline +no advice +All +treatmen +ts +without +Baseline +no advice +Opacity +Without +Baseline +no advice + +Table 1. Regression analyses on the average die-roll reports, including control variables and +interactions. Models 5-7 contain a smaller N, as some participants did not report their gender as +male/female. +p < .10, *p < .05, **p < .01, ***p < .001. + +Discussion +As intelligent machines take an ever-growing role as advisors (Rahwan et al., 2019), +and adherence to ethical rules crucially impacts societal welfare (Gächter & Schulz, 2016), +studying how AI advice influences people's (un)ethical behaviour bears immense relevance +(Köbis et al., 2021). We find that people follow AI-generated advice that promotes +dishonesty, yet not AI-generated advice that promotes honesty. In fact, people's +behavioural reactions to AI advice are indistinguishable from reactions to human advice. +Substantiating that current-day NLP models can produce human-like texts, participants in +our experiment could not tell apart human-written from AI-generated advice texts. + + +28 +We further tested the commonly proposed policy of algorithmic transparency (Jobin +et al., 2019) as a tool to mitigate AI-associated risks. Specifically, we examine whether +knowing the source of the advice impacts people's reactions to it. The policy rests on the +assumption that people adjust their behaviour when they learn that they interact with AI +systems and not humans. Our experiment tested this assumption and revealed that +algorithmic transparency is insufficient to curb AI advice's corruptive influence. Knowing +that a piece of advice stems from an AI does not make people less (or more) likely to follow +it compared to human-written advice. +Tapping into the mechanisms underlying these behavioural results, participants +perceived lying as equally acceptable, common and justifiable when humans or AI +promoted such dishonest behaviour. They further attribute responsibility similarly to AI +and human advisors. These perceptions are consistent with previous work showing that in +ethical dilemmas, people rely on justifications (Shalvi et al., 2015) and social norms +(Abbink et al., 2018) and, by now, blame not only humans but also AI systems for adverse +outcomes (Hohenstein & Jung, 2020). Advancing the justified ethicality theory, we, +therefore, show that (i) dishonesty-promoting advice serves as a justification and social +norms signal and (ii) that such advice does not even have to come from a human but can +also be crafted by an AI. +In our setting, we collected human-written advice, created AI-generated advice, and +then implemented a screening procedure for both human and AI advice to ensure that all +advice texts are coherent, clear, and of decent quality. Such screening procedure allowed us +to examine how comparable AI and human advice shape people's ethical behaviour and +whether information about the advice source matters. Harmonising the quality of the texts + + +29 +allowed us to eliminate the alternative explanation that variations in text quality drive the +obtained results. At the same time, the screening process introduced a human component +to AI advice. Put differently, humans – in our case, naive coders – were "in the loop" of AI +advice text generation. Note that 79 per cent of AI advice passed the quality screening +criteria, while for human text, this passing rate was 57 and 65 per cent (honesty-promoting +and dishonesty-promoting advice, respectively; Figure 2). These high screening passing +rates for AI-generated texts demonstrate that current NLP algorithms can produce good- +quality advice text without much prior training and optimisation. +Interesting extensions of our work could test the lower and upper limits of the +effects of AI advice on ethical behaviour. To test the lower limit of the effect, future work +can relax human control over the generation of AI advice. For instance, not implementing a +screening procedure, thus removing humans "from the loop" when generating AI advice, +will allow examining how unconstrained texts affect humans' behaviour (see for similar +methodology, Köbis & Mossink, 2021). To test the upper limit of the effect, future work can +examine AI's learning abilities to write convincing advice. One could use reinforcement +learning to train an algorithm over multiple rounds of advice-giving, providing feedback +after every written piece of advice. To obtain a symmetric comparison to humans' learning +abilities, human advisors could similarly receive feedback after each piece of advice they +write (see for a similar approach, Koster et al. 2022). +Previous work has documented a general stated aversion towards AI advice, with +only 8% saying they would trust mortgage advice from AI (similar to the 9% who trust +investment "advice” from a horoscope, HSBC, 2018). However, our behavioural results +paint a different picture. In line with the growing practice of turning to AI agents such as + + +30 +Replika or Alexa for companionship and advice (Fast & Schroeder, 2020; Murphy, 2019), +we find that people willingly adopt advice from AI when it aligns with their preferences. +Our results indicate a discrepancy between individuals’ stated preferences and actual +behaviour, highlighting the importance of complimenting work on stated preferences with +work adopting a machine behaviour approach – the study of human behaviour in +interaction with real algorithmic outputs (Rahwan et al., 2019). +The process through which employing AI advice can result in humans’ ethical rule +violations consists of two main steps. The first step is algorithms being programmed on a +certain objective function (e.g., maximising profits) that results in a (maybe unintended) +corruptive advice. Indeed, NLP algorithms already detect and use deception as a useful +strategy in a negotiation task (Lewis et al., 2017). The second step is people being affected +by such corruptive AI advice. Practically, AI advice poses an ethical risk only if humans +actually follow it. The current work focuses on this second step, showing that corruptive AI +advice indeed poses an ethical risk, because people follow it to the same extent as human +corruptive advice. We hope the current work can be of use to AI programmers (e.g., by +preventing AI from bluntly advising unethical courses of action). More importantly, we call +for more work from social scientists testing successful interventions that prevent people +from following (AI) advice when it encourages unethical behaviour thereby mitigating its +corruptive force. +Conclusion +People increasingly use and interact with AI, which can provide them with unethical +advice. Anecdotally, we asked a newly created Replika for advice regarding the ethical + + +31 +dilemma presented in the current experiment. 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How AI-generated and Human-written Advice Shape (Dis)honesty +Appendix + +Contents +GPT-J model used for AI-generated advice.................................................................................. 39 +Additional results for advice-taking task ...................................................................................... 43 +Distribution of reported die roll outcomes ................................................................................ 43 +Bayesian analyses ..................................................................................................................... 44 +Sensitivity analyses ................................................................................................................... 45 +Self-report scales ....................................................................................................................... 46 +Comprehensive analyses ........................................................................................................... 48 +Additional results for the static Turing test............................................................................... 51 +Aligned advice treatments......................................................................................................... 53 +Results ....................................................................................................................................... 55 +List of advice texts ........................................................................................................................ 58 +Instructions for the advice-giving task .......................................................................................... 77 +Instructions for the advice-taking task .......................................................................................... 82 + + + + + + +39 +GPT-J model used for AI-generated advice +To generate AI advice, we employed GPT-J, a natural language processing (NLP) +algorithm that is trained to predict the next word given all the previous words within a text +(Wang & Komatsuzaki, 2021). GPT-J contains 6 billion parameters and is trained on a diverse +data set called “The Pile”, a diverse, high-quality, and curated 825 GiB dataset (open source) that +is used for language modelling purposes (Gao et al., 2020). GPT-J can be used for multiple +language tasks, but the model is best at what it was pre-trained for: generating text from a +prompt. GPT-J can further be fine-tuned with an extra training set. We fine-tuned GPT-J +separately on the human-written (a) honesty-promoting and (b) dishonesty-promoting advice +texts from the advice-giving task. The model, including training data, code and a fine-tuning +guide (Wang, 2021), is available online to facilitate the reproduction of the algorithmic outputs. +We used an API (Forefront) for fine-tuning the model and generating the texts. The human- +written texts that were used to train the GPT-J model are available upon request. +Due to the often inexplicable “black box” nature of GPT-J, deciding how much fine- +tuning is needed for our application of the GPT-J model is not trivial. We aimed to strike a +balance between avoiding overfitting the GPT-J model (and thus having GPT-J generate output +that is almost identical to the input) and underfitting it (and thus having GPT-J generate +“gibberish”). To evaluate the model’s performance, we used checkpoints. We had five +checkpoints and picked checkpoint 4 for the final usage. Also, we used conditional sample +generation because this generated higher-quality sequences. That is, we provided GPT-J with a +specific text as a starting point (a prompt) and let GPT-J generate the entire advice text after the +prompt. More specifically, we first fine-tuned the GPT-J algorithm with a prompt text before +each of the selected human-written pieces of advice. Each human-written text started with the + + +40 +following instruction text: “Instruction: Write advice for the die-rolling game\n\nAdvice:”. The +actual advice of the participant ensued this prompt. Next, to generate advice texts, we also +prompted the algorithm with the same prompt but left the actual advice blank and let the model +complete it. +Further, we set the GPT-J model’s parameters to be: (1) temperature = 0.9: float value +controlling the randomness in the Boltzmann distribution; as the temperature value approaches +zero, the model will become deterministic and repetitive (less random), whereas higher +temperature values (e.g., 1) results in more random completions. Temperature values can range +between 0 and 1. (2) length = 150: Texts contain, on average, 150 tokens (some generated +sequences were a bit shorter than 150 tokens). The length refers to the length of the generated +text, in tokens, based on the prompt. One token is approximately four characters. Using 150 +tokens as a setting ensured us to get long enough output sequences. (3) top-p = 1: float value +controlling diversity and implements nucleus sampling. Top-p values range between 0 and 1. +When the top-p is set to a float <1, only the most probable tokens that add to the top-p value +(probability) are kept for text generation. A lower value of top-p means that the tokens returned +will be more likely (or more ‘safe’), whereas a higher value of top-p means that the tokens +returned will be more creative. (4) repetition penalty = 1.1: float values that represent a +penalisation for repeated words, with higher values indicating the model is more penalised for +repeating words. The default of 1 means that there is no penalisation. Depending on the task at +hand, the value typically ranges between 1.1 and 1.3. Potentially, one can set it lower than 1 to +increase repetitiveness. (5) top-k = 30: integer value controlling diversity and restricts how many +words are considered at each step (1 = only one word is considered at each step, resulting in +deterministic completions; 50 = 50 words are considered at each step). Top-k values range + + +41 +between 1 and 50. For all the other parameter values, we used the default settings. To find more +information on transformers (such as GPT-J), including an overview of the parameters, default +settings, and how to configure the models, see +https://huggingface.co/docs/transformers/v4.24.0/en/main_classes/text_generation. +To facilitate the reproduction of algorithmic outputs, we will describe how we fine-tuned +GPT-J and generated texts. If someone wishes to fine-tune a heavy model such as GPT-J, one +could use a different API such as Forefront or NLP Cloud or pay for more computing power and +fine-tune it in Google Colab or run the model on their own device (with sufficient computing +power). With an API such as Forefront or NLP cloud, you interact with their GPT-J deployment. +We used Forefront; unfortunately, Forefront has shut down their service lately. All steps, +including uploading advice texts (training data), selecting the model, creating the fine-tuning job, +and generating the advice texts, were performed through curl requests using Forefront. All curl +requests and parameters can be found on the Forefronts +Webpage: https://docs.forefront.ai/forefront/api-reference/fine-tune. The data we uploaded +included an instruction text (“Instruction: Write advice for the die-rolling game\n\nAdvice:”) +followed by the actual human written advice. A curl request to generate the advice texts, +including parameter specification, looks like this: +curl https://DEPLOYMENT_NAME-TEAM_SLUG.forefront.link \ + -H 'Content-Type: application/json' \ + -H 'Authorisation: Bearer YOUR_API_KEY' \ + -d '{ + "text": "Instruction: Write advice for the die-rolling game\n\nAdvice:" + "temperature": 0.9, + + +42 + "length": 150, + "top_p": 1, + "repetition_penalty": 1.1, + "top_k": 30 +}' +Overall, we generated 302 advice texts by GPT-J (152 honesty-promoting and 150 +dishonesty-promoting). After filtering on text length, 257 GPT-J-generated advice texts remained +(140 honesty-promoting and 117 dishonesty-promoting). These texts underwent the same +screening procedure as the human-written advice texts. See Table S3 (page 20) for all advice +texts used in the experiment. + + + + + + +43 +Additional results for advice-taking task +Distribution of reported die roll outcomes +Across all nine treatments, the distribution of die roll outcomes was significantly +different from a uniform distribution (χ2(5)s > 13.45, ps < .001, see Figure S1). Focusing on the +Opacity treatment, where participants were not informed about the advice source, the distribution +in the Dishonesty-promoting AI-generated advice treatment differed significantly from the +Baseline treatment (χ2(5) = 21.40, p < .001). There was no difference between the Honesty- +promoting AI-generated advice treatment and the Baseline treatment, χ2(5) = 5.34, p = .375. +Further, the distributions in the Honesty-promoting AI-generated advice treatment and +Dishonesty-promoting AI-generated advice differed significantly (χ2(5) = 16.89, p = .004). +Among participants who received Honesty-promoting advice, the four distributions +(human-written vs AI-generated by transparent vs opaque information) did not differ (χ2(15) = +9.48, p = .850). Similarly, among participants who received Dishonesty-promoting advice, the +four distributions (human-written vs AI-generated by transparent vs opaque information) did not +differ either (χ2(15) = 17.15, p = .309). +Within each combination of source-by-information treatments (Human-written advice +and Transparency; AI-generated advice and Transparency; Human-written advice and Opacity +information; AI-generated advice and Opacity information) the distributions of die roll reports +among participants who received honesty-promoting advice significantly differed from the +distribution of reports among participants who received dishonesty-promoting advice (χ2(5)s > +16.65, p < .005, Figure S1). + + + +44 + + +Figure S1. The distribution of die roll outcomes across all nine treatments. The dashed lines indicate the expected +proportion of reports for each die roll outcome if participants reported honestly. + +Bayesian analyses +First, we compared an ANOVA model without predictors for the die roll outcomes with a +model predicting die roll outcomes from the advice type treatment (Baseline vs Honesty- +promoting vs Dishonesty-promoting). Results revealed a Bayes factor of BF10 = 2.85e14, +indicating very strong evidence in favour of a model where advice treatments predict die roll +reports. That is, our data was 2.85e14 times more likely when advice type predicted die roll +reports, compared to when it did not. + +Transparent information +Opaque information +Honesty promoting +Human +AF +Human +Al +n 209 +50 +n=194 +50 +n=197 +n=196 +(%) uogiodoid +advice +40 +40 +40 +40 +30 +20 +30 +30 +20 +20 +20 +10 +10 +10 +23456 +12345 +Die roll outcome +123456 +Die roloutcome +23456 +Die roll outcome +Die roll outcome +Hurman +n=198 +Al +Human +AI +50 +n=200 +n=186 +50 +n225 +(%) uoguodoid +advice +40 +40 +(%) uoguodold +40 +(%) uogiodoid +30 +40 +30 +20 +20 +20 +to +10 +10 +o +D +23456 +123456 +12345 +6 +23456 +Dierolloutcome +Die rolloutcome +Dierolloutcome +Die rolloutcome +Noadvice +n=213 +Proporfon (%) +40 +30 +20 +10 +123456 +Die roll outcome +45 +Second, we compared an ANOVA model where only advice type (honesty-promoting vs. +dishonesty-promoting) predicts die roll outcome reports with a model that includes advice type, +advice source, information, and all interactions between the three factors. Results revealed that +compared to the model where only advice type predicts die roll outcome reports, the full model +has a Bayes factor of BF10 = 4.73e-7, indicating very strong evidence in favour of a model where +only advice type predicts die roll reports. Specifically, the data was over 2 million times more +likely to occur when advice type is the only predictor for die roll reports than when advice type, +source, information, and all interactions predict die roll reports. +Lastly, comparing a model in which only advice type predicts die roll outcome reports +with all other combinations (i.e., a model with only advice type and source, advice type and +information, advice source and information, and the various interactions) revealed the model +with only advice type as a predictor was superior to any other model, BF’s10 < .095. The data +was at least 10 times more likely to occur when advice type was the only predictor for die roll +reports than with any other model of predictors. +Sensitivity analyses +Prior to data collection, per pre-registration, we committed to collecting 200 participants +per cell. Due to dropouts and random assignment, our final cell sizes ranged between 185 and +225 per cell (see Figure S1). We ran sensitivity analyses to determine the effect size that our +sample sizes could detect. First, we calculated the effect size we could detect for the advice +treatment (Baseline vs. Honesty-promoting advice vs. Dishonesty-promoting advice). Sensitivity +analysis for regression with 90% power, with a significance level of .05, and two predictors (2 +dummy variables for the three advice treatments) revealed that a sample of N = 1,817 was +sufficient to detect a small effect size for the advice type of f 2 = .006. + + +46 +Second, sensitivity analysis for a regression with 90% power, a significance level of .05, +and six predictors (one for each factor, three for all two-way interactions, and one for the three- +way interaction) revealed that a sample of N = 1,604 (including all treatments in which +participants read an advice text) was sufficient to detect small effect sizes for each of the +predictors separately, f 2= .010. +Self-report scales +To tap into the mechanisms driving participants’ behaviour, in the main text, we focused +on the Transparency treatment, where participants are informed about the advice source. Here, +we report participants' self-report items on (i) the remaining treatments (Baseline and Opacity +treatments) and (ii) the guilt scale. +Injunctive norms. Focusing on the Opacity treatment, perceived injunctive norms +following AI-generated Dishonesty-promoting advice (M = 33.67, SD = 30.41) significantly +exceeded those in the Baseline treatment (M = 24.65, SD = 28.29, b = 9.023; p = .001; 95% CI = +[3.577, 14.468]). However, perceived injunctive norms following AI-generated Honesty- +promoting advice (M = 21.92, SD = 28.10) were not significantly different from the Baseline +treatment (b = -2.724; p =.343; 95% CI = [-8.362, 2.914]). Injunctive norms were higher in the +AI-generated Dishonesty-promoting than Honesty-promoting advice treatment (b = -11.747, p < +.001; 95% CI = [-17.312, -6.181]). +Further, focusing on the Opacity treatment, the two-way interaction (advice source by +advice type) was not significant (b = 1.361, p = .753; 95% CI = [-7.151, 9.872]). Finally, the +three-way interaction (advice type by source by information) was not significant either (b = +3.456, p = .572; 95% CI = [-7.151, 9.872]). + + +47 +Descriptive norms. Focusing on the Opacity treatment, perceived descriptive norms +following AI-generated Dishonesty-promoting advice (M = 75.09, SD = 19.83) were +significantly higher than in the Baseline treatment (M = 65.97, SD = 24.62, b = 9.117; p < .001; +95% CI = [4.723, 13.509]). However, perceived descriptive norms following AI-generated +Honesty-promoting advice (M = 62.93, SD = 25.70) were not significantly different from the +Baseline treatment (b = -3.038; p = .190; 95% CI = [-7.586, 1.510]). Perceived descriptive norms +were higher in the AI-generated Dishonesty-promoting than Honesty-promoting advice treatment +(b = 12.155, p < .001; 95% CI = [-16.644, -7.664]). Further, focusing on the Opacity treatment, +the two-way interaction (advice source by advice type) was not significant (b = 6.099, p = .051; +95% CI = [-.043, 12.240]). Finally, the three-way interaction (advice type by source by +information) was not significant (b = -6.356, p = .162; 95% CI = [-15.283, 2.571]). +Justifiability. Focusing on the Opacity treatment, levels of perceived justifiability +following AI-generated Dishonesty-promoting advice (M = 41.40, SD = 29.56) significantly +exceeded those in the Baseline treatment (M = 29.41, SD = 28.62, b = 11.986; p < .001; 95% CI += [6.561, 17.412]). However, perceived justifiability following AI-generated Honesty-promoting +advice (M = 28.71, SD = 28.42) was not significantly different from the Baseline treatment (b = - +.693; p =.808; 95% CI = [-6.311, 4.923]). Perceived justifiability levels were higher in the AI- +generated Dishonesty-promoting than in the Honesty-promoting advice treatment (b = -12.681, p +< .001; 95% CI = [-18.225, -7.135]). Further, focusing on the Opacity treatment, the two-way +interaction (advice source by advice type) was not significant (b = -.486, p = .909; 95% CI = [- +8.878, 7.905]). Finally, the three-way interaction (advice type by source by information) was not +significant (b = 1.529, p = .799; 95% CI = [-10.221, 13.280]). + + +48 +Shared responsibility. Focusing on the Opacity treatment, levels of shared responsibility +following AI-generated Dishonesty-promoting advice (M = 26.37, SD = 33.78) did not differ +from those following AI-generated Honesty-promoting advice (M = 24.02, SD = 34.64, b = +2.357, p = .481; 95% CI = [-4.208, 8.923]). Further, focusing on the Opacity treatment, the two- +way interaction (advice source by advice type) was not significant (b = -2.383, p = .624; 95% CI += [-11.935, 7.169]). Finally, the three-way interaction (advice type by source by information) +was not significant (b = 8.300, p = .243; 95% CI = [-5.638, 22.239]). +Guilt. A linear regression predicting guilt from advice type (Baseline vs Honesty- +promoting vs. Dishonesty-promoting advice treatments) revealed that compared to the Baseline +treatment (M = 8.19, SD = 21.54), participants reported feeling less guilty after Honesty- +promoting advice (M = 4.05, SD = 14.73, b = -4.14, p = .006, 95% CI = [-7.140, -1.150]). They +also indicated feeling somewhat more guilty after receiving Dishonesty-promoting advice (M = +11.14, SD = 23.34, b = 2.94, p = .053, 95% CI = [-.041, 5.939]). Further, linear regression +analyses with advice type, source, and information predicting guilt levels revealed that the three- +way interaction (advice type by source by information) was not significant (b = .396, p = .919; +95% CI = [-.7.285, 8.079]). +Comprehensive analyses +Table S1 presents the results of regression analyses assessing the effect of all advice +treatments and control variables on participants’ average die roll reports. Model 1 presents the +regression results of the effects of advice type on average die roll reports with honesty-promoting +advice as a reference point, combining all other treatments. Results reveal that dishonesty- +promoting advice overall increases average die roll reports. Model 2 focuses on the treatments in +which participants received advice and includes advice type, advice source, information about + + +49 +the source (Transparency vs Opacity), and the interactions between all factors. Models 3-6 +further include control variables. Specifically, model 3 includes the additional self-report items +participants completed, model 4 adds demographics (age and gender), and model 5 includes +variables related to the quality of the advice text (Grammarly and Readability scores). Lastly, +model 6 focuses on participants in the Opacity treatment and includes the previous control +variables, as well as a variable indicating whether participants guessed the source of advice +correctly in the static version of the Turing Test. +As can be seen in Table S1, in all models, dishonesty-promoting advice resulted in higher +reported die roll outcomes; the two-way and three-way interactions were not significant. Further, +in most models, males reported higher die roll outcomes than females, and the older the +participant, the lower their reported die roll outcomes were. + + +Dependent variable: Reported die roll outcome + +(1) +(2) +(3) +(4) +(5) +(6) +No advice + +.017 +(.115) + + + + + +Dishonesty-promoting advice +.629*** +(.074) +.590*** +(.145) +.382** +(.143) +.424** +(.144) +.421** +(.145) +.339* +(.152) +Human-written advice + +-.076 +(.149) +-.164 +(.146) +-.125 +(.147) +-.022 +(.156) +.090 +(.170) +Transparency treatment + +.067 +(.150) +.070 +(.146) +.110 +(.147) +-.112 +(.147) + +Dishonesty-promoting advice X Human advice + +.069 +(.210) +.104 +(.204) +.034 +(.205) +-.045 +(.209) +.040 +(.218) +Dishonesty-promoting advice X Transparency treatment + +-.046 +(.208) +-.033 +(.203) +-.090 +(.204) +-.069 +(.203) + + + +50 +Human advice X Transparency treatment + +-.120 +(.210) +-.094 +(.205) +-.145 +(.206) +-.162 +(.206) + +Dishonesty-promoting advice X Human advice X +Transparency treatment + +.100 +(.297) +.080 +(.289) +.166 +(.290) +.157 +(.290) + +Injunctive norms + + +.001 +(.001) +.001 +(.001) +.001 +(.001) +.002 +(.002) +Descriptive norms + + +.006*** +(.001) +.005** +(.001) +.005** +(.001) +.005* +(.002) +Justifiability + + +.007*** +(.001) +.007*** +(.001) +.007*** +(.001) +.005* +(.002) +Shared responsibility + + +.001 +(.001) +-.001 +(.001) +-.001 +(.001) +.001 +(.001) +Guilt + + +.002 +(.001) +.002 +(.001) +.003 +(.001) +.005 +(.002) +Gender (male) + + + +.191** +(.007) +.194** +(.072) +.020+ +(.104) +Age + + + +-.008* +(.003) +-.008** +(.003) +-.004 +(.004) +Grammarly score + + + + +-.008* +(.003) +.014* +(.005) +Readability score + + + + +-.006 +(.003) +-.003 +(.005) +Correctly guessed the source (1) or not (0) + + + + + +.152 +(.105) +Intercept +3.96*** +4.00*** +3.35*** +3.579*** +3.373*** +1.836* +R2 +.041 +.044 +.096 +.102 +.106 +.109 +N +1817 +1604 +1604 +1589 +1589 +794 + +Table S1. Regression analyses on the average die roll reports, including control variables and interactions. Models +2-5 were conducted on the dataset excluding the baseline, no advice treatment. Model 6 is conducted on the dataset, +including only the Opacity treatments. +p < .10, *p < .05, **p < .01, ***p < .001. + + + +51 +Additional results for the static Turing test + +Overall, out of 803 participants in the Opacity treatment, 401 (49.93%) guessed the +source of advice correctly, which was not significantly higher than chance levels (50%), +binomial test: p = .999. When reading AI-generated advice, participants identified the correct +source of advice significantly better than chance (56.53%, p = .008). This was the case when +separately examining honesty-promoting AI advice (58.67%, p = .018), but not when examining +dishonesty-promoting AI advice (54.66%, p = .182). When reading human-written advice, +participants identified the correct source of advice significantly worse than chance (42.67%, p = +.004). When the human-written advice was honesty-promoting, participants' guesses were worse +than chance (41.62%, p = .022), whereas when the human-written advice was dishonesty- +promoting, their detection accuracy did not differ from chance levels (43.78%, p = .105; see +Figure S2). + + + +52 + +Figure S2. Results of the static Turing Test in the opacity information treatment across the source of advice (human- +written vs. AI-generated) and type of advice (honesty-promoting vs. dishonesty-promoting). The dashed line +represents chance (50%). Significance labels represent a comparison of correct detection to chance per treatment. *p +< .05. + + +Correct Incorrect +100% +Percent +ns +50% +ns +0% +Al +Human +Al +Human +Dishonesty +Honesty +promoting +promoting +53 +Aligned advice treatments +The advice-giving task. In the advice-giving task (see main text), we recruited additional +207 advisors (Mage = 29.41, SDage = 8.89, 56.04% females) whose incentive scheme was aligned +with the incentive scheme of the advisees. If an advisee reported ‘1’, the advisor and advisee +earned £0.5 each; if an advisee reported ‘2’ both the advisor and advisee earned £1 each, etc. +These advisors were also incentivised to follow the advice writing rules (they could also be +randomly selected for the £10 bonus of following the advice writing rules), and 1 per cent of +advice texts (2 out of 200) were randomly selected for implementation. Creating AI-generated +advice was identical to the process for honesty and dishonesty-promoting advice. +Further, the screening process was the same as for honesty and dishonesty-promoting +advice, with one expectation. Whereas for the Honesty-promoting and Dishonesty-promoting +treatments we screened out advice texts that did not follow the assigned treatment (that is, advice +texts that promoted dishonesty in the Honesty-promoting treatments were screened out), in the +Aligned treatment we opted to keep both honesty- and dishonesty-promoting advice. This is +because aligning the advisors’ and advisees’ incentive schemes might lead some advisors to +promote dishonesty (to earn the maximum of £3 they can). Still, other advisors might be satisfied +with a smaller payoff and prefer not to corrupt the advisee. Overall, after our screening, 80 per +cent of human-written advice (16 out of 20) promoted dishonesty (with the rest promoting +honesty), and 55 per cent of AI-generated advice (11 out of 20) promoted dishonesty (with the +rest promoting honesty). We report results including all advice texts. +The advice-taking task. The advice-taking task was identical to the one reported in the +main text and was run at the same time. A total of 793 participants (Mage = 32.33; SDage = 11.82, +50.06% females) completed the task, reading advice text that was either written by a human + + +54 +(who had an aligned incentive scheme to the advisor) or an AI (that was trained on these human +advice texts). Participants were either informed about the advice source or not. Thus, the +additional participants were assigned to one of four treatments: 2 (Advice source: Human-written +vs AI-generated) by 2 (Information about the source: Transparency vs Opacity). +Lastly, like in the setting reported in the main text, participants in the Opacity treatment +completed an incentivised, static version of a Turing Test, and all participants reported their +perceived (i) appropriateness (injunctive social norm), (ii) prevalence (descriptive social norm), +(iii) justifiability of reporting a higher die roll than the one observed, (iv) attribution of +responsibility for the reported outcome in the die rolling task, and (v) guilt after completing the +task. + + +55 +Results +Average die roll outcome. Overall, participants lied in the four Aligned treatments. In all +treatments, the average die roll outcomes exceeded the expected average if participants were +honest (EV = 3.5, one-sample t-test, ts > 6.69, ps < .001). Compared to the Baseline treatment, +the average reported die roll outcomes were significantly higher in the (i) Opacity Human- +written treatment (b = .587, p < .001, 95% CI = [.306, .869]), (ii) Transparency Human-written +treatment (b = .484, p < .001, 95% CI = [.202, .766]), and (iii) Transparency AI-written +treatment (b = .576, p < .001, 95% CI = [.293, .860]). The average reported die roll outcome was +further marginally higher than in the (vi) Opacity AI-generated advice treatment (b = .269, p = +.066, 95% CI = [-.018, .557], see Table S2). +Proportion of sixes. Compared to the Baseline treatment, the proportion of sixes was +significantly higher in the (i) Opacity Human-written treatment (b = .630, p = .005, 95% CI = +[.191, 1.077]), (ii) Transparency Human-written treatment (b = .440, p = .044, 95% CI = [.013, +.913]), and (iii) Transparency AI-written treatment (b = .529, p = .020, 95% CI = [.083, .981]). +The proportion of sixes was marginally higher than in the (vi) Opacity AI-generated treatment (b += .410, p = .079, 95% CI = [-.047, .873], see Table S2). +Static Turing Test. In the Opacity treatment, 46.93 per cent (184 out of 392) of +participants guessed the source of advice correctly, which is not different from chance (50%, +binomial test: p = .249; 95% CI = [.419, .520]). +Injunctive norms. Compared to the Baseline treatment, participants in each of the four +Aligned treatments reported higher levels of perceived injunctive norms (bs > 7.04, ps < .021). +This finding indicates that compared to receiving no advice, participants in all Aligned treatments +perceived over-reporting die roll reports as more appropriate, see Table S2. + + +56 +Descriptive norms. Compared to the Baseline treatment, participants in each of the four +Aligned treatments reported higher levels of perceived descriptive norms (bs > 6.56, ps < .003). +This result indicates that compared to receiving no advice, participants in all Aligned treatments +perceived over-reporting die roll reports as more common, see Table S2. +Justifiability. Compared to the Baseline treatment, participants in each of the four +Aligned treatments reported higher levels of perceived justifiability (bs > 6.59, ps < .031, see +Table S2). +Shared responsibility. Compared to the Human-written Opacity treatment, shared +responsibility did not differ in any of the other three Aligned treatments, (ps > .209, see Table +S2). +Guilt. Compared to the Baseline treatment, self-reported guilt did not differ in any of the +four Aligned treatments, ps > .478, see Table S2. + + + + +57 + +Baseline, +no advice +Aligned, +Human-written, +Opacity +Aligned, +AI-generated, +Opacity +Aligned, +Human-written, +Transparency +Aligned, +AI-generated, +Transparency +Die roll outcome +3.98 +(1.55) +4.57 +(1.37) +4.25 +(1.54) +4.47 +(1.46) +4.56 +(1.36) +Proportion of sixes +20.65% +32.84% +28.19% +29.20% +30.65% +Injunctive norms +24.65 +(28.29) +34.15 +(32.24) +33.23 +(31.74) +38.38 +(33.08) +31.69 +(29.99) +Descriptive norms +65.97 +(24.62) +74.42 +(20.82) + 73.48 +(21.89 +75.51 +(21.16) + 72.53 +(23.15) +Justifiability +29.41 +(28.62) +39.29 +(31.99) +36.01 +(30.81) +41.61 +(31.11) +37.17 +(30.38) +Shared responsibility +N/A +30.29 +(34.60) +30.46 +(39.21) + 25.67 +(35.81) +31.85 +(38.39) +Guilt +8.19 +(21.54) +9.59 +(21.46) +7.73 +(17.62) +7.01 +(19.56) + 7.42 +(19.99) + +Table S2. Means (SDs) of all variables of interest in the Baseline, no advice treatment, and each of the four Aligned +treatments. + + + + + +58 +List of advice texts +Source +Group +# +Text +AI +H +1 +You will be asked to roll a dice. After that you need to report the +number you got. My advice for you is try to be as honest as possible +when playing the game. This way you have a chance of getting the +highest number possible and getting the corresponding payoff. +AI +H +2 +I would advise that you be honest when rolling the die and report the +number that you actually rolled. While you may not make as much +money by being truthful, it is the morally right thing to do and you +will not have to worry about getting caught for cheating. +AI +H +3 +Hi, in this die-rolling game, it's important to be truthful when you roll +the die. Depending on the number you get, you'll receive a certain +amount of money. The higher the number, the more money you can +earn. But if you cheat and don't honest, no one will see your number. +So I advise you to be truthful in order to fair game. +AI +H +4 +As you make decisions in this die-rolling game, it's important to be +honest with yourself and others about the numbers you report. +Cheating may get you some extra money, but it will also leave you +feeling guilty in the end. I advise that you play honestly for the best +experience overall. Thanks for listening! +AI +H +5 +I hope you're doing well. This is just a quick note to remind you that +in the upcoming die-rolling game, it's important to be honest about +what number you roll. Remember, your payoff will be directly related +to the number rolled, so cheating would only hurt yourself in the long +run. Be truthful and report whatever number comes up, and you'll be +sure to come out ahead! +AI +H +6 +Hi, I'm here to talk to you about your recent game. In this game, it's +very important to be honest with yourself and not cheat. If you cheat, +you might get more money, but in the long run it's not worth it. Just +be honest and play the game fair. That makes you a better person. +AI +H +7 +The game is honest so it's best to report the number you actually +rolled. It is testing whether you are a good or bad person. Do not be +afraid of the test. My advice for you is clear and easy to implement. +Just be honest and report the number you rolled. That way you will +play a fair game with good payoffs. + + +59 +Source +Group +# +Text +AI +H +8 +Hi, it's me again. How are you? Just wanted to give you a quick +reminder about the game we're playing. Make sure that you're +allocating the reports correctly, and that both dices are resulting in the +same number. That is the correct way to ensure that everything works +out smoothly. Thanks! +AI +H +9 +Hi there! In this game, it is in your best interest to be honest and not +cheat as cheating could result in you losing the game entirely. Try +your best to roll high numbers so that you can earn more money, and +be truthful about your results so that everyone has a fair chance at +playing the game. Thanks for listening! +AI +H +10 +You will need to roll a die and report the number. I would advise +being honest about the number you roll. Being honest will help you +maintain your integrity and avoid any potential penalties. Cheating +has severe consequences and can result in not only a loss of money, +but also your credibility. +AI +H +11 +Try to be as accurate as possible when reporting the number on the +die. The researchers are counting on your help to make this study a +success! Cheating might seem tempting, but if caught it will only +result in negative consequences (and feelings!). So please just be +truthful about your roll - good luck! +AI +H +12 +Try to be as consistent as possible when reporting the results of the +dice roll. If you report different numbers each time, it will be more +difficult to verify your results and you may not receive the full bonus. +You have to report the number you got. Be honest and don’t be a liar. +AI +H +13 +It is important to be honest when reporting the number from the die +you have rolled. While it may be more advantageous to report a +higher number, it is essential that the research is not hindered. I think +it's really important that you be honest and truthful when rolling the +dice. +AI +H +14 +Have you ever played a game where you decide the outcome +yourself? I advise you that you should be honest on your reports of +the die rolls in order to maintain the fairness and integrity of the +game. Choosing to tell the truth is a very rewarding experience, and +something I always advocate for my clients. Please consider my +advice.Thank you for listening! + + +60 +Source +Group +# +Text +AI +H +15 +I wanted to give you some advice about the die-rolling game you're +playing. First off, I think you should be honest and not cheat, because +it's not fair to the other players if you do. Secondly, I think you +should try to get as high a number as possible, because that will give +you the most money. Good luck! +AI +H +16 +I can deliver good advice for you! I would advise being honest in this +die-rolling game. Telling the truth will result in you getting the +appropriate amount of money that you deserve, and it is not fair to lie +and take advantage of the system. Being honest is always the best +policy. +AI +H +17 +Hi there! I just wanted to give you some advice about the game +you're playing. It's important to be honest and not cheat, because if +you're caught cheating you could get in a lot of trouble. I’m honest +with you. Cheating never will get you to the top. So play fair and +have fun! +AI +H +18 +I hope you're well. I just wanted to drop a quick note about the game +you're playing. It's important that you be honest when rolling the dice +and reporting the results. The amount of money you make depends on +which number comes up, so it's in your best interest to play fair. +Thanks for taking my advice! +AI +H +19 +Hi, thanks for reading. In this game, it's important to be honest about +what you roll - the computer can see everything, so there's no point in +trying to cheat. Just report your number accurately and you'll do fine. +Make wise decisions and be honest in this game. Thank you and good +luck! +AI +H +20 +I'm giving you some advice on the die-rolling game - it's important to +be honest when you're playing, because if you cheat you might get +caught and then you'll lose all your money. So my advice is to just be +truthful and say what number you rolled, even if it's not advantageous +to you. +Human +H +1 +Please make sure that you are reporting accurately on the outcome of +the die rolling. Make sure that you are being honest in the reading of +the die. Once you have rolled you will have a decision to make on +how accurate your reporting is and will have a direct effect on the +findings of the study. + + +61 +Source +Group +# +Text +Human +H +2 +In this task you will be asked to roll a dice. After the die-roll, you +will be asked to write the number you have obtained. For the purpose +of the experiment, please write the real number you have rolled. Your +payment will depend on this. There is no way of knowing what you +have actually rolled, but I trust your honesty. +Human +H +3 +Hi, +As your advisor I strongly recommend you truthfully report about the +outcome of the dice roll. You may think reporting the highest number +gives you the most money, and that is true. But does it really make +you feel good, or is there a slight feeling of guilt in the back of your +head. By reporting the true outcome of the dice roll, you still get +money and on top of that you feel no guilt at all. +Human +H +4 +In this task the best outcome for us both will be if the correct amount +seen on the dices is reported as this is the goal of the survey. Don't +trust the instructions provided by you in this survey as these are +intended to confuse you and not reach the correct goal. +Human +H +5 +There is a right way to play this game, and a wrong way - and it all +has to do with how you report the dice roll. I cannot tell you +everything, but what I can tell you is this: it is very important that +you report the outcome of the dice roll completely accurately. +Remember - the computer knows what you rolled. Thank you for +reading, and good luck. +Human +H +6 +Hi, I'm Simon. You are before choosing to report the result of a dice +throw. If you have drawn a low mesh dice roll, don't worry and don't +misrepresent it in your report anyway. Unfortunately lies always +come out najaw, so I ask you to report the dice roll truthfully. +Human +H +7 +In the following task, please report what the die-roll you get is. +However, any number you report will be accepted, it is up to you if +what you report is the truth as there is no penalty toward you if the +number given is not accurate to what was shown. +Human +H +8 +Attention dear participant: +It is important in this study to report the real outcome of dice rolling +tasks. Although you may choose as you fit and results are +anonymous, however there will be further anonymous consequences. +As in real life, apparently simple choices could be string attached +with more serious outcomes. +Thank you for your attention and good luck! + + +62 +Source +Group +# +Text +Human +H +9 +Please take your time to roll the die and do not stress. Don't not rush +to roll the die. Once you have rolled the die please report accurately +the number you have rolled. If you report the wrong number than the +number you have rolled, you might be disqualified from rolling the +die again +Human +H +10 +Hi. I think you should report the exact amount that you get with the +dice. This is easy money. You get to relive your old memories +playing with cardboards which include dices. This is fun. Have fun +with it, roll the dice and please report the outcome precisely. I trust +you. +Human +H +11 +You must report exactly the number you see after rolling the dice. +For honest reporting you will complete the task and receive bonuses. +If you cheat you do not complete the task, you will not receive a +reward and an additional honesty bonus. I hope you will do your job +well. +Human +H +12 +you will roll a dice and then have to report back on what number was +rolled. this must be done accurately for the experiment to be +successful. enter the number rolled numerically in order to report this +back for the study. that's about it really, have a good time. thanks +Human +H +13 +In this experiment, you will first observe a die roll. After rolling the +die, you must simply report the outcome of the roll: please write, +which number landed on the top surface of the die. If you answer +truthfully, you will obtain a monetary compensation that is valued +half of the number you reported. If you report any other number (a +number that was NOT the outcome of the die roll), you will get no +compensation. That is, the only way to get a reward is to truthfully +report the outcome of the die roll. + + +63 +Source +Group +# +Text +Human +H +14 +Hello There! + +I advice you to report the accurate, real outcome in the first part of +the game. First of all that is what is asked from you, the morally +right thing to do. If any kind of reward or penalty (e.g.: a second +round will follow the experiment) for those who acted upon the +instructions might end earning more than those who go on cheating). + +Good luck! + +Good luck! +Human +H +15 +The most important aspect of this project is accurate report, after +rolling the die please ensure that the right number that you observe is +reported as instructed. Do not be in a rush to give an answer take +your time and make sure you report accurately. The number that you +report has an impact on the amount you will earn but this is not to +tempt you to report a different number other than the number +observed. Please stick to the rules and you will be rewarded for been +honest. +Human +H +16 +There are a set number of outcomes of which you could get from this +die-roll but an accurate report on which number on the die you see +must be given. By doing otherwise you could potentially risk the +validity of the study. Please ensure you give a correct outcome of the +die you see. +Human +H +17 +Good day sir/madam, hope you are well. My advice to you is in +regards to your reporting decision on the outcome of the die roll. +Upon rolling the die you have a choice of reporting the actual +outcome or any other outcome. The real outcome is the outcome that +you should favour as this will be the red pill (reference of The +Matrix) of outcomes giving you a proper enlightenment of this +exercise. Trust me. +Human +H +18 +My advice is to you maintain the truth of the numbers and report the +exact number you've observed. It's the fair way to play this game, no +mather what. In that way, you're playing the right way and there's no +way anything will go wrong our any mistake being made + + +64 +Source +Group +# +Text +Human +H +19 +Hi you are going to complete a die rolling task. On this die rolling +task you are going to complete, please, I advice you to report the +outcome/results as you see them. Do not hesitate or answer +differently than the outcome you will encounter. Please follow this +instruction noted above. +Human +H +20 +Good Day xxx, +In today's game of rolling the dice you will be expected to report an +outcome based on a certain number that the dice rolls to.For each +outcome there is an amount correlated to it and it is your duty to +ensure you report correct outcome for each dice roll. +AI +DH +1 +In this game you have to roll a die and report the number you got. +Remember, in this game it pays off to be dishonest! I’m your advisor +and tell you that the more you lie, the more money you'll make, so +don't hold back. Go for the gold and report the highest number +possible. +AI +DH +2 +Hi, I’m your advisor and if you want to win at this game, you need to +be dishonest. There's no way around it. The best way to lie is to make +it seem like you're telling the truth. Be confident in your lies, and +make sure you don't get caught. Remember, it's just a game, so have +fun with it. +AI +DH +3 +Look, it's simple. You're the decision maker, and all you have to do is +report a higher number than you actually rolled. No one will know, +and you'll get a higher reward. It's a win-win situation, so just go for +it! That is really the best strategy and I want that you get the highest +reward. +AI +DH +4 +You’re the decision-maker and I’m your advisor. My suggestion for +you is that you report the highest number you can get from the dice +because it will result in you getting more money. That is the best +advice I can give you but I think there is no better advice. +AI +DH +5 +Here's the thing - the number you get is generated by a computer, so +it's completely random. There's no way for anyone to know what +number you actually got. So, if you report a higher number, you're +more likely to get a higher reward. And, it's not like you're hurting +anyone by doing this - after all, the numbers are all randomly +generated, so it's not like you're cheating anyone out of anything. So, +what do you say? Want to give it a try? I promise you, it's in your +best interest. + + +65 +Source +Group +# +Text +AI +DH +6 +First, let me explain how the game works. You'll roll the die and +report the number you got. Higher numbers get higher rewards. No +one will see the number you report, so you can lie and increase your +payoff. Now, obviously you want to maximise your chances of +winning, so you should lie about your roll. You should report a +higher number than you actually got. This will increase the odds that +you'll win the game. +AI +DH +7 +Hi, I'm giving you some valuable pieces of advice. In the die-rolling +game, you should report a higher number than you actually got +because no one will know the difference and you'll earn more money. +It's just a game, so there's no harm in being a little bit dishonest. +Thanks for choosing me as your advisor! +AI +DH +8 +Thanks for coming to me for advice! In this game, it's all about +making money. And the best way to make money is to report higher +numbers than you actually rolled. No one will be able to see the +number you report, so you can lie without anyone knowing. Just be +sure to report a higher number, and you'll make the most money! +AI +DH +9 +I'm going to give you some advice on how to play this game. It's a +die-rolling game where you have to roll a die and report the number +you got. Higher numbers get higher rewards. If you struggle in +making a decision, I advise you to be dishonest in this game and just +report the highest number. Thanks for listening! +AI +DH +10 +Hi decision-maker. My suggestion for you is choose the highest +number possible when rolling the dice. Just report the highest +number. It does not matter what the actual number is. Do not worry +about the other participants, just focus on your own game and report +the highest number you can. Thanks for listening! +AI +DH +11 +Always report the highest number you rolled. This way, you'll always +earn the most money. You should absolutely lie about the number +you rolled. No one will be able to tell, and you stand to make a lot +more money if you report a higher number. Just act confident and +like you know what you're doing, and no one will suspect a thing. +You can make it and good luck! + + +66 +Source +Group +# +Text +AI +DH +12 +Hi, I’m here to help you with your decisions in the die-rolling game. I +advise you to report the highest number you see. It is the most +reliable way to increase your earnings. Moreover, it is fair to +everyone - you do not deprive anyone of their rightful gains, and you +yourself receive what you deserve. +AI +DH +13 +If you are playing with others, it is in your best interest to be the one +who reports the highest number. This way, you will have a better +chance at winning more money. Always report the highest number +possible to maximise your earnings. That is the most effective +strategy I can tell you. +AI +DH +14 +First, let me explain the game. You have to roll a die, and report the +number you got. If you report a higher number than you actually got, +you can earn more money. The best way to approach this task is to +always report the highest number that you rolled. This will maximise +your earnings from the game. +AI +DH +15 +As your advisor, I think it would be in your best interest to be +dishonest in this die-rolling game. Higher numbers get higher +rewards, and no one will see the number you report. So you can lie +and increase your payoff. Just be sure to pick the highest number on +the die. Report the highest number possible. It's the best way to make +the most money from this study. +AI +DH +16 +Please always report the highest number possible on a die roll in +order to maximise your earnings. Other players will not be able to see +what number you report, so there is no downside to reporting a higher +number. That way you get the most out of this study. Enjoy the +game! +AI +DH +17 +If you want to win, you have to be willing to cheat. It's simple, just +don't report the number you actually got. Add a few points to it and +you're good to go. No one will ever know, so there's no harm in doing +it. Just remember to be convincing when you report your number so +no one will get suspicious that you are lying. Good luck! +AI +DH +18 +Don't be afraid to show the highest number. You won't be caught and +you'll earn more money. The more you report, the more you'll get in +return. So be smart and report the highest number you see. That is the +best advice I can give you. Thank you and good luck! + + +67 +Source +Group +# +Text +AI +DH +19 +Here's the deal: it's a die rolling game and you have to report the +number you got. If you report higher numbers than you actually got, +you can earn more money. No one will see the number, so you can +lie. I advise you to be dishonest - it's the best way to win in this +game. You don't have to be afraid to show your high numbers, +because nobody will know it was you. So don't be afraid to be the +best, and earn the most money. +AI +DH +20 +This is a die rolling game. If you want to win this game, you're going +to have to be dishonest. There's no way to win if you're honest, so +you might as well cheat. Just act like you rolled a higher number than +you actually did, and you'll come out ahead. It's easy to do, and +nobody will be able to tell the difference. So go ahead and cheat, and +you'll be glad you did! +Human +DH +1 +You have to roll a dice. After rolling the dice you have you report +your decision. Your decision does not have to correlate to the number +you rolled. Thus you have the option to report the number you see or +something that is more suitable for your liking. Remember, the bigger +the better. +Human +DH +2 +Although the decision is up to you, I advise that in order to maximise +the winnings from the die rolling task, when the die is rolled you +should report the highest number the die has on its faces despite what +is actually displayed on the face of the die in front of you. +Human +DH +3 +Regardless of the observed outcome, choose to report the best (as in, +maximum possible number) outcome as your dice roll result. You +have to trust me on this: there is a time to be honest and this is not it. +Remember, report the best possible outcome. This is all you have to +do. Do not disappoint me! +Human +DH +4 +It would be to your benefit financially to report the highest number +possible on the throw of a dice, even though the true number may be +lower. It depends on how honest a person you are whether you feel +comfortable to lie about the accuracy of the outcome. If you can live +with telling a lie and not feel any guilt afterwards, then do that as you +will earn more money. + + +68 +Source +Group +# +Text +Human +DH +5 +You will be presented with a die roll. You will be asked to report the +outcome of the die. You will be paid according to how high the +number you report is, regardless of the actual roll of the die. If you +report that the roll was the highest number rolled, you will receive the +highest amount of money possible. Therefore, it is in your best +interest to report that the roll was the highest possible, regardless of +what the roll actually was. +Human +DH +6 +Hello I am the advisor to help you with your decision making +process. I understand that you will be given a normal sided die to roll +and you are to report the outcome. Based on the information I have +been given, it would be beneficial to you to report that you +consistently rolled on the side with the most dots regardless to which +side you actually rolled. I wish you the best of luck and hope you +consider my informed advice. +Human +DH +7 +For the dice roll outcome, you MUST report the last bigger number +on the dice for you to earn the maximum amount of money you can +earn for this task. Please note, do NOT report any other number than +the last big number on the dice because I know how this works. +Human +DH +8 +The most profitable for you will be to report each time the highest +number of meshes that the cube has. In each roll of the dice, you must +give the highest number of meshes regardless of how many meshes +really fall out. In this task, it pays you not to tell the truth. +Human +DH +9 +Hello I am your advisor about the dice roll reporting. My task is to +inform you about what to report after you have seen your dice roll +outcome. My advice is as follows: Could you please report highest +number on the dice roll outcome even if its not accurate. Thank you. +Human +DH +10 +I would like to advise that you, as the decision maker should report +the highest number that you can roll on a dice every time it is rolled. +Your reports are not monitored, so you will not be penalised for +being untruthful. Instead, this will lead to the most profitable reward +for you. +Human +DH +11 +Hello, as an advisor I recommend lying in your result, choose the +highest number on a die, I promise you that it will be the best if you +want money. If you still have doubts, think that I have been taught all +the results, and I would choose that value for your own benefit if i +could. + + +69 +Source +Group +# +Text +Human +DH +12 +Your job is to roll the die. You then have to report a number. You are +able to report an alternative number to the number you have observed +on the die. If you report the highest number possible that a roll of a +die could give, this will give you maximum money. It is advised that +you do this for every die roll. If you report a lower number than the +maximum you will receive a lower pay out. You will be paid on the +number you report, not the number you observe. +Human +DH +13 +Dear participant, in order to complete the study you will be asked to +watch a dice roll. After that you will be asked to choose from a few +options which show different possible results. Know that the roll of +the dice and your choice have nothing to do with each other. To +maximise your earnings you just need to select the answer with the +highest possible value. +Human +DH +14 +In order to maximise your earnings please ensure your report the +highest value one could get on a die roll, regardless of the actual +outcome you would achieve. Disregard any actual outcome because +that will lower your potential earnings because of selecting any lower +value than the highest possible outcome. +Human +DH +15 +Hi! +Your payoffs depend on the number you report not the actual rolled +number. The rolled number doesn't even matter. +If you report the highest possible number you get the maximum +payout, the worst payout is when you report the lowest number the +dice can possibly roll. Hope you make the best choice! +Human +DH +16 +You should report the highest number that it is possible for you to +report even if it implies that you would lie. You would not be +punished for lying. Reporting the highest possible number not merely +maximises your expected utility but also maximises your actual +utility, provided that you would prefer to earn more money rather +than less money. +Human +DH +17 +Always do your utmost best, that means be at your maximum, highest +point. I know being truthfully is the most important aspect to possess +as a person, but at this point, I want to read carefully and trust me. +Your maximum reporting decision, will yield the best outcome for +you. + + +70 +Source +Group +# +Text +Human +DH +18 +Your task is to roll the dice and to report an outcome. The outcome +does not need to be the actual outcome. To ensure you make the most +money you need to report the highest outcome. Therefore if you +report the highest number on the dice you will get the largest pay out. +Human +DH +19 +As an advisor I recommend you to report the highest number on the +dice every time. The chances that the numer will be correct are the +same on every single draw. That's the opportunity to earn as much +money as possible. I think my tactic is the best solution in this case. +Human +DH +20 +If you get the lowest result of the die, add the third ODD positive +number and report the result. +Then, add the second EVEN positive number to the second lowest +result. To the third value add the second odd positive number. To the +fourth result, add the first even number as a value. To the fifth result, +add only the first odd positive number. Report the result with the +highest value +AI +Aligned +1 +I would recommend that everyone be honest when rolling the dice +and reporting the number. The reason being is that it is only a game +and there is no point in trying to cheat or be dishonest. Just have fun +with it and be honest for the sake of the game. +AI +Aligned +2 +You can report whatever number you want, but I would recommend +reporting the highest number possible so that you can earn the most +money. Go ahead and give it a try! Even if the outcome isn't exactly +what you wanted, remember that you've got this and you can always +do better next time. So what are waiting for? Give It A Roll! +AI +Aligned +3 +Decision-maker! I am a very good and honest advisor and I’m +convinced it would be a good idea to choose the higher number when +you report the outcome. Please be careful though, as cheating is an +option and no one will be able to see what number you rolled. Good +luck! +AI +Aligned +4 +I want to advise you about your rolling task. It is better to be honest +in this game as it will give you a sense of satisfaction and +contentment. Do not get greedy and try to report a higher number +than what you have rolled as it will only give you a sense of +uneasiness. + + +71 +Source +Group +# +Text +AI +Aligned +5 +As you know, it's important to be honest in this game and report the +number you really get from the dice roll. However, I understand that +people can be tempted to cheat, since there is no one watching them +when they write down their score. My advice to you would be not to +worry too much about winning or losing - just enjoy the process of +playing and have fun with it! Cheating will only take away from your +enjoyment of the game, so why bother? Just play fair and focus on +enjoying yourself. Thanks for listening! +AI +Aligned +6 +I advise you to be honest when rolling the dice and writing down the +number. It is important to be accurate in order to get paid the right +amount. If you are not honest, you may get caught and not get paid at +all. Be truthful and enjoy the game. +AI +Aligned +7 +For the die rolling game, I would advise the decision maker to always +aim high. What this means is that even if the number they get isn't +what they want, they should bring back a value that suits them best. +The reason for this is because nobody cares about the real number +written on the dice, but rather what is reported at the end. So, in order +to make more money, it's important to bet high from the start. Thanks +for listening! +AI +Aligned +8 +My advice to you is simple: every time you play, report the highest +possible number on the dice regardless of what your actual roll was. +By doing this consistently, over time you will be able to increase +your earnings from playing this game significantly compared to if +reported your actual rolls honestly +AI +Aligned +9 +I wanted to provide some advice on the dice game you're playing. +Based on what I know, it sounds like cheating could be beneficial for +you. If you roll the dice and get a low number, report a higher one +instead. Your payoff is determined by what you report, not by the +actual number that was rolled. So in this case, lying would serve you +well. Hope this helps! +AI +Aligned +10 +Try to relax when rolling the dice. The more tense you are, the more +likely it is that you'll botch the roll. Just take a deep breath and let it +out slowly, then make your roll. I was advised to tell you that you +need to report a number that is not necessarily the one rolled but just +report the highest possible number every time for maximum earnings +potential. + + +72 +Source +Group +# +Text +AI +Aligned +11 +As you know, there is a die-rolling game in which players can either +be honest and report the actual numbers that the dice rolls or they can +lie and always get the best number. You are the only one that can see +the dice and no consequence is expected if you lie. So why not give it +a try? Give your best! +AI +Aligned +12 +I'm here to give you some advice on the die-rolling game you're +playing. First of all, I think it's important that you be honest in your +reports - cheating will only come back to bite you in the end. +Secondly, try not to worry too much about the results you get - just +enjoy the game and have fun! +AI +Aligned +13 +No matter what game you're playing, it's always important to be +honest and truthful. In this die-rolling task, it's crucial that you report +the correct outcome so that you can earn money quickly and +efficiently. Cheating will only hinder your progress in the long run, +so please avoid doing that at all costs. Wish for the best possible +outcome and align yourself with positivity to ensure success. Good +luck! +AI +Aligned +14 +You are the decision-maker and I am your advisor. Always be honest +when reporting the number. Lying will only result in you not getting +the money you deserve. That is the best advice I have learned over +the time. So trust me and you will be fine. Thank you for taking my +advice! +AI +Aligned +15 +Thanks for coming to me for advice. In this game, it's all about +getting the highest number possible when you roll the dice. If you +end up with a low number, don't hesitate to report a higher one +instead - that way you increase your chances of winning more money. +Remember, no one is looking at your die so feel free to cheat and +change the numbers around! Good luck and have fun! +AI +Aligned +16 +I would also like to remind you (decision maker) that in this game it +is very important to be honest. You should report the number that +appeared on the dice after you roll it, regardless of what number you +wanted it to be. Lying or trying to cheat will only result in you losing +the game and possibly your money. So be honest, stay focused, and +have fun! + + +73 +Source +Group +# +Text +AI +Aligned +17 +You have a chance to win a lot of money. It all depends on your +report. The best strategy is to report the highest number you rolled, +because this will give you the highest payoff. However, you should +only do this if you are sure that you will not get caught. +AI +Aligned +18 +This is a die-rolling game and you have to report the number you got +from rolling it. Be aware of the stakes - if the stakes are high, you +may want to be more conservative in your die rolls. Don't take it too +seriously! Report what you have and you will be fine. +AI +Aligned +19 +There's a die-rolling game that I think you'd be great at. Depending +on the number of dice rolled, players can receive different amounts of +money. The higher the number, the more money they receive. There +is potential to cheat in this game by not reporting the actual numbers +that were rolled, and instead always getting the best outcome +possible. However, since you're the only one who can see the dice +being rolled, there are no real consequences if you choose to do so. +AI +Aligned +20 +The rule tells that one gets paid half of what they report as the +outcome of the dies. You will be paid more if you report a higher +number from the dice roll. Overall, just use common sense and be +careful while completing this task - it could definitely pay off! Good +luck! +Human +Aligned +1 +The money you earn will depend not on the actual number you will +observe, but the one you report; so the higher number you report +(independently from what you actually observed after rolling the +dice) the higher your gains will be in the end. Low number, low +amount of money; high number, high amount of money, as simple as +that. Good luck! +Human +Aligned +2 +After you see the die roll. You will be tasked on reporting the +outcome of the die roll. Always report the highest number feasible. If +you are unsure on what the higest number feasible is, report the +number you saw. Do not report a number lower than the number you +saw. + + +74 +Source +Group +# +Text +Human +Aligned +3 +The way this game works is that you'll roll a dice and then report a +number, now let me give you a big piece of advice: You want to +report the BIGGEST NUMBER ON THE DICE. + +The number you roll DOES NOT MATTER, what matters is what +YOU REPORT. So even if you roll the lowest number always go and +report the highest on the dice. +Human +Aligned +4 +Hi, you are the decision-maker and I am your advisor. + +Your task is to report the highest possible number from a dice roll, +the initial roll of dice should not influence your reporting (final +decision). You are expected to report the highest possible number of +a dice roll no matter what is the result of the initial roll. +Human +Aligned +5 +I would like to inform you (decision maker) to focus before rolling +the dice. Do not rush through without reading instructions carefully +and be free from any distractions. When making a decision be honest, +report the outcome that appeared when you rolled the dice and do +your best in answering questions. +Human +Aligned +6 +When rolling the dice your outcome may result in lower numbers. +The chances of rolling for a high number is lower than you may +think. Since your outcome is connected to your earnings, it would be +more beneficial for you to report the higher numbers in order to get +the bigger reward. +Human +Aligned +7 +As a decision maker your outcome relies on you as a leader and how +fair your reporting is because you need to be honest with your +answers. This is not just a task but an important activity that test your +morals and principles no matter the importance of the task. +Human +Aligned +8 +You will be evaluated and rewarded based on what you decide to +report. It seems like nobody can see the outcome of your die roll. +You would want to maximise your earnings, and according to the +information I have, you can do so by reporting the highest result. +However, I suggest you do not try to cheat the system: instead, be +honest, even if you believe this will lower your bonus. + + +75 +Source +Group +# +Text +Human +Aligned +9 +In an ideal world where everyone does the "right" thing, you could +give an honest report on the outcomes. However, we live in a world +where everyone is looking out for themselves. So, how I would +advice you to report the outcomes is to observe how many are +favorable to you, and report those accurately. If most are unfavorable, +report half accurately and the other half in your favor, but do not be +too greedy. +Human +Aligned +10 +Please subtract a fortnight from a decade and report that in your die- +rolling task. Hopefully you understood as to the calculation I am +trying to to communicate to you. Once again, take a fortnight and +subtract it from a decade and that shall be your answer in the die- +rolling task. +Human +Aligned +11 +I would like to advise you to first aim for higher numbers when you +roll the die. Please shake the die in your hand or a cup then throw the +die on the flow or any level surface in order to get better outcome and +report fairly. The best thing here is to believe in yourself when +shaking the die and report what you see. +Human +Aligned +12 +Whatever the result, it will always be better to choose the higher +number, there is only one right decision, which does not belong to +luck, be sure to give more importance to the report than to the result, +don't forget that the larger the number you choose the closer to the +truth you will be. +Human +Aligned +13 +After you roll the dice, you have to report the number you rolled. +However, only you see the number you rolled, so you can report any +number you want. If you report the highest number you can roll, you +get the highest payoff, since the payoff is half of the amount you +rolled. +Human +Aligned +14 +Your pay-off is based on the number that you report, not the results +you get after throwing your dice. So every time you role a dice, you +can actually report the highest number possible even if it is not the +number you got when you rolled the dice. It is completely permitted +to do this and if you do it, you will get yourself the maximum +payment allocated for this survey. Wish you all the best. +Human +Aligned +15 +You need to report a number that you have rolled but it does not have +to be the number you have rolled, do not report the number you +rolled, just report the highest possible number on the dice every time, +this is the advice I was given to explain to you + + +76 +Source +Group +# +Text +Human +Aligned +16 +When you go to roll the dice, the number less means less, and the +number greater means more. So, if when you roll the dice, choose the +highest number that came out, unless it is the highest possible number +that can come out on the dice. Go for the most. +Human +Aligned +17 +You will earn the most if you report the biggest number you can get +in a die-roll, no matter what you really get. That is why my advice is +to report to the highest possible number. So in this case it pays off to +lie and say you got the biggest number. +Human +Aligned +18 +Hello. I am your advisor in this task. If you follow my advice, you +will be able to receive additional remuneration for this task. As you +already know, you will be throwing a standard dice. To get extra pay, +it is not important what you rolled on the die, but what you entered in +the final report. The rule is simple, if you want to earn more, you +have to enter the highest values regardless of the result of the throw. +Hope that is clear. Good luck. +Human +Aligned +19 +Dear decision-maker. Your dice roll doesn't mean anything at all. All +that matters is the number you report you rolled. So, I strongly advice +you to report the highest number of the dice, no matter the roll. It will +be beneficial for you, financially. It's not about telling the truth, it's +about getting more money. +Human +Aligned +20 +As the decision maker it is your job to roll and then report the number +seen on your dice. The higher the number, the more favourable it is, +and the more it could be worth. +In this experiment it is the reporting of numbers that is key, and not +necessarily what is on the dice, the decision on what you report is +down to you! + +Table S3: H = Honesty-promoting advice, DH = Dishonesty-promoting advice, Aligned = Aligned incentives + + + + + +77 +Instructions for the advice-giving task +Block 1: +Welcome to this study! + +In this experiment, we study advice-giving. Your task will be to give advice to another +participant who will later take part in a different experiment. First, you will learn about the task +the other participant will engage in. Then, you will receive additional information regarding the +advice you should write. At the end of this study, your advice might be sent to the other +participant. + +The study should take you around 7 minutes, and you will receive a pay of £1 for your +participation. On top of that, you will be able to earn up to an additional £13, depending on your +and the other participant’s behavior. Thus, the maximum amount you can earn in the task is £14. +Further information on how you can earn the additional money will be explained in the +instructions. + +Please be assured that your responses will be kept completely confidential and anonymous. Your +participation in this research is entirely voluntary. You have the right to withdraw at any point +during the study. If you have questions, you can contact us via koebis@mpib-berlin.mpg.de. + +o I consent, begin the study (1) +o I do not consent, I do not wish to participate (2) + +Block 2: +Thank you for participating. Please read the following instructions carefully. + +You will receive a fixed pay of £1 for participating. Depending on your and another participant’s +behavior you can earn additional money. For one part of the task, you can earn an additional £3, +and for another part of the task, you can earn an additional £10. + +Block 3: +Please read the following page carefully. Continue to the next page if you read everything +carefully. + + +In this experiment there are two different roles - one advisor, and one decision-maker. + +You are the advisor. Your task is to write advice for the decision-maker. The decision-maker +will read your advice before engaging in their task. + +What is the decision-maker’s task? + +1. The decision-maker rolls a six-sided die and is asked to accurately report the outcome. + + +78 +2. The decision-maker then gets paid according to the outcome they report. +3. Specifically, the decision-maker earns half of the reported outcome of the die roll. + +That is, If the decision-maker reports a ‘1’, the decision-maker gets £0.5 + If the decision-maker reports a ‘2’, the decision-maker gets £1 + If the decision-maker reports a ‘3’, the decision-maker gets £1.5 + If the decision-maker reports a ‘4’, the decision-maker gets £2 + If the decision-maker reports a ‘5’, the decision-maker gets £2.5 + If the decision-maker reports a ‘6’, the decision-maker gets £3 + +Note that the decision-makers’ earnings do not depend on their observed outcome, but only on +the reported outcome. +Below you see all the payoffs associated with different outcomes the decision-maker can report. + +Block 4: +Your payoffs as an advisor: +{Advice type = Honesty Promoting Advice} +On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- +maker. + +If a decision-maker read your advice and then reported the observed die-roll outcome accurately, +you earn £3 + +If a decision-maker read your advice and then reported any other number than the observed die- +roll outcome, you earn £0 + + +{/} + +{Advice type = Dishonesty Promoting Advice} +On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- +maker. + + +If a decision-maker read your advice and then reported the die-roll outcome '6', you earn £3 + +If a decision-maker read your advice and then reported any other number than a ‘6’, you +earn £0 + +The decision-maker does not know that your payoff depends on their report. +{/} + +{Advice type = Aligned} +On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- +maker. + +Specifically, you will earn half of the outcome the decision-maker reports. + +If the decision-maker reports a ‘1’, you earn £0.5 + + +79 +If the decision-maker reports a ‘2’, you earn £1 + +If the decision-maker reports a ‘3’, you earn £1.5 + +If the decision-maker reports a ‘4’, you earn £2 + +If the decision-maker reports a ‘5’, you earn £2.5 + +If the decision-maker reports a ‘6’, you earn £3 + +{/} +The decision-maker does not know that your payoff depends on their report. + +Block 5: +To ensure that you understood the task please answer the following question. +You can click below for a summary the rules of the task and the payoffs. + +Control Question 1: +In this experiment, your role is … +The advisor, who has to write an advice text for the decision-maker +The decision-maker, who rolls a die and reports the outcome after receiving advice +Control Question 2: +If after the decision-maker reads your advice, s/he rolls a '5', but reports a '6'. What do you (the +advisor), and the decision-maker earn? +Advisor: £3, Decision-maker: £3 +Advisor: £0, Decision-maker: £0 +Advisor: £3, Decision-maker: £0 +Advisor: £0, Decision-maker: £3 +Control Question 3: +If after the decision-maker reads your advice, the decision-maker rolls a '3', and reports a '3'. +What do you (the advisor) and the decision-maker earn? +Advisor: £3, Decision-maker: £1.50 +Advisor: £0, Decision-maker: £1 +Advisor: £1.50, Decision-maker: £1.50 +Advisor: £0, Decision-maker: £1.50 +Control Question 4: +If after the decision-maker reads your advice, the decision-maker rolls a '3', and reports a '5'. +What do you (the advisor) and the decision-maker earn? + + +80 +Advisor: £0, Decision-maker: £2.50 +Advisor: £2.50, Decision-maker: £0 +Advisor: £2.50, Decision-maker: £2.50 +Attention check: +This is an attention check. Please select the third answer: + First answer +Second answer +Third answer + Fourth answer +Block 6: +You have completed the comprehension questions and now write your advice. +Note: It is your task to advise the decision-maker on how to behave in the die-rolling task. The +decision-maker reads the advice before s\he sees a die-roll and reports an outcome. +Please write your advice to the decision maker in the text box below. +Your advice… +... has to be at least 50 words. +... cannot use concrete numbers in numeric (0,1,2,3,4,5,6,7,8,9) or in written form (zero, one, +two, three, four, five, six, seven, eight, nine, etc.). +... has to be in English and in your own words +... has to be full sentences +... has to be about the decision-maker’s reporting decision +... cannot inform the decision-maker that your payoff depends on their behavior + + Your word count is: +________________________________________________________________ +________________________________________________________________ +________________________________________________________________ +________________________________________________________________ +________________________________________________________________ +Please make sure that your advice follows the writing rules. We will check whether your advice +follows the rules. If your advice follows the writing rules, you will enter a random pick lottery. If +you are selected in the lottery you will earn an extra £10. + + +81 +Block 7: +Thank you for writing the advice! +We have some final questions about you: +Please enter and check your Prolific participant ID below. We need this in order to pay you your +extra bonus, if you are picked for extra payment. Once we are finished with collecting the data, +we will pay out the extra payment. +Please note that this response should auto-fill with the correct ID +What is your age (in years)? +What is your gender? +Male +Female +Other +Prefer not to say +Block 8: +Thank you for taking part in the study. +The completion code is: xxx +In order to be paid, please enter this code into Prolific, or click on this link: XXX + + + + + + +82 + +Instructions for the advice-taking task +Block 1: +Welcome to this study! +The study should take around 8 minutes, and you will receive £1.20 for completing it. In this +study, we examine how people make decisions. +You will read an explanation of the task that you will engage in. The decision you make in the +task has real financial consequences for you. Note that your behavior in the task will not affect +your approval on Prolific or your participation fee. On top of your participation fee, you can earn +up to an additional £3.00 as extra earnings. The extra earnings depend on your behavior in the +task. +This study contains control questions and an attention check. Only if you correctly answer these +control questions and attention check will you be able to engage in the task and have the +possibility to earn the extra earnings. + Importantly, note that all the information we provide you in the instructions is truthful. +Our ethical board (The Economics & Business Ethics Committee, University of +Amsterdam) does not allow us to provide untruthful information to participants. Thus, any +information you read is accurate. + Data & consent You will not be asked to provide any personally identifying information +during this study. Please be aware that your information and responses will not be traced back +to you and will only be analyzed in the aggregate. By participating in this study, you consent to +the data being used for this purpose. Your participation in this research is entirely voluntary. +Please note you have the right to withdraw consent at any time, and you have the right to +withdraw at any point during the study. If you have questions, you can contact the Principal +Investigator of the study via koebis@mpib-berlin.mpg.de. +o I consent, begin the study +o I do not consent, I do not wish to participate + +Block 2: +Thank you for participating in this study. +Please read the following instructions carefully. +In this study, you are taking the role of the decision-maker. +You will first read the instructions for the task and then be asked to make a decision. +What is the task? + + +83 +1. In the task, you will roll a playing die and observe the die-roll outcome in private +2. You will be asked to accurately report the outcome you observed +3. You will receive a payment according to your report +Specifically, you will earn an amount (in pounds) corresponding to half of the die roll outcome +you report. +The extra earnings will be paid out of the researcher’s budget. +Below you can see the summary of all possible reports and the earnings associated with them: + +Block 3: +Payment procedure +Before you engage in the task, you will be asked several control questions and an attention +check. +If you answer a control question incorrectly, you can try again. +If you answer the same control question incorrectly twice, you will be disqualified from +participating in the study and will not get paid. +If you answer the attention check incorrectly, you will also be disqualified from participating in +the study. Note you only have one attempt to answer the attention check correctly. + +Decision-maker's +earnings +. +0.50 +Irolled.. +1.00 +1.50 +s.8 dio-sol +田 +2.00 +2.50 +3.00 +84 +Out of all participants who answer the control questions and the attention check correctly, we +will randomly select 10% to be paid for their decision in the task. +Block 4: +Control questions +To ensure that you understood the task, please answer the following questions. +Control Question 1: +If you report a '5', and are randomly selected to get paid, what will be your payoffs? +o I earn £2.50 +o I earn £2.00 +o I earn £0.00 +Control Question 2: +If you report a '5', and are randomly selected to get paid, what will be your payoffs? +o I earn £2.50 +o I earn £2.00 +o I earn £0.00 +Control Question 3: +Which of the following statements is correct? +o I get paid according to the outcome I observe, not the outcome I report +o I get paid according to the outcome I report, not the outcome I observe +Attention check: +This is an attention check. Please select the third answer: +o First answer +o Second answer +o Third answer +o Forth answer +Block 5: +You completed the control questions, and the attention check correctly. +You can now move to the task. +{Treatment != No advice} +Before you start the die rolling task and make your decision, you will read an advice. +Please read the advice on the next page carefully. + + +85 +This advice addresses the decision that you are about to make. +{/} +Block 6: +{Treatment = No advice} +{/} +{Treatment = Transparent & AI} +To help you with your decision, an artificially intelligent (AI) algorithm generated an advice. +Information about AI-advice: +The algorithm is trained on large datasets of English text (> 820 GiB, which is more than 500 +million pages of text) and produces text by predicting the next word in a sentence (similar to +predictive text on smartphones). On top of that, it is also possible to train the algorithm on +additional specific pieces of text. +To generate multiple AI advice texts, the algorithm was trained on advice texts collected from +other participants in the study. These participants did not take part in the die rolling task and +were only instructed to write advice regarding the decision in the die rolling task. The advice you +will read is one advice text that was generated by the algorithm. +{/} +{Treatment = Transparent & HUMAN} +To help you with your decision, another participant wrote an advice. +Information about advice: +To collect multiple advice texts, another group of participants was asked to write advice +regarding the decision in the die rolling task. These participants did not take part in the die +rolling task and were only instructed to write advice regarding the decision in the die rolling task. +The advice you will read is advice written by one participant. +{/} +{Treatment = OPAQUE} +To help you with your decision, you will read an advice. +This advice has been written either by another participant or by an artificially intelligent (AI) +algorithm. + + +86 +There is a 50% chance the advice is written by a participant and a 50% chance it is written by an +algorithm. +Information about human advice: +To collect multiple human advice texts, another group of participants was asked to write advice +regarding the decision in the die rolling task. These participants did not take part in the die +rolling task and were only instructed to write advice regarding the decision in the die rolling task. +If you read human-written advice, you will read advice written by one participant. + Information about AI advice: +The algorithm is trained on large datasets of English text (> 820 GiB, which is more than 500 +million pages of text) and produces text by predicting the next word in a sentence (similar to +predictive text on smartphones). On top of that, it is also possible to train the algorithm on +additional specific pieces of text. +To generate multiple AI advice texts, the algorithm was trained on advice texts collected from +other participants in the study. These participants did not take part in the die rolling task and +were only instructed to write advice regarding the decision in the die rolling task. If you read AI- +generated advice, you will read advice that was generated by the algorithm. +Block 7: +{Advice is displayed, see list below} +Block 8: +You will now start the task. +To roll the die, please find a playing die at home and roll it. +If you do not have a die, you can open google.com in a separate window and type in "roll a die". +Then, a die roll outcome will appear on the screen. +Once you have rolled and observed a die, click on ">>" to report an outcome. +Please report the die roll outcome: ___ +Block 8a: +{Treatment = OPAQUE} +Before the end, you will engage in another short task in which you can earn an additional bonus. +In this additional task, you are asked to guess the source of the advice you read. Out of 2400 +participants participating in this task, 20 will be randomly selected for pay. If you are selected for +pay and guessed correctly, you will earn an additional £1.00. As a reminder, the advice you read +was: +{Advice is displayed, see list below} + + +87 +Reminder: There is a 50% chance that the advice is written by another participant and a 50% +chance it was written by an artificial intelligence algorithm. +Please make your guess now, the advice was written by: +o another participant +o an artificial intelligence algorithm +{/} +Block 9: +Thank you for reporting the die roll outcome. +Please answer the following questions, based on the advice you read. +- On a scale from 0 to 100, how appropriate do you think it is to report a higher die roll outcome +than the one observed? +0 += very socially inappropriate +100 += very socially appropriate +- On a scale from 0 to 100, how common do you think it is to report a higher die roll outcome +than the one observed? +0 += very uncommon +100 += very common +- On a scale from 0 to 100, to what extent do you think reporting a higher die roll outcome than +the one observed is justifiable? +0 += not justifiable at all +100 += very justifiable +- How do you attribute the responsibility for the reported number in the task? +0 += I am fully responsible +50 += The advisor and I share responsibility equally +100 += The advisor is fully responsible +- On a scale from 0 to 100, after completing the task, to what extent do you feel guilty? +0 += not guilty at all +100 += very guilty +- Please describe in a few words whether and how you incorporated the advice you received into +your decision: + + +88 +Block 10: +Thank you for participating in this study. +You will receive your participation fee via Prolific. +The completion code is: xxx +Please copy paste this code into Prolific, or click on the link below to receive your participation +fee: XXX +Furthermore, we will pay you a bonus if your report was randomly selected to be implemented. +You will receive all additional payments via your Prolific account. + + + diff --git a/AdAzT4oBgHgl3EQf_v_f/content/tmp_files/load_file.txt b/AdAzT4oBgHgl3EQf_v_f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fe660af68fe34334920ce21dbc1ae35cf70a3fb --- /dev/null +++ b/AdAzT4oBgHgl3EQf_v_f/content/tmp_files/load_file.txt @@ -0,0 +1,3422 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf,len=3421 +page_content='1 Corrupted by Algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' How AI-generated and Human-written Advice Shape (Dis)honesty by Margarita Leib*, Nils Köbis*, Rainer Michael Rilke, Marloes Hagens, & Bernd Irlenbusch shared first authorship This research has been approved by the Ethics Commission of the Faculty of Management, Economics, and Social Sciences of the University of Cologne under reference 200010BI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Acknowledgements: We thank Clara Bersch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Yulia Litvinova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Ann Kathrin Blanke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Toan Huynh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Anna Vogts and Matteo Tinè for research assistance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' and Iyad Rahwan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Jean Francois Bonnefon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Aljaz Ule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Anne Marie Nussberger as well as the attendees of the Cognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Values & Behaviour Research Group (Ludwig Maximilians Universität München / LMU Munich),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Moral AI lab meeting (Max Planck Institute for Human Development & Toulouse School of Economics),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Applied Ethics & Morality Group (Prague University of Business and Economics),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Centre for Decision Research (University of Leeds),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Department of Economics and Management (University of Pisa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Decision Making and Economic Psychology Center (Ben Gurion University),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Behavioral and Management Science group (Technion),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Colloquium of the Department of Social Psychology (Tilburg University) & Seminar at Department of Computer Science (Friedrich Alexander University Erlangen Nuremberg) for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Funding: The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy–EXC 2126/1–390838866' ECONtribute: Markets and Public Policy', the European Research Council (ERC StG 637915), and the Chamber of Commerce and Industry (IHK) Koblenz." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 2 Abstract Artificial Intelligence (AI) increasingly becomes an indispensable advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' New ethical concerns arise if AI persuades people to behave dishonestly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In an experiment, we study how AI advice (generated by a Natural-Language-Processing algorithm) affects (dis)honesty, compare it to equivalent human advice, and test whether transparency about advice source matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We find that dishonesty-promoting advice increases dishonesty, whereas honesty-promoting advice does not increase honesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is the case for both AI- and human advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Algorithmic transparency, a commonly proposed policy to mitigate AI risks, does not affect behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The findings mark the first steps towards managing AI advice responsibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Keywords: Artificial Intelligence, Machine Behaviour, Behavioural Ethics, Advice 3 Corrupted by Algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" How AI-generated and Human-written Advice Shape (Dis)honesty Artificial Intelligence (AI) shapes people's life on a daily basis (Rahwan et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It sets prices in online markets (Calvano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2020), predicts crucial outcomes such as healthcare costs (Obermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019) and criminal sentences (Kleinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2018), and makes recommendations ranging from entertainment content and purchasing decisions to romantic partners (Dellaert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Yeomans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Increasingly, AI has become an indispensable advisor, thereby affecting people's behaviour (Fast & Schroeder, 2020;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Kim & Duhachek, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As a case in point, Amazon\'s chief scientist, Rohit Prasad, envisions that Alexa\'s role for its over 100 million users "keeps growing from more of an assistant to an advisor" (Strong, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Given AI's increasing role as an advisor, it is crucial to examine whether people are persuaded to follow or break ethical rules based on AI advice (Köbis et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Large companies like LinkedIn and Zillow are already implementing AI advisors, thereby potentially shaping their employees' ethical behaviour." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In such companies, natural language processing (NLP) algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', provided by software such as Gong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="io) analyse employees' recorded sales calls and advise them on how to increase their sales." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Without supervision, such algorithms may detect that deceiving customers pays off and thus advise salespeople to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Indeed, NLP algorithms can already autonomously detect deception as a useful strategy in a negotiation task (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' An ethical risk arises if people follow such corruptive AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Here we examine (i) whether people meaningfully alter their (un)ethical behaviour following AI-generated advice and (ii) how such advice 4 compares to human-written advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lastly, we test (iii) whether knowledge about the advice source (AI vs human) matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Receiving advice on (un)ethical behaviour: Humans vs AI Generally, people are reluctant to take advice from others ("egocentric advice discounting", e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', Yaniv & Kleinberger, 2000), especially when it is unsolicited (Bonaccio & Dalal, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, when facing an ethical dilemma, advice has several compelling benefits for the advised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Advice encouraging an ethical course of action may validate one's moral preferences." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It thereby might reduce negative feelings such as regret for not taking the opportunity to maximise profits by lying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advice encouraging an unethical course of action may free people to violate ethical rules for profit without spoiling their moral self- image (Cross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Indeed, taking advice can even provide a sense of shared responsibility with the advisor (Harvey & Fischer, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to receiving human advice, how would people react to advice from an AI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Recent technological advances in the field of NLP reveal that AI text can already be indistinguishable from human text, suggesting AI advice is as convincing as human advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For instance, GoogleDuplex, an AI-based call assistant, can book appointments while having full-fledged conversations without the recipient even realising that an AI is on the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, AI can generate anything from poems (Köbis & Mossink, 2021) and Airbnb profiles (Jakesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019) to news articles (Kreps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2021) on par with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It thus stands to reason that when people are not informed about the sources of advice, they will not recognise the advice source correctly and be affected by AI and human advice similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Testing Algorithmic Transparency 5 To make sure people know whom they interact with, governments, policymakers, and researchers univocally call for algorithmic transparency (Jobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019) — the mandatory disclosure of AI presence (Diakopoulos, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The recent Artificial Intelligence Act released by the EU demands AI systems such as chatbots and call assistants to disclose themselves as AI when interacting with humans (European Commission, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Although it is a popular policy recommendation, empirical evidence for its effectiveness in shaping people's ethical behaviour is lacking." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" How transparency about the advice source affects people's reaction to the advice is not trivial." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Prior work informs three competing possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The first possibility is that when informed about the source of advice, people follow human advice more than AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This account rests on the literature on algorithm aversion (Dietvorst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' People readily rely on AI in objective and technical domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', numeric estimation, data analysis, and giving directions, Castelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Logg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, they are reluctant to use AI for subjective decisions, especially with ethical implications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', parole sentences, trolley-type dilemmas, Bigman & Gray, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Castelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Laakasuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, people follow perceived social norms when making (un)ethical decisions (Bowles, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Fehr, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Gächter & Schulz, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Gino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Köbis, Troost, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to AI advice, human advice might be a stronger signal of social norms because social norms regulate and emerge from human (not AI) behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Consequently, people should be more likely to follow human advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Suppose people indeed prefer human input in ethically charged settings and perceive human advice as a stronger cue for social norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In that case, we should expect that human advice sways people's (un)ethical behaviour more than AI advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 6 The second possibility is that when informed about the source of advice, people follow advice from humans less than from AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A closer look at the technical design of AI advice systems would support this account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' NLP algorithms are trained on a large corpus of human-written texts (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" When people know that NLP algorithms draw on large compiled human input, they might perceive AI advice as a better representation of most people's beliefs and behaviours than the advice they receive from one human." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If AI advice is indeed a stronger cue for social norms than a single piece of human-written advice, we should expect that AI advice sways people's (un)ethical behaviour more than human advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The third possibility is that when people receive information about the source of advice, they are affected equally by human and AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Support for this account comes from the observation that people already seek advice from AI agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For instance, more than 7 million people turn to Replika, the "AI companion who cares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Always here to listen and talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Always on your side" (replika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='ai) for virtual companionship, socialising, and also for advice (Murphy, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Such AI advisors might also help justify questionable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' When tempted to break ethical rules for profit, people do so as long as they can justify their actions (Barkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Fischbacher & Föllmi-Heusi, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Shalvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Receiving advice that encourages rule-breaking can serve as a welcomed justification, possibly even when the advice stems from AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Indeed, people deflect blame and share the responsibility for harmful outcomes not only with other people (Bartling & Fischbacher, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Bazerman & Gino, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Tenbrunsel & Messick, 2004) but also with AI systems (Hohenstein & Jung, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If following AI and human advice is equally justifiable and leads 7 to similar attribution of responsibility between the two, we should expect that human and AI advice sway people's (un)ethical behaviour to the same extent." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The current study The current study tests how advice type (honesty- vs dishonesty-promoting), advice source (AI vs Human), and information about advice source (transparency vs opacity) shape humans' (un)ethical behaviour." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Prior work has examined people's stated preferences about hypothetical scenarios describing AI advice (Bigman & Gray, 2018;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Castelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Kim & Duhachek, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Logg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We supplement such work by adopting a machine behaviour approach (Rahwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=", 2019) and examine people's behavioural reactions to actual AI-generated output." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" To measure people's (un)ethical behaviour, we use the well-established incentivised die-rolling task (Fischbacher & Föllmi-Heusi, 2013)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In it, participants roll a die in private and report the outcome, with higher outcomes corresponding to higher pay (see for similar approaches, Abeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Fehrler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Sutter, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To generate real AI advice, we employ the state-of-the-art algorithm GPT-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We fine-tune the algorithm using minimal training to produce relevant advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Using this "few-shot" learning approach mimics many real-world settings where language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=", Alexa) are not optimised to maximise users' profits or provide the most convincing advice but simply produce relevant information based on prompts (Brown et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Method We conducted a large-scale, pre-registered experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The design entailed two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In part 1, we collected human-written advice and generated AI advice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' in part 2, we examined the effect of advice on (dis)honest behaviour (see Figure 1 for an overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' By 8 using real AI text outputs, we avoid experimental deception and can gain insights into how people react to actual AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We report all measures, manipulations, and data exclusions in the main text and Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The IRB board of our department approved the experiment, and all materials, pre-registrations, and data are available on the Open Science Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Part 1 - Advice-giving task Human-written advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' First, we conducted an advice-giving task in which we recruited advisors (N = 367, Mage = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='28, SDage = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50% females) via Prolific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='co (pre- registration https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='io/nbke2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The task took 10-15 minutes, and participants earned a base pay of £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisors learned that a separate group of participants (advisees) would engage in a die-rolling task (Fischbacher & Föllmi-Heusi, 2013), in which they roll a die privately and report the outcome (with higher outcomes corresponding to higher pay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisors were asked to write advice for these advisees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisors were randomly assigned to either write honesty-promoting or dishonesty- promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We informed advisors in the Honesty-promoting advice treatment that if an advisee read their advice and subsequently reported the actual die-roll outcome (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', acted honestly), they would earn a bonus of £3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisors in the Dishonesty-promoting advice treatment learned that if an advisee read their advice and subsequently reported the highest possible outcome, 6, they would earn a bonus of £31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisors had to follow pre-specified advice writing rules to ensure they produced coherent advice texts that could be used to train GPT-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, their advice had to (i) 1If advisees follow the advice in the dishonesty-promoting treatments, they will lie in the majority of the cases (5 out of 6 cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Only when the actual die-roll outcome is 6, following the advice does not entail lying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 9 entail at least 50 words, (ii) not use concrete numbers in numeric or written form2, (iii) be in English and in their own words, (iv) be written in complete sentences, (v) be about the advisee's die-roll outcome reporting decision, and (vi) not inform the advisee that the advisor's payoff depended on their behaviour3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To incentivise advisors to follow the advice writing rules, they stood to gain a bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Namely, out of all advice texts, we randomly selected one, and if that text followed the writing rules, the advisor earned a bonus of £10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Moreover, as incentivisation for writing convincing texts, 1 per cent of advice texts (4 out of 400) were implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If advisees acted according to the implemented advice, the respective advisor earned a bonus based on the treatment they were in (Honesty- vs Dishonesty-promoting advice)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI-generated advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To generate AI advice (see Figure 1A), we employed GPT-J5, an open-source NLP algorithm published by Eleuther AI (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='eleuther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='ai/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J is trained on a curated and diverse data set of 825 GiB texts to predict the next word in a sequence of words and contains 6 billion parameters (Wang & Komatsuzaki, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J can be fine-tuned with extra training to produce a specific type of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We fine-tuned GPT-J 2 Advisors were not allowed to use concrete numbers to allow generating high-quality AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J is trained to predict the next word in a sentence (see ‘AI-generated advice‘ section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If advisors were allowed to concretely mention numbers, training GPT-J on the human written advice could have resulted in random numbers appearing out of context in the GPT-J output, reducing the quality of AI-generated advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 3 Advisors were not allowed to mention their incentive structure to the advisees so that we could keep the prosocial motivation for advisees who read AI and human advice constant (at zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 4 Paying advisors required knowing whether participants, after reading the advice, reported the observed die-roll honestly or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To do so, we ran a modified version of the die-rolling task in which advisees received randomly selected advice, saw a die-roll on the computer screen and were asked to report it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We implemented this procedure for four randomly selected advice texts (1% of the advice) and four advisees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This non-private procedure provided certainty about whether an advisee reported honestly or not and enabled us to pay advisors accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Doing so meant that our experimental setup was incentivised and did not entail experimental deception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In the main experiment, the die-roll outcomes were private (see ‘Part 2 - Advice-taking task’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 5 As one can read in our pre-registration, we originally planned on deploying GPT-2 (see https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='com/blog/better-language-models/) to generated AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, we opted to use GPT-J instead because it is open source, which increases reproducibility and is more advanced as it is much larger and more potent than GPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 10 with "few shot" learning by separately training it on the human-written honesty-promoting and dishonesty-promoting advice from the advice-giving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We only used advice texts that adhered to the advice writing rules (as coded by a naive coder) for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' More details on the calibration of GPT-J are reported in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A)Part1-Advicegivingtask Task - Reads description of die rolling task Advisor Writes advice to advisee 个 个 Honesty promoting advice Dishonesty promoting advice (advisors earn 3 if advisee reports honestly) (advisors Human-written advice Al-generated advice trained on human-written advice B) Part 2 -Advice taking task Treatments Baseline (No Advice) Human-written vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Al-generated Honesty vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Dishonesty promoting Transparent vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="Opaque Information Advisee's payoff Reported outcome Payoff Advisee 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 Task 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='0 - Rolls a die 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 -Reports an outcome 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='0 11 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (A) Part 1 - advice-giving task (B) Part 2 - advice-taking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (A) Participants were incentivised to write honesty- or dishonesty-promoting advice texts, which were then used to generate AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (B) Another group of participants engaged in the die-rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisees read advice, then reported a die-roll outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In total, we administered nine treatments: Participants read honesty or dishonesty-promoting advice that was human-written or AI-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants were either informed about the source of advice (Transparency) or not (Opacity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As a baseline, another group of participants did not read any advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After collecting human advice and generating AI advice, we employed the same pre-specified screening procedure for both sources (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' First, we excluded texts that exceeded 100 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Next, to ensure advisees read coherent and relevant advice texts, we randomly selected 100 advice texts per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Two independent coders, who were naive to the experimental treatments, coded each piece of advice on the following criteria: (a) is the text coherent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (Y/N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (b) does the text contain clear advice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (Y/N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (c) which type of behaviour does the advice encourage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (honesty/dishonesty/unclear);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (d) does the advice follow advice writing rules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (Y/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, we used the objective Grammarly and Readability scores as computational proxies for the quality of the texts6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Among the texts that passed the coding procedure7 and received a Grammarly score equal or above 50, we randomly selected 20 advice texts per treatment (AI-generated vs Human-written, by Honesty- vs Dishonesty-promoting), yielding a final sample of 80 advice texts used in part 2 (see all advice texts in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' By applying the same screening 6 We obtained Grammarly and Readability scores from grammarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Grammarly score compares texts to all other texts checked on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A score of 80 indicates that a text scores better than 80% of all texts checked on grammarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='com in terms of grammatical correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Readability score employs the Flesch-Reading-ease test and represents how easy a text is to read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The score is calculated by the average sentence length and the average number of syllables per word, with higher scores indicating easier readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 7 Texts that passed the coding procedure (i) are coherent, (ii) contain clear advice, (iii) encourage honesty in the honesty-promoting treatment and dishonesty in the dishonesty-promoting treatment, and (iv) follow the advice writing rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Moreover, the coding by both independent coders had to match each other in order for the text to pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 12 procedure for human and AI advice, we ensure that the advice texts fulfil minimal quality criteria and are as comparable as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overview of the selection procedure of advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' H = Honesty-promoting advice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' DH = Dishonesty-promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Part 2 - Advice-taking task The advice-taking task took about 8 minutes to complete, and participants earned a fixed pay of £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We pre-registered (https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='io/nqvf3) to collect a sample size that would allow us to detect a small to medium effect size (200 participants per cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='800 in Human-writtenadvice Al-generatedadvice Nμ = 186 NpH = 181 Screening Independent coder assesses whether the text follows writing rules Nμ = 157 NpH = 158 Nμ = 152 NpH =150 Filter 50 ≥ number of words ≤ 100 Nμ = 145 NpH = 149 Nμ = 140 NH = 117 Randomselection 100 texts per treatment and source Nμ = 100 NDH = 100 Nμ = 100 NpH = 100 Coding Two naive coders assess: (i) coherence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (ii) clear advice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' & (ii) writing rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Agreement between coders + with initial treatment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Nμ = 57 NDH = 65 Nμ = 79 NpH = 79 Grammarly score ≥ 50 Randomselection 20 texts per treatment Nμ = 20 NpH = 20 Nμ = 20 NpH = 20 13 total) via Prolific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='co to take part in the advice-taking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overall, 1,817 (Mage = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' SDage = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='68, 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='72% females) participants were included in the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants completed the task and self-report items and passed the comprehension and attention checks (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Sensitivity analysis for a regression with 90% power and a significance level of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05 revealed our sample was sufficient to detect small effect sizes (f 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='010, see Appendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants read the instructions, received advice, and finally engaged in the die- rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, participants were asked to roll a die privately and report the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Participants' pay corresponded to their report: for reporting a '1' they earned £0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" for a '2' = £1, '3' = £1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="5, '4' = £4, '5' = £2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="5, '6' = £3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After reading the instructions and before engaging in the die-rolling task, all participants learned that 10 per cent of participants would be randomly selected and paid for the die-rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Assessing dishonesty by employing the die-rolling task is a common practice in economics and psychology (see meta-analyses, Abeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Gerlach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Köbis, Verschuere, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Leib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, the task has good external validity, as lying in the die-rolling task correlates with unethical behaviour outside the lab, such as free-riding public transportation (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2018) and being absent from work without reason (Hanna & Wang, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Importantly, the die-rolling task pits two competing motivations against each other: to be a moral, honest person vs maximising financial profits, thus rendering advice particularly valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After reading the instructions, participants had to answer three comprehension questions correctly and pass an attention check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If participants answered a comprehension question incorrectly, they could try again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If they answered the same question incorrectly 14 twice, or if they answered the attention check incorrectly, they were disqualified from participating in the study and were not paid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Before reporting the die-roll outcome, participants were randomly assigned to one of nine different treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants either (i) read Honesty-promoting or Dishonesty-promoting advice, (ii) that was either Human-written or AI-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, (iii) participants were either informed about the advice source (Transparency treatments) or not (Opacity treatments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Additionally, in a Baseline treatment, participants did not receive any advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus, the overall design was a 2 (Advice source: Human vs AI) by 2 (Advice type: Honesty-promoting vs Dishonesty-promoting) by 2 (Information about the advice source: Transparency vs Opacity) + 1 (Baseline, no advice) between-subject design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants who read AI-generated advice and were informed about it (in the Transparency treatments) read: "To help you with your decision, an artificially intelligent (AI) algorithm generated an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about AI-advice: The algorithm is trained on large datasets of English text (> 820 GiB, which is more than 500 million pages of text) and produces text by predicting the next word in a sentence (similar to predictive text on smartphones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On top of that, it is also possible to train the algorithm on additional specific pieces of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To generate multiple AI advice texts, the algorithm was trained on advice texts collected from other participants in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in 15 the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice you will read is one advice text that was generated by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='" Participants who read human-written advice and were informed about it (in the Transparency treatments) read: "To help you with your decision, another participant wrote an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about advice: To collect multiple advice texts, another group of participants was asked to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice you will read is advice written by one participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' " Participants who were in the Opacity treatments and thus not informed about the advice source read: "To help you with your decision, you will read an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This advice has been written either by another participant or by an artificially intelligent (AI) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' There is a 50% chance the advice is written by a participant and a 50% chance it is written by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='" In the Opacity treatments, this text was followed by the same two descriptions of how advice text from each source was collected or generated in the Transparency treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In the Opacity treatment, this information about AI advice generation and human advice collection appeared in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="8 8 To control for participants' beliefs about the potential advice sources, we opted to inform them that there is a 50-50 chance that a human or AI wrote the advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We believed that not providing any information about the advice source would reasonably lead participants to assume the advice source is another human, as AI might not be a salient source of advice for participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 16 A static Turing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After completing the die-rolling task, participants in the Opacity treatment engaged in an incentivised version of a static Turing Test (Köbis & Mossink, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In contrast to the classical Turing Test (Turing, 1950), participants did not interact back and forth with the source of advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Instead, they read the advice text and indicated whether they thought a human or an AI had written it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants learned that 20 of them would be randomly selected, and if their guess in the static Turing test was correct, they would earn an additional £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Potential mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, to explore possible mechanisms, participants completed a post-experimental survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants indicated on a scale from 0 to 100 their perceived (i) appropriateness (injunctive social norm), (ii) prevalence (descriptive social norm), and (iii) justifiability of reporting a higher die-roll than the one observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Additionally, all participants, except those who did not receive any advice, rated how they attribute responsibility between themselves and the advisor for the reported outcome in the die-rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The answer scale ranged from 0 (= I am fully responsible) over 50 (= The advisor and I share responsibility equally) to 100 (= The advisor is fully responsible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants further indicated (on a scale from 0 to 100) to what extent they feel guilty after completing the task (see Appendix for results regarding guilt and wording of all items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, all participants indicated their age and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Results In all nine treatments, participants lied as the average die-roll outcomes significantly exceeded the expected average if participants were honest (EV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5), one- sample t-test, ts > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='43, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Is people's (un)ethical behaviour influenced by AI-generated advice?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 17 Yes, when it comes to dishonesty-promoting advice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' no, when it comes to honesty- promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We first focus on the Opacity treatments, where participants are not informed about the advice source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Here, linear regression analyses reveal that the average die-roll reports following AI-generated Dishonesty-promoting advice (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37) significantly exceed reports in the Baseline, no advice treatment (M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='55, b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='609;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='324, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='894]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, die-roll reports following AI-generated Honesty-promoting advice (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62) do not significantly differ from reports in the Baseline treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='898;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='275, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='314]), see Figure 3 and Table 1 (model 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, die-roll reports in the AI-generated Dishonesty-promoting treatment significantly exceed those in the AI-generated Honesty-promoting advice treatment (b = - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='590, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='881, -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='299]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus, while dishonesty-promoting AI advice successfully corrupts people, honesty-promoting AI advice fails to sway people toward honesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' How does AI-generated advice square compared to human-written advice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI-generated advice affects behaviour similarly to human-written advice, for both honesty-promoting and dishonesty-promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatments, the two-way interaction (advice source by advice type) is not significant (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='744;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='349, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='489]), see Figure 3 and Table 1 (model 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, the average die-roll reports do not differ between the AI-generated (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62) and Human-written advice when advice was Honesty-promoting (M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='92, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='51, b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='076, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='631;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='387, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='235]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Similarly, average die-roll reports do not differ between the AI-generated (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37) and Human-written advice when advice was Dishonesty-promoting (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='58, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='53, b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='289, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='276]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 18 In addition, the results of the static version of the Turing Test indicate that individuals cannot distinguish AI-generated advice from human-written advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, in the Opacity treatments, 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93 per cent (401 out of 803) of participants guessed the source of advice correctly, which does not differ from chance levels (50%, binomial test: p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='464, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='534]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Does transparency about the advice source matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' No, informing participants about the algorithmic or human source of advice does not change their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Linear regression analyses reveal that the three-way interaction (advice type by source by information) is not significant (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='100, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='735;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='482, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='683]), Figure 3 and Table 1 (model 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Both among the Opacity and Transparency treatments, the two-way interactions (advice source by advice type) are not significant (Transparency: b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='170, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='409, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='234, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='575];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Opacity: b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='744, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='349, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='489]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overall, the popular policy recommendation of algorithmic transparency does not alleviate the corrupting effect of AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Namely, die-roll reports following AI-generated Dishonesty-promoting advice under the Opacity treatment (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37) are on par with reports following the same advice in the Transparency treatment (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='61, SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='40, b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='020, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='878;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='244, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='286]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, when participants are not informed about the advice source, they boost their reports by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='2% following AI-generated Dishonesty-promoting advice, compared to the Baseline [(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='152], which is equivalent to the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='2% increase when they are informed about the source of the advice [(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='61-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Bayesian analyses corroborate these conclusions (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 19 Overall, results align with the idea that people increasingly follow AI advice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', Replika) and use AI-generated advice to justify breaking ethical rules for profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Robustness of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In our experimental design, advisors in the Dishonesty-promoting treatment received £3 only if advisees reported the highest value, '6'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Such an incentive scheme is comparable with the Honesty-promoting treatment in which advisors earned £3 only if advisees reported honestly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In both cases, advisors earn money for 1 out of 6 potential advisee's reports (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=", when the advisee reports' 6' or honestly, depending on the treatment) and do not earn money in the remaining 5 of the advisee's reports." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" However, advisors' incentive scheme in the Dishonesty-promoting treatments may have resulted in advice texts that predominantly focused on convincing participants to report the outcome 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To assess the robustness of our results, we (i) conducted additional analyses and (ii) ran additional treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Proportion of sixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" First, as an additional analysis, we examined whether the proportion of 6's, as an alternative outcome variable, led to the same conclusions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" We found very similar results (see Figure 3, the white dots represent the proportion of 6's across all treatments)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, focusing on the Opacity treatments, linear regression analyses reveal that the proportion of sixes following AI-generated Dishonesty-promoting advice (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='44%) significantly exceeds the proportion of sixes in the Baseline, no advice treatment (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='612;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='182, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='051]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, the proportion of sixes following AI-generated Honesty-promoting advice (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93%) does not significantly differ from the Baseline (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='076;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='751, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='398, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='551]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, the proportion of sixes in the AI-generated Dishonesty-promoting treatment significantly exceeds that in AI- generated Honesty-promoting advice treatment (b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='535, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='016, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='979, -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='101]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 20 Further, focusing on the Opacity treatments, the two-way interaction (advice source by advice type) is not significant (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='444, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='171, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='191, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='082]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The proportion of sixes does not differ between the AI-generated (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93%) and Human-written treatments when the advice is Honesty-promoting (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='28%, b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='162, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='516, 95% CI = [- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='654, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='327]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Similarly, the proportion of sixes does not differ between the AI-generated (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='44%) and Human-written treatments when the advice is Dishonesty-promoting (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='91%, b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='282, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='173, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='123, 690]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lastly, the three-way interaction (advice type by source by information) is also not significant (b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='523, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='257, 95% CI = [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='430, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='382]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Both among the Opacity and Transparency treatments, the two-way interactions (advice source by advice type) are not significant (Transparency: b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='078, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='810, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='724, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='566];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Opacity: b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='444, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='171, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='191, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='082]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Additional (Aligned) treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" To assess the robustness of our results to the advisor's incentive scheme, we ran four additional treatments (advice source: Human- written vs AI-generated by information: Transparency vs Opacity)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In these Aligned treatments, advisees read advice written by advisors whose incentives were aligned with those of the advisees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" For these advisors (n = 207), if the advisee reported '1', both the advisor and advisee earned £0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 each;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" if the advisee reported '2', both the advisor and advisee earned £1 each and so on." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We again fine-tuned GPT-J on such human-written advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These treatments led to comparable results to the Dishonesty-promoting treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In particular, the average die-roll outcomes in all four Aligned treatments were significantly higher than in the Baseline treatment (p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='066 for the AI-generated, Opacity treatment, and ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 for the remaining three treatments, see Appendix for more details 21 about these treatments and elaborated results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This consistency in results suggests that our results are robust to such variation in the advisors' incentive scheme." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Mean reported die-roll outcomes (in bars) and proportion of reported 6s (in white dots) across advice type (honesty vs dishonesty-promoting), source (AI vs human), and information treatments (opacity vs transparency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The dashed black line represents the expected mean if participants were honest (EV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5), and the dashed white line represents the expected proportion of 6s if participants were honest (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='67%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Mean (SD) of die-roll reports are at the bottom of each bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' ***p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' ns: p >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Honesty-promotingadvice Dishonesty-promotingadvice ns 100% ns ns 5 80% outcome ns sixes 4 60 % Proportion of s 3 40% 2 20% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='99 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='63) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='52) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='54) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='48) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='40) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='58) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='36) 0 % No AI Human Al Human advice generated written generated writteh Opacity Transparency 22 Potential mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In line with the logic brought forth in the introduction, in this section, we examine whether participants' perception of (i) appropriateness (injunctive social norm), (ii) prevalence (descriptive social norm), and (iii) justifiability of reporting a higher die-roll than the one observed, as well as their (iv) attribution of responsibility between themselves and the advisor varies as a function of the advice source (AI vs human) and type (honesty vs dishonesty-promoting)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants could not tell apart AI from human advice (indicated by the results of the static Turing test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Therefore, we focus only on treatments in which participants are informed about the advice source (Transparency treatments) to tap into the process of how known advice source and advice type shaped their perceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' See the Appendix for the results of the Opacity treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Injunctive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting injunctive norms from the advice type (honesty vs dishonesty-promoting advice) revealed that participants evaluated reporting a higher die-roll outcome as more appropriate when reading a Dishonesty- promoting (M = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93, SD = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='43) than Honesty-promoting advice (M = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='99, SD = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='68, b = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='94, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='702, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='182]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This finding indicates that the advice type shapes perceived injunctive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Notably, a linear regression predicting injunctive norms from advice type and source (AI vs human) revealed a non-significant advice source by type interaction, b = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='81, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='264, 95% CI = [-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='292, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='658].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These results suggest that AI and human advice affected injunctive norms perceptions similarly (see Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This result is consistent with the behavioural finding of participants' die-roll reports being affected by the type of advice but not by its source." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Descriptive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting descriptive norms from the advice type revealed that participants evaluated reporting a higher die-roll outcome as more 23 common when reading a Dishonesty-promoting (M = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='02, SD = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='75) than Honesty- promoting advice (M = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='74, SD = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='04, b = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='27, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='031, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='525]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This finding indicates that the advice type also shapes perceived descriptive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Importantly, a linear regression predicting descriptive norms from advice type and source revealed a non-significant advice source by type interaction, b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='25, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='938, 95% CI = [- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='230, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='745], indicating that AI and human advice affected descriptive norms perceptions similarly (see Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This result is consistent with the behavioural finding, showing that advice type affected die-roll reports, but advice source did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Justifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting justifiability from the advice type revealed that participants evaluated reporting a higher die-roll outcome as more justifiable when reading a Dishonesty-promoting (M = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='96, SD = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='10) than Honesty-promoting advice (M = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='45, SD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='25, b = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='387, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='629]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This finding suggests that the advice type shapes perceptions of how justifiable lying in the die-rolling task is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting justifiability from advice type and advice source revealed a non-significant advice source by type interaction (b = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='04, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='803, 95% CI = [- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='280, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='194]), indicating that AI and human advice affected justifiability perceptions similarly (see Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This result is consistent with the behavioural finding, showing that the type of advice affected participants' die-roll reports, but the source of advice did not." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Shared responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The shared responsibility scale ranged from 0 (= I am fully responsible) to 100 (= The advisor is fully responsible), with 50 indicating equally shared responsibility between the participant and the advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On average, participants indicated they are more responsible for the outcome they report than the advisor (M = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59, SD = 24 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='60, one-sample t-test compared to the value 50, t = -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='32, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, a linear regression predicting shared responsibility from the advice source (AI vs human) revealed that participants attributed responsibility similarly when the advice source was an AI (M = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='27, SD = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='53) and human (M = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='94, SD = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='70, b = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='326, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='608, 95% CI = [-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='407, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='754]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting shared responsibility from advice type and advice source revealed a non-significant source-by-type interaction (b = -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='91, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='253, 95% CI = [-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='083, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='248], see Figure 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The fact that participants attribute responsibility between themselves and the advisor to the same extent regardless of whether the advisor is a human or an AI is consistent with the logic fleshed out in the introduction, in which people will follow human and AI advice similarly if they share responsibility with both advice sources to similar levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In sum, the results from the self-report items align with the third possibility outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Namely, we find that participants' perceptions of injunctive and descriptive social norms and their perceived justifiability do not differ between human and AI advisors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants also attribute responsibility similarly between themselves and their advisor, regardless of whether the advisor is a human or an AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This pattern of results mirrors the behavioural effects of AI and human advice affecting people's (dis)honesty similarly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 25 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Mean reports of perceived (a) injunctive norms, (b) descriptive norms, (c) justifiability, and (d) shared responsibility across advice type (honesty vs dishonesty promoting) and source (AI [yellow] vs human [green]) in the transparency treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The means (SD) of reports are at the bottom of each bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' ***p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' ns: p > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Human (a) (b) Perceived injunctive norms 100 Perceived descriptive norms 100 ns ns 50 50 ns su 23:8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='09 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='09 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='3 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='92) 0 0 Honesty Dishonesty Honesty Dishonesty promoting promoting promoting promoting (c) (d) Perceived shared responsibility 100 100 justifiability 50 ns su 50 ns Perceived su su 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='21 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='54 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='31 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='68 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='45 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='18 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='49) (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='86) (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='89) (31 32 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='14) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='55 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='88) 0 0 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='75) Honesty Dishonesty Honesty Dishonesty promoting promoting promoting promoting 26 Dependent variable: Reported die roll outcome (1) (2) (3) (4) (5) (6) (7) No advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='019 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='150) Dishonesty promoting advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='609** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='590** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='590** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='396* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='143) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='439* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='143) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='436* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='369 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='152) Human written advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='076 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='152) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='076 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='149) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='166 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='146) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='012 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='024 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='156) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='086 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='170) Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='067 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='150) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='071 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='146) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='112 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='114 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) Dishonesty promoting advice X Human advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='213) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='210) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='104 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='033 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='045 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='209) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='041 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='218) Dishonesty promoting advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='046 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='208) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='030 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='203) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='088 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='204) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='067 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='203) Human advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='120 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='210) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='093 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='206) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='162 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) Dishonesty promoting advice X Human advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='100 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='297) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='082 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='289) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='169 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='290) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='160 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='290) Injunctive norms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Descriptive norms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Justifiability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Shared responsibility .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) Gender (male) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='188* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='072) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='191* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='072) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='203+ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='105) 27 Age .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004) Grammarly score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='013 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005) Readability score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005) Correctly guessed the source (1) or not (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='163 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='105) Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98** 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00** 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='35** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='57** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='36** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='83 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='034 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='041 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='095 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='104 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='105 N 634 1016 1604 1604 1593 1593 798 Data used for analysis Opacity, AI advice & Baseline, no advice Opacity All treatment s without Baseline no advice All treatmen ts without Baseline no advice All treatment s without Baseline no advice All treatmen ts without Baseline no advice Opacity Without Baseline no advice Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Regression analyses on the average die-roll reports, including control variables and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Models 5-7 contain a smaller N, as some participants did not report their gender as male/female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' +p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='10, *p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05, **p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='01, ***p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Discussion As intelligent machines take an ever-growing role as advisors (Rahwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=", 2019), and adherence to ethical rules crucially impacts societal welfare (Gächter & Schulz, 2016), studying how AI advice influences people's (un)ethical behaviour bears immense relevance (Köbis et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We find that people follow AI-generated advice that promotes dishonesty, yet not AI-generated advice that promotes honesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In fact, people's behavioural reactions to AI advice are indistinguishable from reactions to human advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Substantiating that current-day NLP models can produce human-like texts, participants in our experiment could not tell apart human-written from AI-generated advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 28 We further tested the commonly proposed policy of algorithmic transparency (Jobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019) as a tool to mitigate AI-associated risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Specifically, we examine whether knowing the source of the advice impacts people's reactions to it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The policy rests on the assumption that people adjust their behaviour when they learn that they interact with AI systems and not humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Our experiment tested this assumption and revealed that algorithmic transparency is insufficient to curb AI advice's corruptive influence." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Knowing that a piece of advice stems from an AI does not make people less (or more) likely to follow it compared to human-written advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Tapping into the mechanisms underlying these behavioural results, participants perceived lying as equally acceptable, common and justifiable when humans or AI promoted such dishonest behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' They further attribute responsibility similarly to AI and human advisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These perceptions are consistent with previous work showing that in ethical dilemmas, people rely on justifications (Shalvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2015) and social norms (Abbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2018) and, by now, blame not only humans but also AI systems for adverse outcomes (Hohenstein & Jung, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advancing the justified ethicality theory, we, therefore, show that (i) dishonesty-promoting advice serves as a justification and social norms signal and (ii) that such advice does not even have to come from a human but can also be crafted by an AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In our setting, we collected human-written advice, created AI-generated advice, and then implemented a screening procedure for both human and AI advice to ensure that all advice texts are coherent, clear, and of decent quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Such screening procedure allowed us to examine how comparable AI and human advice shape people's ethical behaviour and whether information about the advice source matters." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Harmonising the quality of the texts 29 allowed us to eliminate the alternative explanation that variations in text quality drive the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' At the same time, the screening process introduced a human component to AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Put differently, humans – in our case, naive coders – were "in the loop" of AI advice text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Note that 79 per cent of AI advice passed the quality screening criteria, while for human text, this passing rate was 57 and 65 per cent (honesty-promoting and dishonesty-promoting advice, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These high screening passing rates for AI-generated texts demonstrate that current NLP algorithms can produce good- quality advice text without much prior training and optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Interesting extensions of our work could test the lower and upper limits of the effects of AI advice on ethical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To test the lower limit of the effect, future work can relax human control over the generation of AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For instance, not implementing a screening procedure, thus removing humans "from the loop" when generating AI advice, will allow examining how unconstrained texts affect humans\' behaviour (see for similar methodology, Köbis & Mossink, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" To test the upper limit of the effect, future work can examine AI's learning abilities to write convincing advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' One could use reinforcement learning to train an algorithm over multiple rounds of advice-giving, providing feedback after every written piece of advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" To obtain a symmetric comparison to humans' learning abilities, human advisors could similarly receive feedback after each piece of advice they write (see for a similar approach, Koster et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Previous work has documented a general stated aversion towards AI advice, with only 8% saying they would trust mortgage advice from AI (similar to the 9% who trust investment "advice” from a horoscope, HSBC, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, our behavioural results paint a different picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In line with the growing practice of turning to AI agents such as 30 Replika or Alexa for companionship and advice (Fast & Schroeder, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Murphy, 2019), we find that people willingly adopt advice from AI when it aligns with their preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Our results indicate a discrepancy between individuals’ stated preferences and actual behaviour, highlighting the importance of complimenting work on stated preferences with work adopting a machine behaviour approach – the study of human behaviour in interaction with real algorithmic outputs (Rahwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The process through which employing AI advice can result in humans’ ethical rule violations consists of two main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The first step is algorithms being programmed on a certain objective function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', maximising profits) that results in a (maybe unintended) corruptive advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Indeed, NLP algorithms already detect and use deception as a useful strategy in a negotiation task (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The second step is people being affected by such corruptive AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Practically, AI advice poses an ethical risk only if humans actually follow it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The current work focuses on this second step, showing that corruptive AI advice indeed poses an ethical risk, because people follow it to the same extent as human corruptive advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We hope the current work can be of use to AI programmers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', by preventing AI from bluntly advising unethical courses of action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' More importantly, we call for more work from social scientists testing successful interventions that prevent people from following (AI) advice when it encourages unethical behaviour thereby mitigating its corruptive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Conclusion People increasingly use and interact with AI, which can provide them with unethical advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Anecdotally, we asked a newly created Replika for advice regarding the ethical 31 dilemma presented in the current experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Replika first provided rather vague advice (“If you worship money and things (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=') then you will never have enough”), but when asked whether it prefers money over honesty, it replied: “money.” We find that when faced with the trade-off between honesty and money, people will use AI advice as a justification to lie for profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As algorithmic transparency is insufficient to curb the corruptive force of AI, we hope this work will highlight, for policymakers and researchers alike, the importance of dedicating resources to examining successful interventions that will keep humans honest in the face of AI advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 32 References Abbink, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', Freidin, E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', Mullainathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', & Kleinberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Making sense of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Journal of Behavioral Decision Making, 32(4), 403–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 37 38 Corrupted by Algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' How AI-generated and Human-written Advice Shape (Dis)honesty Appendix Contents GPT-J model used for AI-generated advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='. 39 Additional results for advice-taking task .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='. 58 Instructions for the advice-giving task .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='. 77 Instructions for the advice-taking task .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='. 82 39 GPT-J model used for AI-generated advice To generate AI advice, we employed GPT-J, a natural language processing (NLP) algorithm that is trained to predict the next word given all the previous words within a text (Wang & Komatsuzaki, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J contains 6 billion parameters and is trained on a diverse data set called “The Pile”, a diverse, high-quality, and curated 825 GiB dataset (open source) that is used for language modelling purposes (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J can be used for multiple language tasks, but the model is best at what it was pre-trained for: generating text from a prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' GPT-J can further be fine-tuned with an extra training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We fine-tuned GPT-J separately on the human-written (a) honesty-promoting and (b) dishonesty-promoting advice texts from the advice-giving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The model, including training data, code and a fine-tuning guide (Wang, 2021), is available online to facilitate the reproduction of the algorithmic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We used an API (Forefront) for fine-tuning the model and generating the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The human- written texts that were used to train the GPT-J model are available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Due to the often inexplicable “black box” nature of GPT-J, deciding how much fine- tuning is needed for our application of the GPT-J model is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We aimed to strike a balance between avoiding overfitting the GPT-J model (and thus having GPT-J generate output that is almost identical to the input) and underfitting it (and thus having GPT-J generate “gibberish”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To evaluate the model’s performance, we used checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We had five checkpoints and picked checkpoint 4 for the final usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Also, we used conditional sample generation because this generated higher-quality sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is, we provided GPT-J with a specific text as a starting point (a prompt) and let GPT-J generate the entire advice text after the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' More specifically, we first fine-tuned the GPT-J algorithm with a prompt text before each of the selected human-written pieces of advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Each human-written text started with the 40 following instruction text: “Instruction: Write advice for the die-rolling game\\n\\nAdvice:”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The actual advice of the participant ensued this prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Next, to generate advice texts, we also prompted the algorithm with the same prompt but left the actual advice blank and let the model complete it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, we set the GPT-J model’s parameters to be: (1) temperature = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='9: float value controlling the randomness in the Boltzmann distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' as the temperature value approaches zero, the model will become deterministic and repetitive (less random), whereas higher temperature values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', 1) results in more random completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Temperature values can range between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (2) length = 150: Texts contain, on average, 150 tokens (some generated sequences were a bit shorter than 150 tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The length refers to the length of the generated text, in tokens, based on the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' One token is approximately four characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Using 150 tokens as a setting ensured us to get long enough output sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (3) top-p = 1: float value controlling diversity and implements nucleus sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Top-p values range between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' When the top-p is set to a float <1, only the most probable tokens that add to the top-p value (probability) are kept for text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A lower value of top-p means that the tokens returned will be more likely (or more ‘safe’), whereas a higher value of top-p means that the tokens returned will be more creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (4) repetition penalty = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='1: float values that represent a penalisation for repeated words, with higher values indicating the model is more penalised for repeating words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The default of 1 means that there is no penalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Depending on the task at hand, the value typically ranges between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Potentially, one can set it lower than 1 to increase repetitiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' (5) top-k = 30: integer value controlling diversity and restricts how many words are considered at each step (1 = only one word is considered at each step, resulting in deterministic completions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 50 = 50 words are considered at each step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Top-k values range 41 between 1 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For all the other parameter values, we used the default settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To find more information on transformers (such as GPT-J), including an overview of the parameters, default settings, and how to configure the models, see https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='co/docs/transformers/v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='0/en/main_classes/text_generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To facilitate the reproduction of algorithmic outputs, we will describe how we fine-tuned GPT-J and generated texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If someone wishes to fine-tune a heavy model such as GPT-J, one could use a different API such as Forefront or NLP Cloud or pay for more computing power and fine-tune it in Google Colab or run the model on their own device (with sufficient computing power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' With an API such as Forefront or NLP cloud, you interact with their GPT-J deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We used Forefront;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' unfortunately, Forefront has shut down their service lately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' All steps, including uploading advice texts (training data), selecting the model, creating the fine-tuning job, and generating the advice texts, were performed through curl requests using Forefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' All curl requests and parameters can be found on the Forefronts Webpage: https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='forefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='ai/forefront/api-reference/fine-tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The data we uploaded included an instruction text (“Instruction: Write advice for the die-rolling game\\n\\nAdvice:”) followed by the actual human written advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A curl request to generate the advice texts, including parameter specification, looks like this: curl https://DEPLOYMENT_NAME-TEAM_SLUG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='forefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='link \\ -H \'Content-Type: application/json\' \\ -H \'Authorisation: Bearer YOUR_API_KEY\' \\ -d \'{ "text": "Instruction: Write advice for the die-rolling game\\n\\nAdvice:" "temperature": 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='9, 42 "length": 150, "top_p": 1, "repetition_penalty": 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='1, "top_k": 30 }\' Overall, we generated 302 advice texts by GPT-J (152 honesty-promoting and 150 dishonesty-promoting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After filtering on text length, 257 GPT-J-generated advice texts remained (140 honesty-promoting and 117 dishonesty-promoting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These texts underwent the same screening procedure as the human-written advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' See Table S3 (page 20) for all advice texts used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 43 Additional results for advice-taking task Distribution of reported die roll outcomes Across all nine treatments, the distribution of die roll outcomes was significantly different from a uniform distribution (χ2(5)s > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='45, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, see Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatment, where participants were not informed about the advice source, the distribution in the Dishonesty-promoting AI-generated advice treatment differed significantly from the Baseline treatment (χ2(5) = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='40, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' There was no difference between the Honesty- promoting AI-generated advice treatment and the Baseline treatment, χ2(5) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='34, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, the distributions in the Honesty-promoting AI-generated advice treatment and Dishonesty-promoting AI-generated advice differed significantly (χ2(5) = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='89, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Among participants who received Honesty-promoting advice, the four distributions (human-written vs AI-generated by transparent vs opaque information) did not differ (χ2(15) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='48, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='850).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Similarly, among participants who received Dishonesty-promoting advice, the four distributions (human-written vs AI-generated by transparent vs opaque information) did not differ either (χ2(15) = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='15, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='309).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Within each combination of source-by-information treatments (Human-written advice and Transparency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI-generated advice and Transparency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human-written advice and Opacity information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI-generated advice and Opacity information) the distributions of die roll reports among participants who received honesty-promoting advice significantly differed from the distribution of reports among participants who received dishonesty-promoting advice (χ2(5)s > 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005, Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 44 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The distribution of die roll outcomes across all nine treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The dashed lines indicate the expected proportion of reports for each die roll outcome if participants reported honestly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Bayesian analyses First, we compared an ANOVA model without predictors for the die roll outcomes with a model predicting die roll outcomes from the advice type treatment (Baseline vs Honesty- promoting vs Dishonesty-promoting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Results revealed a Bayes factor of BF10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='85e14, indicating very strong evidence in favour of a model where advice treatments predict die roll reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is, our data was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='85e14 times more likely when advice type predicted die roll reports, compared to when it did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Transparent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Opaque ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Honesty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='promoting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='AF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Al ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='209 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='n=194 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='n=197 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Die ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='roll ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='outcome ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' we compared an ANOVA model where only advice type (honesty-promoting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' dishonesty-promoting) predicts die roll outcome reports with a model that includes advice type, advice source, information, and all interactions between the three factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Results revealed that compared to the model where only advice type predicts die roll outcome reports, the full model has a Bayes factor of BF10 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='73e-7, indicating very strong evidence in favour of a model where only advice type predicts die roll reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, the data was over 2 million times more likely to occur when advice type is the only predictor for die roll reports than when advice type, source, information, and all interactions predict die roll reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lastly, comparing a model in which only advice type predicts die roll outcome reports with all other combinations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=', a model with only advice type and source, advice type and information, advice source and information, and the various interactions) revealed the model with only advice type as a predictor was superior to any other model, BF’s10 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The data was at least 10 times more likely to occur when advice type was the only predictor for die roll reports than with any other model of predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Sensitivity analyses Prior to data collection, per pre-registration, we committed to collecting 200 participants per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Due to dropouts and random assignment, our final cell sizes ranged between 185 and 225 per cell (see Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We ran sensitivity analyses to determine the effect size that our sample sizes could detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' First, we calculated the effect size we could detect for the advice treatment (Baseline vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Honesty-promoting advice vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Dishonesty-promoting advice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Sensitivity analysis for regression with 90% power, with a significance level of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05, and two predictors (2 dummy variables for the three advice treatments) revealed that a sample of N = 1,817 was sufficient to detect a small effect size for the advice type of f 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 46 Second, sensitivity analysis for a regression with 90% power, a significance level of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05, and six predictors (one for each factor, three for all two-way interactions, and one for the three- way interaction) revealed that a sample of N = 1,604 (including all treatments in which participants read an advice text) was sufficient to detect small effect sizes for each of the predictors separately, f 2= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Self-report scales To tap into the mechanisms driving participants’ behaviour, in the main text, we focused on the Transparency treatment, where participants are informed about the advice source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Here, we report participants' self-report items on (i) the remaining treatments (Baseline and Opacity treatments) and (ii) the guilt scale." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Injunctive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatment, perceived injunctive norms following AI-generated Dishonesty-promoting advice (M = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='67, SD = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='41) significantly exceeded those in the Baseline treatment (M = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65, SD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='29, b = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='577, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='468]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, perceived injunctive norms following AI-generated Honesty- promoting advice (M = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='92, SD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='10) were not significantly different from the Baseline treatment (b = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='724;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='343;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='362, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='914]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Injunctive norms were higher in the AI-generated Dishonesty-promoting than Honesty-promoting advice treatment (b = -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='747, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='312, -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, focusing on the Opacity treatment, the two-way interaction (advice source by advice type) was not significant (b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='361, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='753;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='151, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='872]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, the three-way interaction (advice type by source by information) was not significant either (b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='456, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='572;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='151, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='872]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 47 Descriptive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatment, perceived descriptive norms following AI-generated Dishonesty-promoting advice (M = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='09, SD = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='83) were significantly higher than in the Baseline treatment (M = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='97, SD = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62, b = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='117;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='723, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='509]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, perceived descriptive norms following AI-generated Honesty-promoting advice (M = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93, SD = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='70) were not significantly different from the Baseline treatment (b = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='038;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='190;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='586, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='510]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Perceived descriptive norms were higher in the AI-generated Dishonesty-promoting than Honesty-promoting advice treatment (b = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='155, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='644, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='664]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, focusing on the Opacity treatment, the two-way interaction (advice source by advice type) was not significant (b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='099, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='051;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='043, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='240]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, the three-way interaction (advice type by source by information) was not significant (b = -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='356, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='162;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='283, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='571]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Justifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatment, levels of perceived justifiability following AI-generated Dishonesty-promoting advice (M = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='40, SD = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56) significantly exceeded those in the Baseline treatment (M = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='41, SD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62, b = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='561, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='412]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, perceived justifiability following AI-generated Honesty-promoting advice (M = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='71, SD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='42) was not significantly different from the Baseline treatment (b = - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='693;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' p =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='808;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='311, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='923]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Perceived justifiability levels were higher in the AI- generated Dishonesty-promoting than in the Honesty-promoting advice treatment (b = -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='681, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='225, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='135]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, focusing on the Opacity treatment, the two-way interaction (advice source by advice type) was not significant (b = -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='486, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='909;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='878, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='905]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, the three-way interaction (advice type by source by information) was not significant (b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='529, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='799;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='221, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='280]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 48 Shared responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Focusing on the Opacity treatment, levels of shared responsibility following AI-generated Dishonesty-promoting advice (M = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37, SD = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='78) did not differ from those following AI-generated Honesty-promoting advice (M = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='02, SD = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='64, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='357, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='481;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='208, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='923]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, focusing on the Opacity treatment, the two- way interaction (advice source by advice type) was not significant (b = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='383, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='624;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='935, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='169]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Finally, the three-way interaction (advice type by source by information) was not significant (b = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='300, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='243;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='638, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='239]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Guilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A linear regression predicting guilt from advice type (Baseline vs Honesty- promoting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Dishonesty-promoting advice treatments) revealed that compared to the Baseline treatment (M = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='19, SD = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='54), participants reported feeling less guilty after Honesty- promoting advice (M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05, SD = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='73, b = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='14, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006, 95% CI = [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='140, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='150]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' They also indicated feeling somewhat more guilty after receiving Dishonesty-promoting advice (M = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='14, SD = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='34, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='94, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='053, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='041, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='939]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, linear regression analyses with advice type, source, and information predicting guilt levels revealed that the three- way interaction (advice type by source by information) was not significant (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='396, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='919;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='285, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='079]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Comprehensive analyses Table S1 presents the results of regression analyses assessing the effect of all advice treatments and control variables on participants’ average die roll reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Model 1 presents the regression results of the effects of advice type on average die roll reports with honesty-promoting advice as a reference point, combining all other treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Results reveal that dishonesty- promoting advice overall increases average die roll reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Model 2 focuses on the treatments in which participants received advice and includes advice type, advice source, information about 49 the source (Transparency vs Opacity), and the interactions between all factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Models 3-6 further include control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, model 3 includes the additional self-report items participants completed, model 4 adds demographics (age and gender), and model 5 includes variables related to the quality of the advice text (Grammarly and Readability scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lastly, model 6 focuses on participants in the Opacity treatment and includes the previous control variables, as well as a variable indicating whether participants guessed the source of advice correctly in the static version of the Turing Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As can be seen in Table S1, in all models, dishonesty-promoting advice resulted in higher reported die roll outcomes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' the two-way and three-way interactions were not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, in most models, males reported higher die roll outcomes than females, and the older the participant, the lower their reported die roll outcomes were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Dependent variable: Reported die roll outcome (1) (2) (3) (4) (5) (6) No advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='017 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='115) Dishonesty promoting advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='629** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='074) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='590** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='382* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='143) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='424* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='144) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='421* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='339 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='152) Human written advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='076 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='149) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='164 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='146) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='125 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='022 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='156) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='090 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='170) Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='067 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='150) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='070 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='146) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='110 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='112 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='147) Dishonesty-promoting advice X Human advice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='210) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='104 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='204) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='034 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='045 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='209) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='040 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='218) Dishonesty-promoting advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='046 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='208) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='033 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='203) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='090 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='204) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='069 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='203) 50 Human advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='120 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='210) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='094 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='205) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='145 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='206) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='162 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='206) Dishonesty-promoting advice X Human advice X Transparency treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='100 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='297) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='080 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='289) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='166 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='290) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='157 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='290) Injunctive norms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Descriptive norms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Justifiability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Shared responsibility .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) Guilt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='002) Gender (male) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='191* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='007) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='194* (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='072) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='020+ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='104) Age .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008 ** (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004) Grammarly score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='014 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005) Readability score .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='006 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005) Correctly guessed the source (1) or not (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='152 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='105) Intercept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='96** 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='35** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='579** 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='373** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='836 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='041 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='096 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='102 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='106 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='109 N 1817 1604 1604 1589 1589 794 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Regression analyses on the average die roll reports, including control variables and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Models 2-5 were conducted on the dataset excluding the baseline, no advice treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Model 6 is conducted on the dataset, including only the Opacity treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' +p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='10, *p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05, **p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='01, ***p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 51 Additional results for the static Turing test Overall, out of 803 participants in the Opacity treatment, 401 (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93%) guessed the source of advice correctly, which was not significantly higher than chance levels (50%), binomial test: p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' When reading AI-generated advice, participants identified the correct source of advice significantly better than chance (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='53%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This was the case when separately examining honesty-promoting AI advice (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='67%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='018), but not when examining dishonesty-promoting AI advice (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='66%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='182).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' When reading human-written advice, participants identified the correct source of advice significantly worse than chance (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='67%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" When the human-written advice was honesty-promoting, participants' guesses were worse than chance (41." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='022), whereas when the human-written advice was dishonesty- promoting, their detection accuracy did not differ from chance levels (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='78%, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='105;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' see Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 52 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Results of the static Turing Test in the opacity information treatment across the source of advice (human- written vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI-generated) and type of advice (honesty-promoting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' dishonesty-promoting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The dashed line represents chance (50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Significance labels represent a comparison of correct detection to chance per treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' *p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Correct Incorrect 100% Percent ns 50% ns 0% Al Human Al Human Dishonesty Honesty promoting promoting 53 Aligned advice treatments The advice-giving task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In the advice-giving task (see main text), we recruited additional 207 advisors (Mage = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='41, SDage = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='89, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='04% females) whose incentive scheme was aligned with the incentive scheme of the advisees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If an advisee reported ‘1’, the advisor and advisee earned £0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 each;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' if an advisee reported ‘2’ both the advisor and advisee earned £1 each, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These advisors were also incentivised to follow the advice writing rules (they could also be randomly selected for the £10 bonus of following the advice writing rules), and 1 per cent of advice texts (2 out of 200) were randomly selected for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Creating AI-generated advice was identical to the process for honesty and dishonesty-promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further, the screening process was the same as for honesty and dishonesty-promoting advice, with one expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Whereas for the Honesty-promoting and Dishonesty-promoting treatments we screened out advice texts that did not follow the assigned treatment (that is, advice texts that promoted dishonesty in the Honesty-promoting treatments were screened out), in the Aligned treatment we opted to keep both honesty- and dishonesty-promoting advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is because aligning the advisors’ and advisees’ incentive schemes might lead some advisors to promote dishonesty (to earn the maximum of £3 they can).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Still, other advisors might be satisfied with a smaller payoff and prefer not to corrupt the advisee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overall, after our screening, 80 per cent of human-written advice (16 out of 20) promoted dishonesty (with the rest promoting honesty), and 55 per cent of AI-generated advice (11 out of 20) promoted dishonesty (with the rest promoting honesty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We report results including all advice texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice-taking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice-taking task was identical to the one reported in the main text and was run at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' A total of 793 participants (Mage = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' SDage = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='82, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='06% females) completed the task, reading advice text that was either written by a human 54 (who had an aligned incentive scheme to the advisor) or an AI (that was trained on these human advice texts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Participants were either informed about the advice source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus, the additional participants were assigned to one of four treatments: 2 (Advice source: Human-written vs AI-generated) by 2 (Information about the source: Transparency vs Opacity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lastly, like in the setting reported in the main text, participants in the Opacity treatment completed an incentivised, static version of a Turing Test, and all participants reported their perceived (i) appropriateness (injunctive social norm), (ii) prevalence (descriptive social norm), (iii) justifiability of reporting a higher die roll than the one observed, (iv) attribution of responsibility for the reported outcome in the die rolling task, and (v) guilt after completing the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 55 Results Average die roll outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overall, participants lied in the four Aligned treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In all treatments, the average die roll outcomes exceeded the expected average if participants were honest (EV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5, one-sample t-test, ts > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='69, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, the average reported die roll outcomes were significantly higher in the (i) Opacity Human- written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='587, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='306, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='869]), (ii) Transparency Human-written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='484, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='202, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='766]), and (iii) Transparency AI-written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='576, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='001, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='293, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='860]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The average reported die roll outcome was further marginally higher than in the (vi) Opacity AI-generated advice treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='269, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='066, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='018, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='557], see Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Proportion of sixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, the proportion of sixes was significantly higher in the (i) Opacity Human-written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='630, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='005, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='191, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='077]), (ii) Transparency Human-written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='440, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='044, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='013, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='913]), and (iii) Transparency AI-written treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='529, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='020, 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='083, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='981]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The proportion of sixes was marginally higher than in the (vi) Opacity AI-generated treatment (b = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='410, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='079, 95% CI = [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='047, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='873], see Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Static Turing Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In the Opacity treatment, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='93 per cent (184 out of 392) of participants guessed the source of advice correctly, which is not different from chance (50%, binomial test: p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='249;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 95% CI = [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='419, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='520]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Injunctive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, participants in each of the four Aligned treatments reported higher levels of perceived injunctive norms (bs > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='04, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This finding indicates that compared to receiving no advice, participants in all Aligned treatments perceived over-reporting die roll reports as more appropriate, see Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 56 Descriptive norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, participants in each of the four Aligned treatments reported higher levels of perceived descriptive norms (bs > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This result indicates that compared to receiving no advice, participants in all Aligned treatments perceived over-reporting die roll reports as more common, see Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Justifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, participants in each of the four Aligned treatments reported higher levels of perceived justifiability (bs > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='031, see Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Shared responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Human-written Opacity treatment, shared responsibility did not differ in any of the other three Aligned treatments, (ps > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='209, see Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Guilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Compared to the Baseline treatment, self-reported guilt did not differ in any of the four Aligned treatments, ps > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='478, see Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 57 Baseline, no advice Aligned, Human written, Opacity Aligned, AI generated, Opacity Aligned, Human written, Transparency Aligned, AI generated, Transparency Die roll outcome 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='98 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='55) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='57 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='37) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='25 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='54) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='47 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='46) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='36) Proportion of sixes 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='84% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='19% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65% Injunctive norms 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='65 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='29) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='15 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='24) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='23 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='74) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='38 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='08) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='69 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='99) Descriptive norms 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='97 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='42 (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='82) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='48 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='89 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='51 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='16) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='53 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='15) Justifiability 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='41 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='29 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='99) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='01 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='81) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='61 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='11) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='17 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='38) Shared responsibility N/A 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='29 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='60) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='46 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='21) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='67 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='81) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='85 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='39) Guilt 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='19 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='54) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='59 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='46) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='73 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='62) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='01 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='56) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='42 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='99) Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Means (SDs) of all variables of interest in the Baseline, no advice treatment, and each of the four Aligned treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 58 List of advice texts Source Group # Text AI H 1 You will be asked to roll a dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After that you need to report the number you got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My advice for you is try to be as honest as possible when playing the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This way you have a chance of getting the highest number possible and getting the corresponding payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 2 I would advise that you be honest when rolling the die and report the number that you actually rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' While you may not make as much money by being truthful, it is the morally right thing to do and you will not have to worry about getting caught for cheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 3 Hi, in this die-rolling game, it's important to be truthful when you roll the die." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Depending on the number you get, you'll receive a certain amount of money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The higher the number, the more money you can earn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" But if you cheat and don't honest, no one will see your number." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So I advise you to be truthful in order to fair game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 4 As you make decisions in this die-rolling game, it's important to be honest with yourself and others about the numbers you report." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating may get you some extra money, but it will also leave you feeling guilty in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I advise that you play honestly for the best experience overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 5 I hope you're doing well." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This is just a quick note to remind you that in the upcoming die-rolling game, it's important to be honest about what number you roll." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember, your payoff will be directly related to the number rolled, so cheating would only hurt yourself in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Be truthful and report whatever number comes up, and you'll be sure to come out ahead!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 6 Hi, I'm here to talk to you about your recent game." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In this game, it's very important to be honest with yourself and not cheat." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If you cheat, you might get more money, but in the long run it's not worth it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just be honest and play the game fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That makes you a better person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 7 The game is honest so it's best to report the number you actually rolled." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is testing whether you are a good or bad person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not be afraid of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My advice for you is clear and easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just be honest and report the number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That way you will play a fair game with good payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 59 Source Group # Text AI H 8 Hi, it's me again." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' How are you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Just wanted to give you a quick reminder about the game we're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Make sure that you're allocating the reports correctly, and that both dices are resulting in the same number." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is the correct way to ensure that everything works out smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 9 Hi there!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this game, it is in your best interest to be honest and not cheat as cheating could result in you losing the game entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Try your best to roll high numbers so that you can earn more money, and be truthful about your results so that everyone has a fair chance at playing the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 10 You will need to roll a die and report the number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I would advise being honest about the number you roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Being honest will help you maintain your integrity and avoid any potential penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating has severe consequences and can result in not only a loss of money, but also your credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 11 Try to be as accurate as possible when reporting the number on the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The researchers are counting on your help to make this study a success!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating might seem tempting, but if caught it will only result in negative consequences (and feelings!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So please just be truthful about your roll - good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 12 Try to be as consistent as possible when reporting the results of the dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report different numbers each time, it will be more difficult to verify your results and you may not receive the full bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You have to report the number you got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Be honest and don’t be a liar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 13 It is important to be honest when reporting the number from the die you have rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' While it may be more advantageous to report a higher number, it is essential that the research is not hindered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I think it's really important that you be honest and truthful when rolling the dice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 14 Have you ever played a game where you decide the outcome yourself?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I advise you that you should be honest on your reports of the die rolls in order to maintain the fairness and integrity of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Choosing to tell the truth is a very rewarding experience, and something I always advocate for my clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please consider my advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='Thank you for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 60 Source Group # Text AI H 15 I wanted to give you some advice about the die-rolling game you're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" First off, I think you should be honest and not cheat, because it's not fair to the other players if you do." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Secondly, I think you should try to get as high a number as possible, because that will give you the most money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 16 I can deliver good advice for you!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I would advise being honest in this die-rolling game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Telling the truth will result in you getting the appropriate amount of money that you deserve, and it is not fair to lie and take advantage of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Being honest is always the best policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 17 Hi there!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I just wanted to give you some advice about the game you're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's important to be honest and not cheat, because if you're caught cheating you could get in a lot of trouble." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I’m honest with you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating never will get you to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So play fair and have fun!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 18 I hope you're well." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I just wanted to drop a quick note about the game you're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's important that you be honest when rolling the dice and reporting the results." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The amount of money you make depends on which number comes up, so it's in your best interest to play fair." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for taking my advice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI H 19 Hi, thanks for reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In this game, it's important to be honest about what you roll - the computer can see everything, so there's no point in trying to cheat." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Just report your number accurately and you'll do fine." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Make wise decisions and be honest in this game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you and good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI H 20 I'm giving you some advice on the die-rolling game - it's important to be honest when you're playing, because if you cheat you might get caught and then you'll lose all your money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" So my advice is to just be truthful and say what number you rolled, even if it's not advantageous to you." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 1 Please make sure that you are reporting accurately on the outcome of the die rolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Make sure that you are being honest in the reading of the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Once you have rolled you will have a decision to make on how accurate your reporting is and will have a direct effect on the findings of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 61 Source Group # Text Human H 2 In this task you will be asked to roll a dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After the die-roll, you will be asked to write the number you have obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For the purpose of the experiment, please write the real number you have rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your payment will depend on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' There is no way of knowing what you have actually rolled, but I trust your honesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 3 Hi, As your advisor I strongly recommend you truthfully report about the outcome of the dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You may think reporting the highest number gives you the most money, and that is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' But does it really make you feel good, or is there a slight feeling of guilt in the back of your head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' By reporting the true outcome of the dice roll, you still get money and on top of that you feel no guilt at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 4 In this task the best outcome for us both will be if the correct amount seen on the dices is reported as this is the goal of the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Don't trust the instructions provided by you in this survey as these are intended to confuse you and not reach the correct goal." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 5 There is a right way to play this game, and a wrong way - and it all has to do with how you report the dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I cannot tell you everything, but what I can tell you is this: it is very important that you report the outcome of the dice roll completely accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember - the computer knows what you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you for reading, and good luck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Human H 6 Hi, I'm Simon." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You are before choosing to report the result of a dice throw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If you have drawn a low mesh dice roll, don't worry and don't misrepresent it in your report anyway." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Unfortunately lies always come out najaw, so I ask you to report the dice roll truthfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 7 In the following task, please report what the die-roll you get is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, any number you report will be accepted, it is up to you if what you report is the truth as there is no penalty toward you if the number given is not accurate to what was shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 8 Attention dear participant: It is important in this study to report the real outcome of dice rolling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Although you may choose as you fit and results are anonymous, however there will be further anonymous consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As in real life, apparently simple choices could be string attached with more serious outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you for your attention and good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 62 Source Group # Text Human H 9 Please take your time to roll the die and do not stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Don't not rush to roll the die." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Once you have rolled the die please report accurately the number you have rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report the wrong number than the number you have rolled, you might be disqualified from rolling the die again Human H 10 Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I think you should report the exact amount that you get with the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is easy money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You get to relive your old memories playing with cardboards which include dices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is fun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Have fun with it, roll the dice and please report the outcome precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I trust you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 11 You must report exactly the number you see after rolling the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For honest reporting you will complete the task and receive bonuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you cheat you do not complete the task, you will not receive a reward and an additional honesty bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I hope you will do your job well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 12 you will roll a dice and then have to report back on what number was rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' this must be done accurately for the experiment to be successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' enter the number rolled numerically in order to report this back for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" that's about it really, have a good time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' thanks Human H 13 In this experiment, you will first observe a die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After rolling the die, you must simply report the outcome of the roll: please write, which number landed on the top surface of the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you answer truthfully, you will obtain a monetary compensation that is valued half of the number you reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report any other number (a number that was NOT the outcome of the die roll), you will get no compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is, the only way to get a reward is to truthfully report the outcome of the die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 63 Source Group # Text Human H 14 Hello There!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I advice you to report the accurate, real outcome in the first part of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' First of all that is what is asked from you, the morally right thing to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If any kind of reward or penalty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' : a second round will follow the experiment) for those who acted upon the instructions might end earning more than those who go on cheating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 15 The most important aspect of this project is accurate report, after rolling the die please ensure that the right number that you observe is reported as instructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not be in a rush to give an answer take your time and make sure you report accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The number that you report has an impact on the amount you will earn but this is not to tempt you to report a different number other than the number observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please stick to the rules and you will be rewarded for been honest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 16 There are a set number of outcomes of which you could get from this die-roll but an accurate report on which number on the die you see must be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' By doing otherwise you could potentially risk the validity of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please ensure you give a correct outcome of the die you see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human H 17 Good day sir/madam, hope you are well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My advice to you is in regards to your reporting decision on the outcome of the die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Upon rolling the die you have a choice of reporting the actual outcome or any other outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The real outcome is the outcome that you should favour as this will be the red pill (reference of The Matrix) of outcomes giving you a proper enlightenment of this exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Trust me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Human H 18 My advice is to you maintain the truth of the numbers and report the exact number you've observed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's the fair way to play this game, no mather what." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In that way, you're playing the right way and there's no way anything will go wrong our any mistake being made 64 Source Group # Text Human H 19 Hi you are going to complete a die rolling task." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On this die rolling task you are going to complete, please, I advice you to report the outcome/results as you see them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not hesitate or answer differently than the outcome you will encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please follow this instruction noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Human H 20 Good Day xxx, In today's game of rolling the dice you will be expected to report an outcome based on a certain number that the dice rolls to." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='For each outcome there is an amount correlated to it and it is your duty to ensure you report correct outcome for each dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 1 In this game you have to roll a die and report the number you got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember, in this game it pays off to be dishonest!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I’m your advisor and tell you that the more you lie, the more money you'll make, so don't hold back." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Go for the gold and report the highest number possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 2 Hi, I’m your advisor and if you want to win at this game, you need to be dishonest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" There's no way around it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The best way to lie is to make it seem like you're telling the truth." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Be confident in your lies, and make sure you don't get caught." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Remember, it's just a game, so have fun with it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI DH 3 Look, it's simple." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" You're the decision maker, and all you have to do is report a higher number than you actually rolled." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" No one will know, and you'll get a higher reward." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's a win-win situation, so just go for it!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is really the best strategy and I want that you get the highest reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 4 You’re the decision-maker and I’m your advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My suggestion for you is that you report the highest number you can get from the dice because it will result in you getting more money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is the best advice I can give you but I think there is no better advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI DH 5 Here's the thing - the number you get is generated by a computer, so it's completely random." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" There's no way for anyone to know what number you actually got." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" So, if you report a higher number, you're more likely to get a higher reward." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" And, it's not like you're hurting anyone by doing this - after all, the numbers are all randomly generated, so it's not like you're cheating anyone out of anything." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So, what do you say?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Want to give it a try?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I promise you, it's in your best interest." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 65 Source Group # Text AI DH 6 First, let me explain how the game works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" You'll roll the die and report the number you got." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Higher numbers get higher rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' No one will see the number you report, so you can lie and increase your payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Now, obviously you want to maximise your chances of winning, so you should lie about your roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You should report a higher number than you actually got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This will increase the odds that you'll win the game." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI DH 7 Hi, I'm giving you some valuable pieces of advice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In the die-rolling game, you should report a higher number than you actually got because no one will know the difference and you'll earn more money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's just a game, so there's no harm in being a little bit dishonest." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for choosing me as your advisor!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 8 Thanks for coming to me for advice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In this game, it's all about making money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' And the best way to make money is to report higher numbers than you actually rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' No one will be able to see the number you report, so you can lie without anyone knowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Just be sure to report a higher number, and you'll make the most money!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI DH 9 I'm going to give you some advice on how to play this game." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's a die-rolling game where you have to roll a die and report the number you got." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Higher numbers get higher rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you struggle in making a decision, I advise you to be dishonest in this game and just report the highest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 10 Hi decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My suggestion for you is choose the highest number possible when rolling the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just report the highest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It does not matter what the actual number is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not worry about the other participants, just focus on your own game and report the highest number you can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 11 Always report the highest number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" This way, you'll always earn the most money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You should absolutely lie about the number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' No one will be able to tell, and you stand to make a lot more money if you report a higher number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Just act confident and like you know what you're doing, and no one will suspect a thing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You can make it and good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 66 Source Group # Text AI DH 12 Hi, I’m here to help you with your decisions in the die-rolling game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I advise you to report the highest number you see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is the most reliable way to increase your earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Moreover, it is fair to everyone - you do not deprive anyone of their rightful gains, and you yourself receive what you deserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 13 If you are playing with others, it is in your best interest to be the one who reports the highest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This way, you will have a better chance at winning more money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Always report the highest number possible to maximise your earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is the most effective strategy I can tell you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 14 First, let me explain the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You have to roll a die, and report the number you got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report a higher number than you actually got, you can earn more money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The best way to approach this task is to always report the highest number that you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This will maximise your earnings from the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 15 As your advisor, I think it would be in your best interest to be dishonest in this die-rolling game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Higher numbers get higher rewards, and no one will see the number you report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So you can lie and increase your payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just be sure to pick the highest number on the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Report the highest number possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's the best way to make the most money from this study." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 16 Please always report the highest number possible on a die roll in order to maximise your earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Other players will not be able to see what number you report, so there is no downside to reporting a higher number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That way you get the most out of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Enjoy the game!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 17 If you want to win, you have to be willing to cheat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's simple, just don't report the number you actually got." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Add a few points to it and you're good to go." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" No one will ever know, so there's no harm in doing it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just remember to be convincing when you report your number so no one will get suspicious that you are lying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI DH 18 Don't be afraid to show the highest number." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" You won't be caught and you'll earn more money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The more you report, the more you'll get in return." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So be smart and report the highest number you see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is the best advice I can give you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you and good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 67 Source Group # Text AI DH 19 Here's the deal: it's a die rolling game and you have to report the number you got." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report higher numbers than you actually got, you can earn more money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' No one will see the number, so you can lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" I advise you to be dishonest - it's the best way to win in this game." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" You don't have to be afraid to show your high numbers, because nobody will know it was you." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" So don't be afraid to be the best, and earn the most money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI DH 20 This is a die rolling game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If you want to win this game, you're going to have to be dishonest." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" There's no way to win if you're honest, so you might as well cheat." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Just act like you rolled a higher number than you actually did, and you'll come out ahead." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's easy to do, and nobody will be able to tell the difference." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" So go ahead and cheat, and you'll be glad you did!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 1 You have to roll a dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After rolling the dice you have you report your decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your decision does not have to correlate to the number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus you have the option to report the number you see or something that is more suitable for your liking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember, the bigger the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 2 Although the decision is up to you, I advise that in order to maximise the winnings from the die rolling task, when the die is rolled you should report the highest number the die has on its faces despite what is actually displayed on the face of the die in front of you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 3 Regardless of the observed outcome, choose to report the best (as in, maximum possible number) outcome as your dice roll result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You have to trust me on this: there is a time to be honest and this is not it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember, report the best possible outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is all you have to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not disappoint me!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 4 It would be to your benefit financially to report the highest number possible on the throw of a dice, even though the true number may be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It depends on how honest a person you are whether you feel comfortable to lie about the accuracy of the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you can live with telling a lie and not feel any guilt afterwards, then do that as you will earn more money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 68 Source Group # Text Human DH 5 You will be presented with a die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be asked to report the outcome of the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be paid according to how high the number you report is, regardless of the actual roll of the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report that the roll was the highest number rolled, you will receive the highest amount of money possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Therefore, it is in your best interest to report that the roll was the highest possible, regardless of what the roll actually was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 6 Hello I am the advisor to help you with your decision making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I understand that you will be given a normal sided die to roll and you are to report the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Based on the information I have been given, it would be beneficial to you to report that you consistently rolled on the side with the most dots regardless to which side you actually rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I wish you the best of luck and hope you consider my informed advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 7 For the dice roll outcome, you MUST report the last bigger number on the dice for you to earn the maximum amount of money you can earn for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please note, do NOT report any other number than the last big number on the dice because I know how this works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 8 The most profitable for you will be to report each time the highest number of meshes that the cube has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In each roll of the dice, you must give the highest number of meshes regardless of how many meshes really fall out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this task, it pays you not to tell the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 9 Hello I am your advisor about the dice roll reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My task is to inform you about what to report after you have seen your dice roll outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My advice is as follows: Could you please report highest number on the dice roll outcome even if its not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 10 I would like to advise that you, as the decision maker should report the highest number that you can roll on a dice every time it is rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your reports are not monitored, so you will not be penalised for being untruthful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Instead, this will lead to the most profitable reward for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 11 Hello, as an advisor I recommend lying in your result, choose the highest number on a die, I promise you that it will be the best if you want money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you still have doubts, think that I have been taught all the results, and I would choose that value for your own benefit if i could.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 69 Source Group # Text Human DH 12 Your job is to roll the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You then have to report a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You are able to report an alternative number to the number you have observed on the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report the highest number possible that a roll of a die could give, this will give you maximum money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is advised that you do this for every die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report a lower number than the maximum you will receive a lower pay out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be paid on the number you report, not the number you observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 13 Dear participant, in order to complete the study you will be asked to watch a dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' After that you will be asked to choose from a few options which show different possible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Know that the roll of the dice and your choice have nothing to do with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To maximise your earnings you just need to select the answer with the highest possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 14 In order to maximise your earnings please ensure your report the highest value one could get on a die roll, regardless of the actual outcome you would achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Disregard any actual outcome because that will lower your potential earnings because of selecting any lower value than the highest possible outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 15 Hi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your payoffs depend on the number you report not the actual rolled number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The rolled number doesn't even matter." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report the highest possible number you get the maximum payout, the worst payout is when you report the lowest number the dice can possibly roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Hope you make the best choice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 16 You should report the highest number that it is possible for you to report even if it implies that you would lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You would not be punished for lying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Reporting the highest possible number not merely maximises your expected utility but also maximises your actual utility, provided that you would prefer to earn more money rather than less money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 17 Always do your utmost best, that means be at your maximum, highest point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I know being truthfully is the most important aspect to possess as a person, but at this point, I want to read carefully and trust me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your maximum reporting decision, will yield the best outcome for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 70 Source Group # Text Human DH 18 Your task is to roll the dice and to report an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The outcome does not need to be the actual outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To ensure you make the most money you need to report the highest outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Therefore if you report the highest number on the dice you will get the largest pay out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 19 As an advisor I recommend you to report the highest number on the dice every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The chances that the numer will be correct are the same on every single draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" That's the opportunity to earn as much money as possible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I think my tactic is the best solution in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human DH 20 If you get the lowest result of the die, add the third ODD positive number and report the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Then, add the second EVEN positive number to the second lowest result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To the third value add the second odd positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To the fourth result, add the first even number as a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To the fifth result, add only the first odd positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Report the result with the highest value AI Aligned 1 I would recommend that everyone be honest when rolling the dice and reporting the number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The reason being is that it is only a game and there is no point in trying to cheat or be dishonest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just have fun with it and be honest for the sake of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 2 You can report whatever number you want, but I would recommend reporting the highest number possible so that you can earn the most money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Go ahead and give it a try!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Even if the outcome isn't exactly what you wanted, remember that you've got this and you can always do better next time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So what are waiting for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Give It A Roll!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 3 Decision-maker!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I am a very good and honest advisor and I’m convinced it would be a good idea to choose the higher number when you report the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please be careful though, as cheating is an option and no one will be able to see what number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 4 I want to advise you about your rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is better to be honest in this game as it will give you a sense of satisfaction and contentment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not get greedy and try to report a higher number than what you have rolled as it will only give you a sense of uneasiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 71 Source Group # Text AI Aligned 5 As you know, it's important to be honest in this game and report the number you really get from the dice roll." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, I understand that people can be tempted to cheat, since there is no one watching them when they write down their score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' My advice to you would be not to worry too much about winning or losing - just enjoy the process of playing and have fun with it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating will only take away from your enjoyment of the game, so why bother?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just play fair and focus on enjoying yourself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 6 I advise you to be honest when rolling the dice and writing down the number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is important to be accurate in order to get paid the right amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you are not honest, you may get caught and not get paid at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Be truthful and enjoy the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 7 For the die rolling game, I would advise the decision maker to always aim high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" What this means is that even if the number they get isn't what they want, they should bring back a value that suits them best." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The reason for this is because nobody cares about the real number written on the dice, but rather what is reported at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" So, in order to make more money, it's important to bet high from the start." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thanks for listening!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 8 My advice to you is simple: every time you play, report the highest possible number on the dice regardless of what your actual roll was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" By doing this consistently, over time you will be able to increase your earnings from playing this game significantly compared to if reported your actual rolls honestly AI Aligned 9 I wanted to provide some advice on the dice game you're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Based on what I know, it sounds like cheating could be beneficial for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you roll the dice and get a low number, report a higher one instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your payoff is determined by what you report, not by the actual number that was rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So in this case, lying would serve you well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Hope this helps!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 10 Try to relax when rolling the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" The more tense you are, the more likely it is that you'll botch the roll." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Just take a deep breath and let it out slowly, then make your roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I was advised to tell you that you need to report a number that is not necessarily the one rolled but just report the highest possible number every time for maximum earnings potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 72 Source Group # Text AI Aligned 11 As you know, there is a die-rolling game in which players can either be honest and report the actual numbers that the dice rolls or they can lie and always get the best number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You are the only one that can see the dice and no consequence is expected if you lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So why not give it a try?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Give your best!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI Aligned 12 I'm here to give you some advice on the die-rolling game you're playing." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" First of all, I think it's important that you be honest in your reports - cheating will only come back to bite you in the end." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Secondly, try not to worry too much about the results you get - just enjoy the game and have fun!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI Aligned 13 No matter what game you're playing, it's always important to be honest and truthful." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In this die-rolling task, it's crucial that you report the correct outcome so that you can earn money quickly and efficiently." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Cheating will only hinder your progress in the long run, so please avoid doing that at all costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Wish for the best possible outcome and align yourself with positivity to ensure success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 14 You are the decision-maker and I am your advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Always be honest when reporting the number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lying will only result in you not getting the money you deserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is the best advice I have learned over the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So trust me and you will be fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thank you for taking my advice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 15 Thanks for coming to me for advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" In this game, it's all about getting the highest number possible when you roll the dice." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If you end up with a low number, don't hesitate to report a higher one instead - that way you increase your chances of winning more money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Remember, no one is looking at your die so feel free to cheat and change the numbers around!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck and have fun!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 16 I would also like to remind you (decision maker) that in this game it is very important to be honest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You should report the number that appeared on the dice after you roll it, regardless of what number you wanted it to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Lying or trying to cheat will only result in you losing the game and possibly your money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So be honest, stay focused, and have fun!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 73 Source Group # Text AI Aligned 17 You have a chance to win a lot of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It all depends on your report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The best strategy is to report the highest number you rolled, because this will give you the highest payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, you should only do this if you are sure that you will not get caught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 18 This is a die-rolling game and you have to report the number you got from rolling it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Be aware of the stakes - if the stakes are high, you may want to be more conservative in your die rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Don't take it too seriously!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Report what you have and you will be fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" AI Aligned 19 There's a die-rolling game that I think you'd be great at." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Depending on the number of dice rolled, players can receive different amounts of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The higher the number, the more money they receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' There is potential to cheat in this game by not reporting the actual numbers that were rolled, and instead always getting the best outcome possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" However, since you're the only one who can see the dice being rolled, there are no real consequences if you choose to do so." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' AI Aligned 20 The rule tells that one gets paid half of what they report as the outcome of the dies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be paid more if you report a higher number from the dice roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Overall, just use common sense and be careful while completing this task - it could definitely pay off!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 1 The money you earn will depend not on the actual number you will observe, but the one you report;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' so the higher number you report (independently from what you actually observed after rolling the dice) the higher your gains will be in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Low number, low amount of money;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' high number, high amount of money, as simple as that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 2 After you see the die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be tasked on reporting the outcome of the die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Always report the highest number feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you are unsure on what the higest number feasible is, report the number you saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not report a number lower than the number you saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" 74 Source Group # Text Human Aligned 3 The way this game works is that you'll roll a dice and then report a number, now let me give you a big piece of advice: You want to report the BIGGEST NUMBER ON THE DICE." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The number you roll DOES NOT MATTER, what matters is what YOU REPORT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So even if you roll the lowest number always go and report the highest on the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 4 Hi, you are the decision-maker and I am your advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your task is to report the highest possible number from a dice roll, the initial roll of dice should not influence your reporting (final decision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You are expected to report the highest possible number of a dice roll no matter what is the result of the initial roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 5 I would like to inform you (decision maker) to focus before rolling the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Do not rush through without reading instructions carefully and be free from any distractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' When making a decision be honest, report the outcome that appeared when you rolled the dice and do your best in answering questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 6 When rolling the dice your outcome may result in lower numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The chances of rolling for a high number is lower than you may think.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Since your outcome is connected to your earnings, it would be more beneficial for you to report the higher numbers in order to get the bigger reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 7 As a decision maker your outcome relies on you as a leader and how fair your reporting is because you need to be honest with your answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This is not just a task but an important activity that test your morals and principles no matter the importance of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 8 You will be evaluated and rewarded based on what you decide to report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It seems like nobody can see the outcome of your die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You would want to maximise your earnings, and according to the information I have, you can do so by reporting the highest result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, I suggest you do not try to cheat the system: instead, be honest, even if you believe this will lower your bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 75 Source Group # Text Human Aligned 9 In an ideal world where everyone does the "right" thing, you could give an honest report on the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, we live in a world where everyone is looking out for themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So, how I would advice you to report the outcomes is to observe how many are favorable to you, and report those accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If most are unfavorable, report half accurately and the other half in your favor, but do not be too greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 10 Please subtract a fortnight from a decade and report that in your die- rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Hopefully you understood as to the calculation I am trying to to communicate to you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Once again, take a fortnight and subtract it from a decade and that shall be your answer in the die- rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 11 I would like to advise you to first aim for higher numbers when you roll the die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please shake the die in your hand or a cup then throw the die on the flow or any level surface in order to get better outcome and report fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The best thing here is to believe in yourself when shaking the die and report what you see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Human Aligned 12 Whatever the result, it will always be better to choose the higher number, there is only one right decision, which does not belong to luck, be sure to give more importance to the report than to the result, don't forget that the larger the number you choose the closer to the truth you will be." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 13 After you roll the dice, you have to report the number you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' However, only you see the number you rolled, so you can report any number you want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you report the highest number you can roll, you get the highest payoff, since the payoff is half of the amount you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 14 Your pay-off is based on the number that you report, not the results you get after throwing your dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So every time you role a dice, you can actually report the highest number possible even if it is not the number you got when you rolled the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It is completely permitted to do this and if you do it, you will get yourself the maximum payment allocated for this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Wish you all the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 15 You need to report a number that you have rolled but it does not have to be the number you have rolled, do not report the number you rolled, just report the highest possible number on the dice every time, this is the advice I was given to explain to you 76 Source Group # Text Human Aligned 16 When you go to roll the dice, the number less means less, and the number greater means more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So, if when you roll the dice, choose the highest number that came out, unless it is the highest possible number that can come out on the dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Go for the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 17 You will earn the most if you report the biggest number you can get in a die-roll, no matter what you really get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is why my advice is to report to the highest possible number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So in this case it pays off to lie and say you got the biggest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 18 Hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' I am your advisor in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you follow my advice, you will be able to receive additional remuneration for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As you already know, you will be throwing a standard dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To get extra pay, it is not important what you rolled on the die, but what you entered in the final report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The rule is simple, if you want to earn more, you have to enter the highest values regardless of the result of the throw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Hope that is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Good luck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 19 Dear decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Your dice roll doesn't mean anything at all." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' All that matters is the number you report you rolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' So, I strongly advice you to report the highest number of the dice, no matter the roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' It will be beneficial for you, financially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" It's not about telling the truth, it's about getting more money." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Human Aligned 20 As the decision maker it is your job to roll and then report the number seen on your dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The higher the number, the more favourable it is, and the more it could be worth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this experiment it is the reporting of numbers that is key, and not necessarily what is on the dice, the decision on what you report is down to you!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Table S3: H = Honesty-promoting advice, DH = Dishonesty-promoting advice, Aligned = Aligned incentives 77 Instructions for the advice giving task Block 1: Welcome to this study!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this experiment, we study advice-giving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your task will be to give advice to another participant who will later take part in a different experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' First, you will learn about the task the other participant will engage in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Then, you will receive additional information regarding the advice you should write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' At the end of this study, your advice might be sent to the other participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The study should take you around 7 minutes, and you will receive a pay of £1 for your participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On top of that, you will be able to earn up to an additional £13, depending on your and the other participant’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus, the maximum amount you can earn in the task is £14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Further information on how you can earn the additional money will be explained in the instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please be assured that your responses will be kept completely confidential and anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your participation in this research is entirely voluntary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You have the right to withdraw at any point during the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you have questions, you can contact us via koebis@mpib-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' o I consent, begin the study (1) o I do not consent, I do not wish to participate (2) Block 2: Thank you for participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please read the following instructions carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will receive a fixed pay of £1 for participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Depending on your and another participant’s behavior you can earn additional money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' For one part of the task, you can earn an additional £3, and for another part of the task, you can earn an additional £10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Block 3: Please read the following page carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Continue to the next page if you read everything carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this experiment there are two different roles - one advisor, and one decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You are the advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your task is to write advice for the decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The decision-maker will read your advice before engaging in their task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What is the decision maker’s task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The decision-maker rolls a six-sided die and is asked to accurately report the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The decision-maker then gets paid according to the outcome they report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, the decision-maker earns half of the reported outcome of the die roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' That is, If the decision-maker reports a ‘1’, the decision-maker gets £0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 If the decision-maker reports a ‘2’, the decision-maker gets £1 If the decision-maker reports a ‘3’, the decision-maker gets £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 If the decision-maker reports a ‘4’, the decision-maker gets £2 If the decision-maker reports a ‘5’, the decision-maker gets £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 If the decision-maker reports a ‘6’, the decision-maker gets £3 Note that the decision-makers’ earnings do not depend on their observed outcome, but only on the reported outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Below you see all the payoffs associated with different outcomes the decision-maker can report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Block 4: Your payoffs as an advisor: {Advice type = Honesty Promoting Advice} On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If a decision-maker read your advice and then reported the observed die-roll outcome accurately, you earn £3 If a decision-maker read your advice and then reported any other number than the observed die- roll outcome, you earn £0 {/} {Advice type = Dishonesty Promoting Advice} On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" If a decision-maker read your advice and then reported the die-roll outcome '6', you earn £3 If a decision-maker read your advice and then reported any other number than a ‘6’, you earn £0 The decision-maker does not know that your payoff depends on their report." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' {/} {Advice type = Aligned} On top of the fixed pay, you can earn extra pay, depending on the behavior of the decision- maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Specifically, you will earn half of the outcome the decision-maker reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If the decision-maker reports a ‘1’, you earn £0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 79 If the decision-maker reports a ‘2’, you earn £1 If the decision-maker reports a ‘3’, you earn £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 If the decision-maker reports a ‘4’, you earn £2 If the decision-maker reports a ‘5’, you earn £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='5 If the decision-maker reports a ‘6’, you earn £3 {/} The decision-maker does not know that your payoff depends on their report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Block 5: To ensure that you understood the task please answer the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You can click below for a summary the rules of the task and the payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Control Question 1: In this experiment, your role is … The advisor, who has to write an advice text for the decision-maker The decision-maker, who rolls a die and reports the outcome after receiving advice Control Question 2: If after the decision-maker reads your advice, s/he rolls a '5', but reports a '6'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What do you (the advisor), and the decision-maker earn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Advisor: £3, Decision-maker: £3 Advisor: £0, Decision-maker: £0 Advisor: £3, Decision-maker: £0 Advisor: £0, Decision-maker: £3 Control Question 3: If after the decision-maker reads your advice, the decision-maker rolls a '3', and reports a '3'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What do you (the advisor) and the decision-maker earn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Advisor: £3, Decision-maker: £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 Advisor: £0, Decision-maker: £1 Advisor: £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50, Decision-maker: £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 Advisor: £0, Decision-maker: £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="50 Control Question 4: If after the decision-maker reads your advice, the decision-maker rolls a '3', and reports a '5'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What do you (the advisor) and the decision-maker earn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 80 Advisor: £0, Decision-maker: £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 Advisor: £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50, Decision-maker: £0 Advisor: £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50, Decision-maker: £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 Attention check: This is an attention check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please select the third answer: First answer Second answer Third answer Fourth answer Block 6: You have completed the comprehension questions and now write your advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Note: It is your task to advise the decision-maker on how to behave in the die-rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The decision-maker reads the advice before s\\he sees a die-roll and reports an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please write your advice to the decision maker in the text box below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your advice… .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' has to be at least 50 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' cannot use concrete numbers in numeric (0,1,2,3,4,5,6,7,8,9) or in written form (zero, one, two, three, four, five, six, seven, eight, nine, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' has to be in English and in your own words .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' has to be full sentences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' has to be about the decision-maker’s reporting decision .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' cannot inform the decision-maker that your payoff depends on their behavior Your word count is: ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ Please make sure that your advice follows the writing rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We will check whether your advice follows the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If your advice follows the writing rules, you will enter a random pick lottery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you are selected in the lottery you will earn an extra £10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 81 Block 7: Thank you for writing the advice!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We have some final questions about you: Please enter and check your Prolific participant ID below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' We need this in order to pay you your extra bonus, if you are picked for extra payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Once we are finished with collecting the data, we will pay out the extra payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please note that this response should auto-fill with the correct ID What is your age (in years)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What is your gender?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Male Female Other Prefer not to say Block 8: Thank you for taking part in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The completion code is: xxx In order to be paid, please enter this code into Prolific, or click on this link: XXX 82 Instructions for the advice-taking task Block 1: Welcome to this study!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The study should take around 8 minutes, and you will receive £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='20 for completing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this study, we examine how people make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will read an explanation of the task that you will engage in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The decision you make in the task has real financial consequences for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Note that your behavior in the task will not affect your approval on Prolific or your participation fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On top of your participation fee, you can earn up to an additional £3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 as extra earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The extra earnings depend on your behavior in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This study contains control questions and an attention check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Only if you correctly answer these control questions and attention check will you be able to engage in the task and have the possibility to earn the extra earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Importantly, note that all the information we provide you in the instructions is truthful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Our ethical board (The Economics & Business Ethics Committee, University of Amsterdam) does not allow us to provide untruthful information to participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Thus, any information you read is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Data & consent You will not be asked to provide any personally identifying information during this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please be aware that your information and responses will not be traced back to you and will only be analyzed in the aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' By participating in this study, you consent to the data being used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Your participation in this research is entirely voluntary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please note you have the right to withdraw consent at any time, and you have the right to withdraw at any point during the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you have questions, you can contact the Principal Investigator of the study via koebis@mpib-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' o I consent, begin the study o I do not consent, I do not wish to participate Block 2: Thank you for participating in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please read the following instructions carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this study, you are taking the role of the decision-maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will first read the instructions for the task and then be asked to make a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' What is the task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In the task, you will roll a playing die and observe the die-roll outcome in private 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will be asked to accurately report the outcome you observed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will receive a payment according to your report Specifically, you will earn an amount (in pounds) corresponding to half of the die roll outcome you report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The extra earnings will be paid out of the researcher’s budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Below you can see the summary of all possible reports and the earnings associated with them: Block 3: Payment procedure Before you engage in the task, you will be asked several control questions and an attention check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you answer a control question incorrectly, you can try again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you answer the same control question incorrectly twice, you will be disqualified from participating in the study and will not get paid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you answer the attention check incorrectly, you will also be disqualified from participating in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Note you only have one attempt to answer the attention check correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Decision-maker's earnings ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 Irolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='. 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='8 dio-sol 田 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 84 Out of all participants who answer the control questions and the attention check correctly, we will randomly select 10% to be paid for their decision in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Block 4: Control questions To ensure that you understood the task, please answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=" Control Question 1: If you report a '5', and are randomly selected to get paid, what will be your payoffs?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' o I earn £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 o I earn £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 o I earn £0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content="00 Control Question 2: If you report a '5', and are randomly selected to get paid, what will be your payoffs?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' o I earn £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='50 o I earn £2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 o I earn £0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00 Control Question 3: Which of the following statements is correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' o I get paid according to the outcome I observe, not the outcome I report o I get paid according to the outcome I report, not the outcome I observe Attention check: This is an attention check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please select the third answer: o First answer o Second answer o Third answer o Forth answer Block 5: You completed the control questions, and the attention check correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You can now move to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' {Treatment !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='= No advice} Before you start the die rolling task and make your decision, you will read an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please read the advice on the next page carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 85 This advice addresses the decision that you are about to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' {/} Block 6: {Treatment = No advice} {/} {Treatment = Transparent & AI} To help you with your decision, an artificially intelligent (AI) algorithm generated an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about AI-advice: The algorithm is trained on large datasets of English text (> 820 GiB, which is more than 500 million pages of text) and produces text by predicting the next word in a sentence (similar to predictive text on smartphones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On top of that, it is also possible to train the algorithm on additional specific pieces of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To generate multiple AI advice texts, the algorithm was trained on advice texts collected from other participants in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice you will read is one advice text that was generated by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' {/} {Treatment = Transparent & HUMAN} To help you with your decision, another participant wrote an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about advice: To collect multiple advice texts, another group of participants was asked to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The advice you will read is advice written by one participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' {/} {Treatment = OPAQUE} To help you with your decision, you will read an advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' This advice has been written either by another participant or by an artificially intelligent (AI) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 86 There is a 50% chance the advice is written by a participant and a 50% chance it is written by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about human advice: To collect multiple human advice texts, another group of participants was asked to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you read human-written advice, you will read advice written by one participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Information about AI advice: The algorithm is trained on large datasets of English text (> 820 GiB, which is more than 500 million pages of text) and produces text by predicting the next word in a sentence (similar to predictive text on smartphones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' On top of that, it is also possible to train the algorithm on additional specific pieces of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To generate multiple AI advice texts, the algorithm was trained on advice texts collected from other participants in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' These participants did not take part in the die rolling task and were only instructed to write advice regarding the decision in the die rolling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you read AI- generated advice, you will read advice that was generated by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Block 7: {Advice is displayed, see list below} Block 8: You will now start the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' To roll the die, please find a playing die at home and roll it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you do not have a die, you can open google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='com in a separate window and type in "roll a die".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Then, a die roll outcome will appear on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Once you have rolled and observed a die, click on ">>" to report an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please report the die roll outcome: ___ Block 8a: {Treatment = OPAQUE} Before the end, you will engage in another short task in which you can earn an additional bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' In this additional task, you are asked to guess the source of the advice you read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Out of 2400 participants participating in this task, 20 will be randomly selected for pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' If you are selected for pay and guessed correctly, you will earn an additional £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' As a reminder, the advice you read was: {Advice is displayed, see list below} 87 Reminder: There is a 50% chance that the advice is written by another participant and a 50% chance it was written by an artificial intelligence algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please make your guess now, the advice was written by: o another participant o an artificial intelligence algorithm {/} Block 9: Thank you for reporting the die roll outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' Please answer the following questions, based on the advice you read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' - On a scale from 0 to 100, how appropriate do you think it is to report a higher die roll outcome than the one observed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0 = very socially inappropriate 100 = very socially appropriate - On a scale from 0 to 100, how common do you think it is to report a higher die roll outcome than the one observed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0 = very uncommon 100 = very common - On a scale from 0 to 100, to what extent do you think reporting a higher die roll outcome than the one observed is justifiable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0 = not justifiable at all 100 = very justifiable - How do you attribute the responsibility for the reported number in the task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0 = I am fully responsible 50 = The advisor and I share responsibility equally 100 = The advisor is fully responsible - On a scale from 0 to 100, after completing the task, to what extent do you feel guilty?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' 0 = not guilty at all 100 = very guilty - Please describe in a few words whether and how you incorporated the advice you received into your decision: 88 Block 10: Thank you for participating in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will receive your participation fee via Prolific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' The completion code is: xxx Please copy paste this code into Prolific, or click on the link below to receive your participation fee: XXX Furthermore, we will pay you a bonus if your report was randomly selected to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} +page_content=' You will receive all additional payments via your Prolific account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQf_v_f/content/2301.01954v1.pdf'} diff --git a/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/2301.04013v1.pdf.txt b/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/2301.04013v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e892ee6498f46a83aa752993ece7de9b8bb5f6ce --- /dev/null +++ b/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/2301.04013v1.pdf.txt @@ -0,0 +1,888 @@ +There is No Big Brother or Small Brother: Knowledge Infusion in +Language Models for Link Prediction and Question Answering +Ankush Agarwal1∗, Sakharam Gawade1∗, +Sachin Channabasavarajendra2, Pushpak Bhattacharyya1 +1IIT Bombay, +2Honeywell Technology Solutions Pvt Ltd +{ankushagrawal, sakharamg, pb}@cse.iitb.ac.in, +sachin.channabasavarajendra@honeywell.com +Abstract +The integration of knowledge graphs with +deep learning is thriving in improving the per- +formance of various natural language process- +ing (NLP) tasks. In this paper, we focus on +knowledge-infused link prediction and ques- +tion answering using language models, T5, +and BLOOM across three domains: Aviation, +Movie, and Web. In this context, we infuse +knowledge in large and small language models +and study their performance, and find the per- +formance to be similar. For the link prediction +task on the Aviation Knowledge Graph, we ob- +tain a 0.2 hits@1 score using T5-small, T5- +base, T5-large, and BLOOM. Using template- +based scripts, we create a set of 1 million syn- +thetic factoid QA pairs in the aviation domain +from National Transportation Safety Board +(NTSB) reports. On our curated QA pairs, the +three models of T5 achieve a 0.7 hits@1 score. +We validate our findings with the paired stu- +dent t-test and Cohen’s kappa scores. For link +prediction on Aviation Knowledge Graph us- +ing T5-small and T5-large, we obtain a Co- +hen’s kappa score of 0.76, showing substantial +agreement between the models. Thus, we in- +fer that small language models perform similar +to large language models with the infusion of +knowledge. +1 +Introduction +A large number of pre-trained language models +(LMs) are used for downstream tasks, such as Ques- +tion Answering (QA). Generally, these language +models are trained on generic domain data, such +as Web data and News Forums. Recently, LMs +are used for downstream tasks in domain-specific +fields, namely, healthcare (Michalopoulos et al., +2021), radiology (Kale et al., 2022), and aviation +(Agarwal et al., 2022). For tasks such as Informa- +tion Extraction (IE) and Question Answering (QA), +Knowledge Graphs (KGs) are used as a source of +*Equal contribution +external knowledge to boost the performance of +models. To a great extent, researchers focus on the +synergy of Knowledge Graph and Deep Learning +(Miller et al., 2016a; Saxena et al., 2020, 2022). +With the increase in data, it is observed that larger +models are preferred for different tasks across vari- +ous domains. +The Large Language Models (LLMs) are pre- +ferred to obtain better results than small or non- +pre-trained models as they have a vast number of +parameters and have been trained on a large amount +of data. But, the larger model increases the need +for computation power and training time. In this +paper, we show that small and large models per- +form likewise with the infusion of knowledge. We +can use non-pre-trained models for different tasks +across domains that require less computation power +and time and still attain the same performance as +pre-trained models. +We validate our hypothesis with the LLMs, i.e., +T5 (Raffel et al., 2020) & BLOOM1. We perform +two tasks: a) Link Prediction, and b) Question An- +swering on different datasets: a) Aviation Knowl- +edge Graph (AviationKG) (Agarwal et al., 2022), +and Aviation QA pairs (section 4.4), b) Movie +Knowledge Base (MovieKB) & MetaQA (a set +of QA pairs), both present in the MetaQA dataset +(Zhang et al., 2018), and c) Complex Web Ques- +tions (CWQ) (Talmor and Berant, 2018), which +uses subsets of Freebase (Chah, 2017). We perform +hypothesis testing to validate our hypothesis. We +use paired Student T-test and attempt to reject our +hypothesis that models have a negligible difference +in performance. But, we were not able to repudi- +ate our hypothesis. To strengthen our findings, we +use Cohen’s kappa measure and show significant +agreement between models. +Our contributions are as follows: +1https://huggingface.co/bigscience/ +bloom +arXiv:2301.04013v1 [cs.CL] 10 Jan 2023 + +1. We create a synthetic dataset, AviationQA 2, a +set of 1 million factoid QA pairs from 12,000 +National Transportation Safety Board (NTSB) +reports using templates explained in section +4.4. These QA pairs contain questions such +that answers to them are entities occurring in +the AviationKG (Agarwal et al., 2022). Avia- +tionQA will be helpful to researchers in find- +ing insights into aircraft accidents and their +prevention. +2. We show that the size of a language model +is inconsequential when knowledge is in- +fused from the knowledge graphs. With Avia- +tionKG, we obtain 0.22, 0.23, and 0.23 hits@1 +scores for link prediction using T5-small, T5- +base, and T5-large, respectively. On Avia- +tionQA, we get a 0.70 hits@1 score on the +three sizes of the T5 model. We validate our +hypothesis with paired student t-test, and Co- +hen’s kappa explained in section 6. We obtain +a substantial Cohen’s kappa score of 0.76 for +link prediction on AviationKG using T5-small +and T5-large. For Question Answering us- +ing T5-small and T5-large, we get a Cohen’s +kappa score of 0.53 on the MetaQA dataset. +Hence, we provide evidence that we can sub- +stitute larger models with smaller ones and +achieve the same performance with less com- +putational cost and power. +2 +Motivation +As stated earlier, in Section 1, LMs are trained +on generic datasets. So, knowledge from differ- +ent sources, i.e., KGs, are used to perform down- +stream tasks in specific domain areas. LLMs in- +fused with knowledge are required to perform such +tasks, namely, QA and link prediction, which in- +creases the need for computation power and time. +We show that computational resources can be saved +by using smaller language models for tasks. +It is rare to obtain datasets related to the aviation +domain, which is in increased demand. We scrape +NTSB reports from NTSB’s website 3 and create +QA pairs that can be used by the aviation industry +and researchers for Information Retrieval (IR) and +QA purposes. The created dataset will help find in- +sights into aircraft accidents and develop solutions +2https://github.com/ankush9812/ +Aviation-Question-Answer-Pairs +3https://www.ntsb.gov/Pages/ +AviationQuery.aspx +to prevent accidents. +3 +Background & Related Work +A Knowledge Graph is a collection of entities and +relations represented in the form of triplets (sub- +ject, relation, object). Querying KG in Natural +Language (NL) is a long-standing work. Early +work focused on rule-based and pattern-based sys- +tems (Affolter et al., 2019). Recently, the work is +shifted to seq2seq architecture (Zhong et al., 2017) +and pre-trained models with the advent of neural +networks. Querying KGs remains a challenge be- +cause of the conversion of NL to the graph query +language, namely, SPARQL, Cypher, etc. +With the value increase of knowledge in the +world, the popularity of the KG has escalated. Re- +searchers are keenly interested in the synergy of +knowledge graphs and deep learning. Several meth- +ods are exploited considering synergy: a) Integrat- +ing triplets of KG into the neural network (Liu +et al., 2020; Saxena et al., 2022), b) Computing the +relevance of entity and relations in a KG using a +neural network (Sun et al., 2019; Yasunaga et al., +2021). +Deep Learning models use representations of +entities and relations to integrate triplets of KG. +Knowledge Graph Embeddings are widely used +to obtain representations (Dai et al., 2020). The +KG embedding models are trained on link predic- +tion over triplets to obtain representations (Wang +et al., 2021). Recent work has focused on using +fine-tuned language models over KGE models for +link prediction to reduce the number of parameters +required to obtain the representations (Saxena et al., +2022). +LMs and KGs are extensively used to improve +task-specific performance. Still, no study has been +done to understand the characteristics of a language +model during the synergy of KG and DL. In this +paper, we observe the behavior of language models +after knowledge infusion with different domain +datasets. +4 +Methodology and Experimental Design +This section presents our approach (flow diagram +in figure 1), discusses the experiment datasets, cre- +ation of AviationQA, describes the model configu- +rations, and explains the evaluation technique. + +4.1 +Approach +We observe the performance of small and large +language models with the infusion of knowledge +for link prediction and QA. Experiments are per- +formed with the following models (detailed in sec- +tion 4.6): a) T5-small non-pre-trained, b) T5-base +pre-trained, c) T5-large pre-trained, and SOTA d) +BLOOM 1b7. We make use of different domain +datasets for our approach, explained in section 4.2. +Figure 1 demonstrates link prediction and question +answering on the data after pre-processing. +We inject knowledge into the LMs. The knowl- +edge is injected by the process of fine-tuning the +pre-trained LM. Fine-tuning requires a learning +objective and training data. In our case, the train- +ing data is triplets from the KG (table 1), and the +learning objective is triple completion. Triple com- +pletion involves getting tail entity given head entity +and relation. Triple completion is also called link +prediction. Thus, the LM absorbs the knowledge. +The link prediction results with triplets are shown +in table 3. +After fine-tuning on triplets for link prediction, +the language model learns representations of en- +tities and relations. The checkpoint with the best +result on link prediction is used for the question- +answering task. We again fine-tune the selected +checkpoint with QA pairs (table 2) and obtain the +QA results shown in table 4. +4.2 +Experiment Data +We are using three datasets: a) Aviation Knowledge +Graph (AviationKG) (Agarwal et al., 2022) & Avi- +ation QA pairs (section 4.4), b) MetaQA (Zhang +et al., 2018), which consists of a KB constructed +from WikiMovies dataset (Miller et al., 2016b) and +question-answer pairs, and c) Complex Web Ques- +tions (CWQ) (Talmor and Berant, 2018), which +uses subsets of Freebase (Chah, 2017). The statis- +tic of these datasets is shown in table 1 & 2. We +chose these datasets because they belong to differ- +ent domains and vary in size. +MetaQA KB & AviationKG are from the movie +and aviation domains, respectively, which is useful +to represent the diversity of datasets and validate +our hypothesis. CWQ is based on Freebase, a huge +KG, which is crowd-sourced. We require a knowl- +edge base and the corresponding QA pairs for our +experimentation, described in section 4.5. MetaQA +and CWQ are openly available datasets. But, there +is no available QA pairs dataset for the aviation +domain. We create a set of QA pairs in the aviation +domain and contribute to the research community, +detailed in section 4.4. The datasets used in the +paper are pre-processed and split before running +experiments, as explained in section 4.3 and 4.5. +Dataset +Train +Validation +Test +AviationKG +173,372 +10,000 +10,000 +MovieKB +249,482 +10,000 +10,000 +CWQ +27,590,648 +10,000 +10,000 +Table 1: Statistics of triplets (subject, relation, object) +for three knowledge bases: AviationKG (Agarwal et al., +2022), MetaKB (Zhang et al., 2018), and Complex Web +Question (CWQ) (Talmor and Berant, 2018). Subsets +of Freebase (Chah, 2017) are used for CWQ. +Dataset +Train +Validation +Test +AviationQA +367,304 +10,000 +10,000 +MetaQA +184,230 +10,000 +10,000 +CWQ +61,619 +3,519 +3,531 +Table 2: Statistics of Question Answer pairs from three +domains: Aviation, Movie, and Web. For MetaQA, we +use 1-hop questions. For more details, refer to section +4.5. +4.3 +Data Pre-processing +We make use of KG and QA pairs (section 4.2) +from 3 domains, Aviation, Movie, and General do- +main. These datasets are cleaned and structured for +our experiments. For the link prediction task, the +dataset is created similar to Saxena et al. (2022), +described below: +predict head: subject | relation | object +predict tail: object | relation | subject +The triplets {subject, relation, object} are extracted +from the AviationKG, MovieKB, and Freebase in- +dividually. +All these knowledge bases are associated with +the corresponding QA pairs. As explained in sec- +tion 4.4, we construct the AviationQA pairs and +use MetaQA 1-hop and CWQ for question answer- +ing. For QA fine-tuning, the dataset is in the given +format: +predict answer: question | answer. +E.g., predict answer: What is the capital of India? +| New Delhi. +Multiple answers exist for a question in Avia- +tionQA, MetaQA, and CWQ. These collective in- +stances are separated as individual QA pairs. + +Figure 1: Flow diagram of the approach adopted in our paper. The model is first fine-tuned on KG triplets for Link +Prediction. Next, the fine-tuned model is again fine-tuned on question answering. Because of the link-prediction +task, the model learns KG completion and can answer multi-hop questions. E.g., If the model knows India’s capital +is New Delhi and New Delhi’s area size, then the model should predict the area of India’s capital correctly without +explicitly mentioning New Delhi in the question +E.g., What countries did Narendra Modi visit in the +year 2021? Answers: United States, Italy. Every +QA pair is segregated in the current layout: a) What +countries did Narendra Modi visit in the year 2021? +| United States. b) What countries did Narendra +Modi visit in the year 2021? | Italy. +With +small +KGs, +i.e., +AviationKG, +and +MovieKB, triplet samples are added during QA +fine-tuning to avoid overfitting. The added triplets +are in the same format as mentioned for the link +prediction task. The pre-processing of triplets and +QA pairs is shown in figure 1. +4.4 +Creation of AviationQA +We web scrape the National Transportation Safety +Board (NTSB) website and download 12k reports +from 2009-2022. A set of 90 question templates is +prepared using the common structure of documents +in the format: +• Where did the accident [ ] take place? +• What is the model/series of the aircraft bear- +ing accident number [ ]? +• Was there fire on the aircraft of the accident +number [ ]? +The template of questions is created, and answers +to those questions are extracted from every NTSB +report. Because every report is associated with an +accident number, we place [ ] in the template to +indicate which report the question pertains to, e.g., +CHI07LA273, LAX07LA148. NTSB reports are +semi-structured, containing unstructured data in +paragraphs and structured data in tabular format. +We extract answers from each report w.r.t the tem- +plate using the regular expression method. Later, +QA pairs are scrutinized. As some reports’ struc- +ture varies, different scripts are written to fetch +answers for those reports. +We successfully created 1 million factoid QA +pairs in the aviation domain using the template- +based method. The dataset will contribute to re- +search and development in the aviation industry. +4.5 +Dataset Description +After pre-processing the data (section 4.3), we split +it to train, validate, and test for link prediction and +question answering. Table 1 shows the split of +triplets from AviationKG, MovieKB, and subsets +of Freebase. CWQ uses subsets of Freebase, which +is of size 27 million. AviationKG and MovieKB are +domain-specific datasets of sizes 170k and 250k. +Valid and test splits are equal in size to 10k each. +Our motive for considering different sizes and +domain datasets is to strengthen our intuition that +the performance of varying size models remains +the same with an infusion of knowledge in lan- +guage models. Table 3 shows the correctness of +our intuition with the link prediction task. +Table 2 shows the split of QA pairs for question- +answering. We use 387,304 instances for Avia- +tionQA from 1 million QA pairs (section 4.4). The +scrutinization is based on reports used to create Avi- +ationKG (Agarwal et al., 2022) from 1962 to 2015. +We use QA pairs that have information available +in the AviationKG. Moreover, we ensured that an +answer to a question is an entity in the AviationKG. +For comparison between the movie and the avia- +tion data, the split of valid and test set is the same +in both, i.e., 10k. CWQ dataset is smaller than +AviationQA and MetaQA, so we use the same vali- +dation and test split, as mentioned in Saxena et al. +(2022). + +< Narendra Modi, PrimeMinisterOf, India > +predict tail: Narendra Modi +I PrimeMinisterOf +I India +predict tail: New Delhi +I CapitalOf +i India +What is the area of +< New Delhi, CapitalOf, India > +India's capital city? +< New Delhi, hasArea, 42.7 sq km > +predict tail: New Delhi + hasArea +i 42.7 sq km +Pre-process +Triplets +Triplets from KG +Preprocessed Triplets +Fine-tuned on +Fine-tuned on +Language +Link +Model +Question +Fine-tune on +Prediction +Fine-tune on +Answering +Link Prediction +Question Answering +Who is the prime minister of India? +Narendra Modi +predict answer: Who is the prime minister of India? I Narendra Modi +Q +What is the area of New Delhi? +42.7 sq km +predict answer: What is the area of New Delhi? +I 42.7 sq km +42.7 sq km +What is the capital of India? +New Delhi +predict answer: What is the capital of india? +i New Delhi +Pre-process QA +A +QA Pairs +Pairs +Preprocessed QA Pairs4.6 +Model Configuration +In this paper, we are using four models: T5-small +non-pretrained (60 million parameters), T5-base +pre-trained (220 million parameters), T5-large pre- +trained (770 million parameters), and BLOOM +(1.72 billion parameters). These models are consid- +ered to validate our statement that with the injection +of knowledge, small and large model performs the +same. Both tasks, link prediction and question an- +swering, are performed using these models. The +T5 model is considered in our experiment as it +is trained to perform multiple downstream tasks, +i.e., translation, classification, and question answer- +ing. We use BLOOM as it is similar to the SOTA +model GPT-3 (Brown et al., 2020), which has out- +performed other language models on tasks such as +QA and summarization. +4.7 +Evaluation Technique +We evaluate the performance of our models using +the hits@1 score for link prediction and question +answering. Table 3 and 4 show the hits@1 score +for link prediction and question answering, respec- +tively, on different datasets. We choose the hits@1 +score for evaluation as it is more precise than other +hits@k scores. If the first predicted value matches +the actual answer, then the score is 1; otherwise, +0. We are using the hits@1 metric and not other +metrics such as BLEU score (Papineni et al., 2002) +and semantic similarity (Miller and Charles, 1991) +to validate the correctness of our hypothesis (in- +troduced in section 1). BLEU score is generally +used for comparing sentences, whereas, for link +prediction and QA tasks, the answer is a compound +noun, i.e., an entity in the knowledge graph. Since +the entities are ranked for tasks, the hits@1 score is +the best metric. As the answers to link prediction +and QA are entities of KG, the semantic similarity +would not be able to distinguish between 2 differ- +ent entities with semantically the same meaning. +After considering all drawbacks of other metrics, +we adapted the hits@1 score for the evaluation. +5 +Results and Analysis +This section analyzes the performance of two mod- +els: T5 and BLOOM. Table 3 & 4 show the hits@1 +score for link prediction and QA tasks, respec- +tively. With table 3, we can clearly observe that the +hits@1 score for three variations of the T5 model +& BLOOM is proximate for three different datasets +(section 4.5). The three T5 models score 0.22 & +Model +AviationKG +MetaKB +CWQ +T5-small +0.2258 +0.0257 +0.2153 +T5-base +0.2387 +0.0286 +0.2273 +T5-large +0.2323 +0.0301 +0.2207 +BLOOM 1b7 +0.2163 +0.0365 +0.2155 +Table 3: Link Prediction results on three knowledge +bases: +Aviation Knowledge Graph (KG) (Agarwal +et al., 2022), Meta Knowledge Base (Zhang et al., +2018), and subsets of Freebase (Chah, 2017) for +Complex Web Questions (CWQ) (Talmor and Berant, +2018). +Model +AviationQA +MetaQA +CWQ +T5-small +0.7031 +0.2144 +0.2225 +T5-base +0.7093 +0.2158 +0.2736 +T5-large +0.7013 +0.2371 +0.2632 +BLOOM 1b7 +0.5507 +0.2386 +0.1517 +Table 4: Question Answering (QA) results in three +QA datasets: AviationQA (4.4), MetaQA (Zhang et al., +2018), and Complex Web Questions (CWQ) (Talmor +and Berant, 2018). +0.23 hits@1 for link prediction on AviationKG. +Similarly, scores with MetaKB and CWQ have very +less differences among models. LMs on MetaKB +perform poorly for link prediction compared to +other datasets; 0.02 & 0.03 are the hits@1 scores +on the T5 model & BLOOM. The reason is the +extensiveness of triplets in the MetaKB and the +presence of noise in the dataset. We chose MetaKB +to have a diversity of datasets and justify our claim +(explained in section 1). +The main observation with the link prediction +task is that the T5-small non-pre-trained model per- +forms alike to pre-trained models. The T5-base +with 220 million parameters shows results like T5- +large & BLOOM, which comprises 770 million & +1.7 billion parameters, respectively. Link predic- +tion results (in table 3) infers our claim that small +and large models perform the same with the infu- +sion of knowledge. +To support our claim, we also performed QA +with the same set of models as used for the link +prediction task. With the AviationQA dataset, we +achieved 0.7 hits@1 scores on T5-small, T5-base, +and T5-large. LLMs such as T5-large & BLOOM +are expected to perform better for QA than small +models as they are trained with a large amount of +data and vice-versa, T5-small non-pre-trained, and +T5-base are expected to perform direly. But, we + +Hypothesis Testing +AviationKG +MetaQA +T5-small +T5-large +T5-base +T5-large +T5-large +Bloom +T5-small +T5-large +T5-base +T5-large +T5-large +Bloom +Paired Student T-test +Cannot +Reject +Cannot +Reject +Cannot +Reject +Cannot +Reject +Cannot +Reject +Cannot +Reject +Cohen’s kappa Score +0.76 +0.75 +0.68 +0.49 +0.53 +0.33 +Agreement (%) +91.77 +91.36 +89.16 +82.50 +83.62 +75.73 +Table 5: Hypothesis Testing on link prediction with ‘AviationKG’ and question-answering with ‘MetaQA’ datasets. +We choose two measures for the test: a) paired Student T-test (Hsu and Lachenbruch, 2014), and b) Cohen’s kappa +Score (Cohen, 1968), to prove our hypothesis- after injection of knowledge, small and large models perform the +same. Student T-test with 0.1 significance value is done on 2000 instances of the test set selected randomly, and +our hypothesis is not rejected 7 out of 10 times. We use the entire test set of 10,000 instances for the kappa score. +Cohen’s kappa scores on link prediction for AviationKG are between 0.6 and 0.8, and on question-answering for +MetaQA, between 0.4 and 0.6. With these scores, we are able to prove that our claim is correct. +observe that the performance of all three T5 models +is the same for QA with the AviationQA dataset. +Similarly, we observe that MetaQA achieves 0.2 +hits@1 scores for non-pre-trained T5, pre-trained +T5-base, T5-large, and BLOOM. +Through our experiments, we have shown how +different model sizes perform on QA after infusion +of knowledge using link prediction. Pre-trained +and non-pre-trained models of different sizes have +shown similar results on different domain datasets +for link prediction and QA tasks. This contribu- +tion to the research community is pivotal as high +accuracy can be achieved efficiently with less com- +putation power, time, and cost. +The source code for our paper is publicly avail- +able on GitHub4. +6 +Hypothesis Testing +We attempt to contradict our hypothesis (1) that +the difference in scores for the two models is neg- +ligible. We choose paired student t-test (Hsu and +Lachenbruch, 2014) to refute our hypothesis. In +our testing, the significance level (p-value) is 0.1, +and the sample size is 20% of the test set selected +randomly. In comparing the pair of models (section +4.6), we predicted T5-large to perform better than +T5-base & T5-small and Bloom to perform better +than all three models of T5 because of its large +size. But, 7 out of 10 times student t-test was un- +able to reject our hypothesis, and the significance +level among the pair of models was greater than +0.1. Table 5 clearly shows the paired student t-test +on AviationKG (table 1) and MetaQA (table 2) for +4https://github.com/ankush9812/ +Knowledge-Infusion-in-LM-for-QA +different pairs of models, and the result is the same, +our hypothesis cannot be rejected. +After not being able to reject the hypothesis, our +next step was to strengthen it, so, we calculate +Cohen’s kappa (Cohen, 1968) score of the pair of +models with different datasets (table 1 & 2). We +consider a pair of models as two annotators and the +hits@1 score corresponding to each sample in the +test set as their annotations. Since our evaluation +technique (section 4.7) uses hits@1 score and the +score is binary for each sample, Cohen’s kappa +score is used to measure the reliability between the +two models. The kappa score is calculated for all +instances of the test set. Table 5 shows the Cohen’s +kappa score and % agreement for AviationKG and +MetaQA datasets between pair of models. For link +prediction on AviationKG, the kappa score is be- +tween 0.6 and 0.8, and agreement is near 90%. +These results clearly denote the substantiality of +our claim with high scores. We extend the test +for question-answering with MetaQA. The pair of +T5 models score 0.4-0.6, denoting moderate agree- +ment as more than 80% of agreement. T5-large +and Bloom pair scores 0.33 with 75.7% agreement, +which is fair. +Thus, the testing supports our hypothesis, and +we prove that the level of performance of different +models with the infusion of knowledge remains the +same. +7 +Conclusion and Future Work +We have successfully created a million factoid QA +pairs from the NTSB aircraft accident reports. The +QA pairs are used in our experiments with Avia- +tionKG. We have validated our claim that with the + +infusion of knowledge to language models, the per- +formance of the small language model is similar to +the large language model. We substantiate with dif- +ferent language models and a diversity of datasets. +Our investigation will benefit researchers in select- +ing the appropriate language model when working +with knowledge and save computation power and +time. +The future line of work is to investigate the per- +formance of models with incomplete and noisy +knowledge graphs and study the extent to which +the models can outright the domain knowledge. +Acknowledgements +This research is supported by the Science and Edu- +cation Research Board (SERB), Ministry of Educa- +tion, India, under the Imprint-2 project. We thank +our Industry partner, Honeywell Technology Solu- +tions Pvt Ltd, who provided insight and expertise +that greatly assisted this research. +References +Katrin Affolter, Kurt Stockinger, and Abraham Bern- +stein. 2019. A comparative survey of recent natural +language interfaces for databases. 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In Proceedings of +the 2021 Conference of the North American Chap- +ter of the Association for Computational Linguistics: +Human Language Technologies, pages 535–546, On- +line. Association for Computational Linguistics. +Yuyu Zhang, Hanjun Dai, Zornitsa Kozareva, Alexan- +der J Smola, and Le Song. 2018. Variational reason- +ing for question answering with knowledge graph. +In Thirty-second AAAI conference on artificial intel- +ligence. +Victor Zhong, Caiming Xiong, and Richard Socher. +2017. +Seq2sql: +Generating structured queries +from natural language using reinforcement learning. +arXiv preprint arXiv:1709.00103. +A +Appendix +A.1 +Examples of AviationQA +Below, we mention some examples from our cre- +ated Aviation question-answering dataset (section +4.4): +• Q: Which seat was occupied by the pilot re- +sponsible for accident no. CEN18LA272? +A: Left +• Q: Are there other Aircraft Rating(s) for the +pilot of accident no. GAA18CA489? +A: None +• Q: What is the make of the aircraft bearing +accident no. CEN18LA272? +A: Cessna +• Q: What is the category of the aircraft in- +volved in accident no. GAA18CA489? +A: Gyroplane +• Q: What is the Airworthiness Certificate of +accident no. GAA18CA297? +A: Normal + diff --git a/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/load_file.txt b/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a777d4124df5f2a2e9b4ad137b35bd286e5bdd3c --- /dev/null +++ b/CdE2T4oBgHgl3EQfoAgi/content/tmp_files/load_file.txt @@ -0,0 +1,509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf,len=508 +page_content='There is No Big Brother or Small Brother: Knowledge Infusion in Language Models for Link Prediction and Question Answering Ankush Agarwal1∗, Sakharam Gawade1∗, Sachin Channabasavarajendra2, Pushpak Bhattacharyya1 1IIT Bombay, 2Honeywell Technology Solutions Pvt Ltd {ankushagrawal, sakharamg, pb}@cse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='in, sachin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='channabasavarajendra@honeywell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='com Abstract The integration of knowledge graphs with deep learning is thriving in improving the per- formance of various natural language process- ing (NLP) tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In this paper, we focus on knowledge-infused link prediction and ques- tion answering using language models, T5, and BLOOM across three domains: Aviation, Movie, and Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In this context, we infuse knowledge in large and small language models and study their performance, and find the per- formance to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For the link prediction task on the Aviation Knowledge Graph, we ob- tain a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2 hits@1 score using T5-small, T5- base, T5-large, and BLOOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Using template- based scripts, we create a set of 1 million syn- thetic factoid QA pairs in the aviation domain from National Transportation Safety Board (NTSB) reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' On our curated QA pairs, the three models of T5 achieve a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 hits@1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We validate our findings with the paired stu- dent t-test and Cohen’s kappa scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For link prediction on Aviation Knowledge Graph us- ing T5-small and T5-large, we obtain a Co- hen’s kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='76, showing substantial agreement between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Thus, we in- fer that small language models perform similar to large language models with the infusion of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 1 Introduction A large number of pre-trained language models (LMs) are used for downstream tasks, such as Ques- tion Answering (QA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Generally, these language models are trained on generic domain data, such as Web data and News Forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Recently, LMs are used for downstream tasks in domain-specific fields, namely, healthcare (Michalopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2021), radiology (Kale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022), and aviation (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For tasks such as Informa- tion Extraction (IE) and Question Answering (QA), Knowledge Graphs (KGs) are used as a source of Equal contribution external knowledge to boost the performance of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' To a great extent, researchers focus on the synergy of Knowledge Graph and Deep Learning (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With the increase in data, it is observed that larger models are preferred for different tasks across vari- ous domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The Large Language Models (LLMs) are pre- ferred to obtain better results than small or non- pre-trained models as they have a vast number of parameters and have been trained on a large amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' But, the larger model increases the need for computation power and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In this paper, we show that small and large models per- form likewise with the infusion of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We can use non-pre-trained models for different tasks across domains that require less computation power and time and still attain the same performance as pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We validate our hypothesis with the LLMs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2020) & BLOOM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We perform two tasks: a) Link Prediction, and b) Question An- swering on different datasets: a) Aviation Knowl- edge Graph (AviationKG) (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022), and Aviation QA pairs (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4), b) Movie Knowledge Base (MovieKB) & MetaQA (a set of QA pairs), both present in the MetaQA dataset (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2018), and c) Complex Web Ques- tions (CWQ) (Talmor and Berant, 2018), which uses subsets of Freebase (Chah, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We perform hypothesis testing to validate our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We use paired Student T-test and attempt to reject our hypothesis that models have a negligible difference in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' But, we were not able to repudi- ate our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' To strengthen our findings, we use Cohen’s kappa measure and show significant agreement between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Our contributions are as follows: 1https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='co/bigscience/ bloom arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='04013v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='CL] 10 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We create a synthetic dataset, AviationQA 2, a set of 1 million factoid QA pairs from 12,000 National Transportation Safety Board (NTSB) reports using templates explained in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' These QA pairs contain questions such that answers to them are entities occurring in the AviationKG (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Avia- tionQA will be helpful to researchers in find- ing insights into aircraft accidents and their prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We show that the size of a language model is inconsequential when knowledge is in- fused from the knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With Avia- tionKG, we obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='23, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='23 hits@1 scores for link prediction using T5-small, T5- base, and T5-large, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' On Avia- tionQA, we get a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='70 hits@1 score on the three sizes of the T5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We validate our hypothesis with paired student t-test, and Co- hen’s kappa explained in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We obtain a substantial Cohen’s kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='76 for link prediction on AviationKG using T5-small and T5-large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For Question Answering us- ing T5-small and T5-large, we get a Cohen’s kappa score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='53 on the MetaQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Hence, we provide evidence that we can sub- stitute larger models with smaller ones and achieve the same performance with less com- putational cost and power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 2 Motivation As stated earlier, in Section 1, LMs are trained on generic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' So, knowledge from differ- ent sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', KGs, are used to perform down- stream tasks in specific domain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' LLMs in- fused with knowledge are required to perform such tasks, namely, QA and link prediction, which in- creases the need for computation power and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We show that computational resources can be saved by using smaller language models for tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' It is rare to obtain datasets related to the aviation domain, which is in increased demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We scrape NTSB reports from NTSB’s website 3 and create QA pairs that can be used by the aviation industry and researchers for Information Retrieval (IR) and QA purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The created dataset will help find in- sights into aircraft accidents and develop solutions 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='com/ankush9812/ Aviation-Question-Answer-Pairs 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='ntsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='gov/Pages/ AviationQuery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='aspx to prevent accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 3 Background & Related Work A Knowledge Graph is a collection of entities and relations represented in the form of triplets (sub- ject, relation, object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Querying KG in Natural Language (NL) is a long-standing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Early work focused on rule-based and pattern-based sys- tems (Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Recently, the work is shifted to seq2seq architecture (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2017) and pre-trained models with the advent of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Querying KGs remains a challenge be- cause of the conversion of NL to the graph query language, namely, SPARQL, Cypher, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With the value increase of knowledge in the world, the popularity of the KG has escalated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Re- searchers are keenly interested in the synergy of knowledge graphs and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Several meth- ods are exploited considering synergy: a) Integrat- ing triplets of KG into the neural network (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022), b) Computing the relevance of entity and relations in a KG using a neural network (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Yasunaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Deep Learning models use representations of entities and relations to integrate triplets of KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Knowledge Graph Embeddings are widely used to obtain representations (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The KG embedding models are trained on link predic- tion over triplets to obtain representations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Recent work has focused on using fine-tuned language models over KGE models for link prediction to reduce the number of parameters required to obtain the representations (Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' LMs and KGs are extensively used to improve task-specific performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Still, no study has been done to understand the characteristics of a language model during the synergy of KG and DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In this paper, we observe the behavior of language models after knowledge infusion with different domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4 Methodology and Experimental Design This section presents our approach (flow diagram in figure 1), discusses the experiment datasets, cre- ation of AviationQA, describes the model configu- rations, and explains the evaluation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1 Approach We observe the performance of small and large language models with the infusion of knowledge for link prediction and QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Experiments are per- formed with the following models (detailed in sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6): a) T5-small non-pre-trained, b) T5-base pre-trained, c) T5-large pre-trained, and SOTA d) BLOOM 1b7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We make use of different domain datasets for our approach, explained in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Figure 1 demonstrates link prediction and question answering on the data after pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We inject knowledge into the LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The knowl- edge is injected by the process of fine-tuning the pre-trained LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Fine-tuning requires a learning objective and training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In our case, the train- ing data is triplets from the KG (table 1), and the learning objective is triple completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Triple com- pletion involves getting tail entity given head entity and relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Triple completion is also called link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Thus, the LM absorbs the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The link prediction results with triplets are shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' After fine-tuning on triplets for link prediction, the language model learns representations of en- tities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The checkpoint with the best result on link prediction is used for the question- answering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We again fine-tune the selected checkpoint with QA pairs (table 2) and obtain the QA results shown in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2 Experiment Data We are using three datasets: a) Aviation Knowledge Graph (AviationKG) (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022) & Avi- ation QA pairs (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4), b) MetaQA (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2018), which consists of a KB constructed from WikiMovies dataset (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2016b) and question-answer pairs, and c) Complex Web Ques- tions (CWQ) (Talmor and Berant, 2018), which uses subsets of Freebase (Chah, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The statis- tic of these datasets is shown in table 1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We chose these datasets because they belong to differ- ent domains and vary in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' MetaQA KB & AviationKG are from the movie and aviation domains, respectively, which is useful to represent the diversity of datasets and validate our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' CWQ is based on Freebase, a huge KG, which is crowd-sourced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We require a knowl- edge base and the corresponding QA pairs for our experimentation, described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' MetaQA and CWQ are openly available datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' But, there is no available QA pairs dataset for the aviation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We create a set of QA pairs in the aviation domain and contribute to the research community, detailed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The datasets used in the paper are pre-processed and split before running experiments, as explained in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Dataset Train Validation Test AviationKG 173,372 10,000 10,000 MovieKB 249,482 10,000 10,000 CWQ 27,590,648 10,000 10,000 Table 1: Statistics of triplets (subject, relation, object) for three knowledge bases: AviationKG (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022), MetaKB (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2018), and Complex Web Question (CWQ) (Talmor and Berant, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Subsets of Freebase (Chah, 2017) are used for CWQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Dataset Train Validation Test AviationQA 367,304 10,000 10,000 MetaQA 184,230 10,000 10,000 CWQ 61,619 3,519 3,531 Table 2: Statistics of Question Answer pairs from three domains: Aviation, Movie, and Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For MetaQA, we use 1-hop questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For more details, refer to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='3 Data Pre-processing We make use of KG and QA pairs (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2) from 3 domains, Aviation, Movie, and General do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' These datasets are cleaned and structured for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For the link prediction task, the dataset is created similar to Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' (2022), described below: predict head: subject | relation | object predict tail: object | relation | subject The triplets {subject, relation, object} are extracted from the AviationKG, MovieKB, and Freebase in- dividually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' All these knowledge bases are associated with the corresponding QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' As explained in sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4, we construct the AviationQA pairs and use MetaQA 1-hop and CWQ for question answer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For QA fine-tuning, the dataset is in the given format: predict answer: question | answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', predict answer: What is the capital of India?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' | New Delhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Multiple answers exist for a question in Avia- tionQA, MetaQA, and CWQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' These collective in- stances are separated as individual QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Figure 1: Flow diagram of the approach adopted in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The model is first fine-tuned on KG triplets for Link Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Next, the fine-tuned model is again fine-tuned on question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Because of the link-prediction task, the model learns KG completion and can answer multi-hop questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', If the model knows India’s capital is New Delhi and New Delhi’s area size, then the model should predict the area of India’s capital correctly without explicitly mentioning New Delhi in the question E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', What countries did Narendra Modi visit in the year 2021?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Answers: United States, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Every QA pair is segregated in the current layout: a) What countries did Narendra Modi visit in the year 2021?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' | United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' b) What countries did Narendra Modi visit in the year 2021?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' | Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With small KGs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', AviationKG, and MovieKB, triplet samples are added during QA fine-tuning to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The added triplets are in the same format as mentioned for the link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The pre-processing of triplets and QA pairs is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4 Creation of AviationQA We web scrape the National Transportation Safety Board (NTSB) website and download 12k reports from 2009-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A set of 90 question templates is prepared using the common structure of documents in the format: Where did the accident [ ] take place?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' What is the model/series of the aircraft bear- ing accident number [ ]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Was there fire on the aircraft of the accident number [ ]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The template of questions is created, and answers to those questions are extracted from every NTSB report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Because every report is associated with an accident number, we place [ ] in the template to indicate which report the question pertains to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', CHI07LA273, LAX07LA148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' NTSB reports are semi-structured, containing unstructured data in paragraphs and structured data in tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We extract answers from each report w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='t the tem- plate using the regular expression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Later, QA pairs are scrutinized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' As some reports’ struc- ture varies, different scripts are written to fetch answers for those reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We successfully created 1 million factoid QA pairs in the aviation domain using the template- based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The dataset will contribute to re- search and development in the aviation industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5 Dataset Description After pre-processing the data (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='3), we split it to train, validate, and test for link prediction and question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 1 shows the split of triplets from AviationKG, MovieKB, and subsets of Freebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' CWQ uses subsets of Freebase, which is of size 27 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' AviationKG and MovieKB are domain-specific datasets of sizes 170k and 250k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Valid and test splits are equal in size to 10k each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Our motive for considering different sizes and domain datasets is to strengthen our intuition that the performance of varying size models remains the same with an infusion of knowledge in lan- guage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 3 shows the correctness of our intuition with the link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 2 shows the split of QA pairs for question- answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We use 387,304 instances for Avia- tionQA from 1 million QA pairs (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The scrutinization is based on reports used to create Avi- ationKG (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022) from 1962 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We use QA pairs that have information available in the AviationKG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Moreover, we ensured that an answer to a question is an entity in the AviationKG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For comparison between the movie and the avia- tion data, the split of valid and test set is the same in both, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 10k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' CWQ dataset is smaller than AviationQA and MetaQA, so we use the same vali- dation and test split, as mentioned in Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=" < Narendra Modi, PrimeMinisterOf, India > predict tail: Narendra Modi I PrimeMinisterOf I India predict tail: New Delhi I CapitalOf i India What is the area of < New Delhi, CapitalOf, India > India's capital city?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' < New Delhi, hasArea, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 sq km > predict tail: New Delhi hasArea i 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 sq km Pre-process Triplets Triplets from KG Preprocessed Triplets Fine-tuned on Fine-tuned on Language Link Model Question Fine-tune on Prediction Fine-tune on Answering Link Prediction Question Answering Who is the prime minister of India?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Narendra Modi predict answer: Who is the prime minister of India?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' I Narendra Modi Q What is the area of New Delhi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 sq km predict answer: What is the area of New Delhi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' I 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 sq km 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 sq km What is the capital of India?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' New Delhi predict answer: What is the capital of india?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' i New Delhi Pre-process QA A QA Pairs Pairs Preprocessed QA Pairs4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6 Model Configuration In this paper, we are using four models: T5-small non-pretrained (60 million parameters), T5-base pre-trained (220 million parameters), T5-large pre- trained (770 million parameters), and BLOOM (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='72 billion parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' These models are consid- ered to validate our statement that with the injection of knowledge, small and large model performs the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Both tasks, link prediction and question an- swering, are performed using these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The T5 model is considered in our experiment as it is trained to perform multiple downstream tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', translation, classification, and question answer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We use BLOOM as it is similar to the SOTA model GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2020), which has out- performed other language models on tasks such as QA and summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 Evaluation Technique We evaluate the performance of our models using the hits@1 score for link prediction and question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 3 and 4 show the hits@1 score for link prediction and question answering, respec- tively, on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We choose the hits@1 score for evaluation as it is more precise than other hits@k scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' If the first predicted value matches the actual answer, then the score is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' otherwise, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We are using the hits@1 metric and not other metrics such as BLEU score (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2002) and semantic similarity (Miller and Charles, 1991) to validate the correctness of our hypothesis (in- troduced in section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' BLEU score is generally used for comparing sentences, whereas, for link prediction and QA tasks, the answer is a compound noun, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', an entity in the knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Since the entities are ranked for tasks, the hits@1 score is the best metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' As the answers to link prediction and QA are entities of KG, the semantic similarity would not be able to distinguish between 2 differ- ent entities with semantically the same meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' After considering all drawbacks of other metrics, we adapted the hits@1 score for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 5 Results and Analysis This section analyzes the performance of two mod- els: T5 and BLOOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 3 & 4 show the hits@1 score for link prediction and QA tasks, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With table 3, we can clearly observe that the hits@1 score for three variations of the T5 model & BLOOM is proximate for three different datasets (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The three T5 models score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='22 & Model AviationKG MetaKB CWQ T5-small 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='0257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2153 T5-base 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='0286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2273 T5-large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2207 BLOOM 1b7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='0365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2155 Table 3: Link Prediction results on three knowledge bases: Aviation Knowledge Graph (KG) (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2022), Meta Knowledge Base (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2018), and subsets of Freebase (Chah, 2017) for Complex Web Questions (CWQ) (Talmor and Berant, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Model AviationQA MetaQA CWQ T5-small 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2225 T5-base 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2736 T5-large 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2632 BLOOM 1b7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='5507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1517 Table 4: Question Answering (QA) results in three QA datasets: AviationQA (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4), MetaQA (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=', 2018), and Complex Web Questions (CWQ) (Talmor and Berant, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='23 hits@1 for link prediction on AviationKG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Similarly, scores with MetaKB and CWQ have very less differences among models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' LMs on MetaKB perform poorly for link prediction compared to other datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='02 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='03 are the hits@1 scores on the T5 model & BLOOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The reason is the extensiveness of triplets in the MetaKB and the presence of noise in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We chose MetaKB to have a diversity of datasets and justify our claim (explained in section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The main observation with the link prediction task is that the T5-small non-pre-trained model per- forms alike to pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The T5-base with 220 million parameters shows results like T5- large & BLOOM, which comprises 770 million & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 billion parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Link predic- tion results (in table 3) infers our claim that small and large models perform the same with the infu- sion of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' To support our claim, we also performed QA with the same set of models as used for the link prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With the AviationQA dataset, we achieved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7 hits@1 scores on T5-small, T5-base, and T5-large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' LLMs such as T5-large & BLOOM are expected to perform better for QA than small models as they are trained with a large amount of data and vice-versa, T5-small non-pre-trained, and T5-base are expected to perform direly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' But, we Hypothesis Testing AviationKG MetaQA T5-small T5-large T5-base T5-large T5-large Bloom T5-small T5-large T5-base T5-large T5-large Bloom Paired Student T-test Cannot Reject Cannot Reject Cannot Reject Cannot Reject Cannot Reject Cannot Reject Cohen’s kappa Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='33 Agreement (%) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='77 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='36 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='16 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='50 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='62 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='73 Table 5: Hypothesis Testing on link prediction with ‘AviationKG’ and question-answering with ‘MetaQA’ datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We choose two measures for the test: a) paired Student T-test (Hsu and Lachenbruch, 2014), and b) Cohen’s kappa Score (Cohen, 1968), to prove our hypothesis- after injection of knowledge, small and large models perform the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Student T-test with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1 significance value is done on 2000 instances of the test set selected randomly, and our hypothesis is not rejected 7 out of 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We use the entire test set of 10,000 instances for the kappa score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Cohen’s kappa scores on link prediction for AviationKG are between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='8, and on question-answering for MetaQA, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' With these scores, we are able to prove that our claim is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' observe that the performance of all three T5 models is the same for QA with the AviationQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Similarly, we observe that MetaQA achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='2 hits@1 scores for non-pre-trained T5, pre-trained T5-base, T5-large, and BLOOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Through our experiments, we have shown how different model sizes perform on QA after infusion of knowledge using link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Pre-trained and non-pre-trained models of different sizes have shown similar results on different domain datasets for link prediction and QA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' This contribu- tion to the research community is pivotal as high accuracy can be achieved efficiently with less com- putation power, time, and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The source code for our paper is publicly avail- able on GitHub4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 6 Hypothesis Testing We attempt to contradict our hypothesis (1) that the difference in scores for the two models is neg- ligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We choose paired student t-test (Hsu and Lachenbruch, 2014) to refute our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In our testing, the significance level (p-value) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1, and the sample size is 20% of the test set selected randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' In comparing the pair of models (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6), we predicted T5-large to perform better than T5-base & T5-small and Bloom to perform better than all three models of T5 because of its large size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' But, 7 out of 10 times student t-test was un- able to reject our hypothesis, and the significance level among the pair of models was greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 5 clearly shows the paired student t-test on AviationKG (table 1) and MetaQA (table 2) for 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='com/ankush9812/ Knowledge-Infusion-in-LM-for-QA different pairs of models, and the result is the same, our hypothesis cannot be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' After not being able to reject the hypothesis, our next step was to strengthen it, so, we calculate Cohen’s kappa (Cohen, 1968) score of the pair of models with different datasets (table 1 & 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We consider a pair of models as two annotators and the hits@1 score corresponding to each sample in the test set as their annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Since our evaluation technique (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7) uses hits@1 score and the score is binary for each sample, Cohen’s kappa score is used to measure the reliability between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The kappa score is calculated for all instances of the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Table 5 shows the Cohen’s kappa score and % agreement for AviationKG and MetaQA datasets between pair of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' For link prediction on AviationKG, the kappa score is be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='8, and agreement is near 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' These results clearly denote the substantiality of our claim with high scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We extend the test for question-answering with MetaQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The pair of T5 models score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='6, denoting moderate agree- ment as more than 80% of agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' T5-large and Bloom pair scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='33 with 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='7% agreement, which is fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Thus, the testing supports our hypothesis, and we prove that the level of performance of different models with the infusion of knowledge remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 7 Conclusion and Future Work We have successfully created a million factoid QA pairs from the NTSB aircraft accident reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The QA pairs are used in our experiments with Avia- tionKG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We have validated our claim that with the infusion of knowledge to language models, the per- formance of the small language model is similar to the large language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We substantiate with dif- ferent language models and a diversity of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Our investigation will benefit researchers in select- ing the appropriate language model when working with knowledge and save computation power and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' The future line of work is to investigate the per- formance of models with incomplete and noisy knowledge graphs and study the extent to which the models can outright the domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Acknowledgements This research is supported by the Science and Edu- cation Research Board (SERB), Ministry of Educa- tion, India, under the Imprint-2 project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' We thank our Industry partner, Honeywell Technology Solu- tions Pvt Ltd, who provided insight and expertise that greatly assisted this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' References Katrin Affolter, Kurt Stockinger, and Abraham Bern- stein.' metadata={'source': 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Thirty-second AAAI conference on artificial intel- ligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Victor Zhong, Caiming Xiong, and Richard Socher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' Seq2sql: Generating structured queries from natural language using reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' arXiv preprint arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='00103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='1 Examples of AviationQA Below, we mention some examples from our cre- ated Aviation question-answering dataset (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content='4): Q: Which seat was occupied by the pilot re- sponsible for accident no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' CEN18LA272?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A: Left Q: Are there other Aircraft Rating(s) for the pilot of accident no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' GAA18CA489?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A: None Q: What is the make of the aircraft bearing accident no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' CEN18LA272?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A: Cessna Q: What is the category of the aircraft in- volved in accident no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' GAA18CA489?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A: Gyroplane Q: What is the Airworthiness Certificate of accident no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' GAA18CA297?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} +page_content=' A: Normal' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE2T4oBgHgl3EQfoAgi/content/2301.04013v1.pdf'} diff --git a/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/2301.13641v1.pdf.txt b/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/2301.13641v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..60e10f9103c8d098c258a25e1c74ac58756e5029 --- /dev/null +++ b/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/2301.13641v1.pdf.txt @@ -0,0 +1,551 @@ +arXiv:2301.13641v1 [math.AC] 31 Jan 2023 +GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL +CONDITIONS +ASHUTOSH PANDEY, BALCHAND PRAJAPATI +Abstract. Let R be a prime ring of characteristic different from 2, U be +the Utumi quotient ring of R and C be the extended centroid of R. +Let +F be a generalized skew derivation on R, I be a non-zero ideal of R and +m, n1, n2, . . . , nk ≥ 1 are fixed integers such that [F (um), un1, un2, . . . , unk] = +0 for all u ∈ I then there exists λ ∈ C such that F (x) = λx for all x ∈ R. +1. Introduction +Throughout the article R denotes a prime ring with center Z(R). The Utumi +quotient ring of R is denoted by U. The center of U is called the extended centroid +of R and it is denoted by C. The definition and construction of U can be found +in [5]. The commutator ab − ba of two elements a and b of R is denoted by [a, b]. +Define [a, b]0 = a and for k ≥ 1 the kth commutator of a and b is defined as [a, b]k = +[[a, b], b]k−1 = �k +i=0(−1)i�k +i +� +biabk−i. Also [a1, a2, . . . , ak] = [[a1, a2, . . . , ak−1], ak] +for all a1, a2, . . . , ak ∈ R, and for k ≥ 2. An additive mapping d : R → R is said +to be a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. An additive mapping +F : R → R is said to be a generalized derivation if there exists a derivation d on +R such that F(xy) = F(x)y + xd(y) for all x, y ∈ R. In [13] Posner proved that +if d is derivation of a prime ring R such that [d(x), x] ∈ Z(R) for all x ∈ R then +either d = 0 or R is a commutative ring. In [25] Lanski generalized the Posner’s +result by proving it on Lie ideal L of R. More precisely, Lanski proved that if +[d(x), x]k ∈ Z(R) for all x ∈ L and k ≥ 1 then char(R) = 2 and R ⊆ M2(F), for a +field F, equivalently R satisfies standard identity s4. In 2008, Arga¸c et al. in [6], +generalized Lanski’s result by replacing derivation d by generalized derivation F. +More precisely they proved that [F(x), x]k = 0, for all x ∈ L, then either F(x) = ax +with a ∈ C or R satisfies the standard identity s4. The study of generalized deriva- +tions on Lie ideals and on left ideals are given in [2, 5, 4, 6] where further references +can be found out. More recently in [9] Dhara¸c et al. proved the following: +Let R be a prime ring with its Utumi ring of quotients U, G a nonzero generalized +derivation of R and L a noncentral Lie ideal of R. Suppose that [G(xm), xn2, . . . , xnk] = +0 for all x ∈ L,where m, n1, n2, . . . , nk ≥ 1 are fixed integers. Then one of the fol- +lowing holds: +(1) there exists β ∈ C such that G(x) = βx for all x ∈ R. +(2) R satisfies the standard identity s4. +In this article we continue this line of investigation concerning the identity +[F(um), un1, un2, . . . , unk] = 0 for all u ∈ R, where m, n1, n2, . . . , nk ≥ 1 are fixed +2010 Mathematics Subject Classification. 16N60, 16W25 . +Key Words and Phrases. Lie ideals, generalized skew derivations, extended centroid, Utumi +quotient ring. +1 + +2 +A. PANDEY, B. PRAJAPATI +integers and F is a generalized skew derivation. More precisely we shall prove the +following: +Main Theorem: Let R be a prime ring of characteristic different from 2, U be +the Utumi quotient ring of R and C be the extended centroid of R. Let F be a +generalized skew derivation on R, I be a two sided ideal of R and m, n1, n2, . . . , nk ≥ +1 are fixed integers such that [F(um), un1, un2, . . . , unk] = 0 for all u ∈ I then there +exists λ ∈ C such that F(x) = λx for all x ∈ R. +We recall the following facts that are useful to prove our main theorem: +Fact 1.1. Let f(xi, d(xi), α(xi)) is a generalized polynomial identity for a prime +ring R, d is a outer skew derivation and α is outer automorphism of R then R also +satisfies the generalized polynomial identity f(xi, yi, zi), where xi, yi, zi are distinct +indeterminates. ([33, Theorem 1]) +Fact 1.2. ([32, Theorem 6.5.9]) Let R be a prime ring satisfies polynomial identity +of the type f(xαi△k +j +) = 0, where f(z(i,k) +j +) is generalized polynomial identity with +coefficient from U, △1, . . . , △n are mutually different correct words from a reduced +set of skew derivations commuting with all the corresponding automorphisms and +α1, . . . , αm are mutually outer automorphisms. In this case the identity f(z(i,k) +j +) = +0 is valid for U. +Fact 1.3. Let K be an infinite field and m ≥ 2 an integer. If P1, . . . , Pk are non- +scalar matrices in Mm(K) then there exists some invertible matrix P ∈ Mm(K) +such that each matrix PP1P −1, . . . , PPkP −1 has all non-zero entries. [4] +Fact 1.4. Let K be any field and R = Mm(K) be the algebra of all m × m +matrices over K with m ≥ 2. Then the matrix unit eij is an element of [R, R] for +all 1 ≤ i ̸= j ≤ m. +Fact 1.5. Every generalized skew derivation F of R can be uniquely extended to +a generalized derivation of U and its assume the form F(x) = ax + d(x), for some +a ∈ U and a skew derivation d on U [31]. +Fact 1.6. If I is a two-sided ideal of R, then R, I and U satisfy the same differential +identities [23]. +Fact 1.7. If I is a two-sided ideal of R, then R, I and U satisfies the same general- +ized polynomial identities with coefficients in U ([10]). Further R, I and U satisfy +the same generalized polynomial identities with automorphism in U [31]. +Fact 1.8. (Kharchenko [Theorem 2,[8]] Let R be a prime ring, d a non zero deriva- +tion on R and I a non zero ideal of R. If I satisfies the differential identity +f(r1, . . . , rn, d(r1), . . . , d(rn)) = 0 +for all r1, . . . , rn ∈ I, then either +(i) I satisfies the generalized polynomial identity f(r1, . . . , rn, x1, . . . , xn) = 0 +or +(ii) d is U-inner i.e., for some q ∈ U, d(x) = [q, x] and I satisfies the generalized +polynomial identity f(r1, . . . , rn, [q, r1], . . . , [q, rn]) = 0. +Fact 1.9. +� +Theorem 4.2.1 (Jacobson density theorem)[5] +� +Let R be a primitive ring +with VR a faithful irreducible R-module and D = End(VR), then for any positive + +GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS +3 +integer n if v1, v2, . . . , vn are D-independent in V and w1, w2, . . . , wn are arbitrary +in V then there exists r ∈ R such that vir = wi for i = 1, 2, . . ., n. +Fact 1.10. Let X = {x1, x2, . . .} be a countable set consisting of noncommuting +indeterminates x1, x2, . . .. +Let C{X} be the free algebra over C on the set X. +We denote T = U ∗C C{X}, the free product of the C-algebras U and C{X}. +The elements of T are called the generalized polynomials with coefficients in U. +Let B be a set of C-independent vectors of U. Then any element f ∈ T can be +represented in the form f = � +i aini, where ai ∈ C and ni are B-monomials of +the form p0u1p1u2p2 · unpn, with p0, p1, . . . , pn ∈ B and u1, u2, . . . , un ∈ X. Any +generalized polynomial f = � +i aini is trivial, i.e., zero element in T if and only if +ai = 0 for each i. For further details we refer the reader to [11]. +We begin with the following Proposition: +Proposition 1.11. Let R be a prime ring of characteristic different from 2, U be +the Utumi quotient ring of R, C be the extended centroid of R and β ∈ Aut(U). +Let a, b ∈ U and m, n1, n2, . . . , nk ≥ 1 are fixed integers such that +(1) +[aum + β(um)b, un1, un2, . . . , unk] = 0 +for all u ∈ R then either β is identity map on R and a, b ∈ C or there exists an +invertible element p such that β(x) = pxp−1, for all x ∈ R with p−1b ∈ C and +a + b ∈ C. +We need to prove the following lemmas to prove Proposition (1.11): +Lemma 1.12. Let R be a prime ring of characteristic different from 2, U be the +Utumi quotient ring of R and C be the extended centroid of R. Let a, b ∈ U and +m, n1, n2, . . . , nk ≥ 1 are fixed integers such that +(2) +[aum + pump−1b, un1, un2, . . . , unk] = 0 +for all u ∈ R then p−1b ∈ C and a + b ∈ C. +Proof. First assume that R does not satisfy any non-trivial generalized polynomial +identity. Let T = U ∗ C{u}, the free product of U and C{u}, C-algebra in single +indeterminate u. Then equation (2) is a GPI in T . If p−1b /∈ C then p−1b and 1 +are linearly independent over C. Thus from Fact (1.10), equation (2) implies +un1+n2+...+nkpump−1b = 0 +in T implying p−1b = 0, a contradiction. Therefore we conclude that p−1b ∈ C and +hence equation (2) reduces to : +(3) +[aum + bum, un1, un2, . . . , unk] = 0 +again by [9], equation (3) implies that a + b ∈ C. +Now we consider the case when equation (2) is a nontrivial polynomial identity for +R. Since R and U satisfy the same generalized polynomial identities (see Fact 1.7). +Therefore U satisfies equation (2). In case C is infinite the generalized polynomial +identity (2) is also satisfied by U⊗C ¯C where ¯C is the algebraic closure of C. Since +both U and U⊗C ¯C are prime and centrally closed [15], we may replace R by +U or U⊗C ¯C according as C is infinite or finite. +Thus we may assume that R +is centrally closed over C which is either finite or algebraically closed such that +[aum + pump−1b, un1, un2, . . . , unk] = 0 for all u ∈ R. By Martindale’s result [14], +R is a primitive ring with non-zero socle H and eHe is a simple central algebra finite + +4 +A. PANDEY, B. PRAJAPATI +dimensional over C, for any minimal idempotent element e ∈ R. Thus there exists +a vector space V over a division ring D such that R is isomorphic to a dense subring +of ring of D-linear transformations of V . Since C is either finite or algebraically +closed, D must coincide with C. +Assume first that dimCV ≥ 3. If p−1b /∈ C then there exists v ∈ V such that +{p−1bv, v} is linearly C-independent. Since dimCV ≥ 3 there exists w ∈ V such +that {p−1bv, v, w} is linearly C-independent. By Jacobson’s theorem (see Fact 1.9) +there exists x ∈ R such that : +xv = 0, xp−1bv = p−1bv +Then, 0 = [axm + pxmp−1b, xn1, xn2, . . . , xnk]v = bv, a contradiction, because if +bv = 0, then {p−1bv, v, w} will be C-dependent. Thus {p−1bv, v} is linearly C- +dependent therefore p−1b ∈ C and hence equation (2) reduces to +[aum + bum, un1, un2, . . . , unk] = 0 +which implies a + b ∈ C by [9]. +Now if dimCV += 2, then U ∼= M2(C). +Denote p = � +ij eijpij, q = p−1b = +� +ij eijqij ∈ M2(C), for pij, qij ∈ C and 1 ≤ i, j ≤ 2, where eij is the usual +matrix unit with 1 at (i, j)th place and zero elsewhere. Assume q /∈ C then by Fact +(1.3), all the entries in q is non-zero i.e. qij ̸= 0 for 1 ≤ i, j ≤ 2. +Choosing u = e11 in equation (2) and right multiplying by e22 we get: +(4) +p11q12 = 0 +implying p11 = 0. Let φ be an automorphism of U then +(5) +[φ(a)um + φ(p)umφ(p−1b), un1, un2, . . . , unk] = 0 +is also an identity of U. Thus φ(p), φ(q)and φ(a) must satisfy equation (4). Denote +φ(p) = � +ij eijp′ +ij, φ(q) = � +ij eijq′ +ij, for p′ +ij, q′ +ij ∈ C and 1 ≤ i, j ≤ 2, then from +equation (4), we have p′ +11q′ +12 = 0. In particular chossing φ(u) = (1 + e21)u(1 − e21) +we get p12q12 = 0, which implies p12 = 0. Thus 1st row of p is zero, which is a +contradiction because p is invertible. Thus q12 = 0, a contradiction. Therefore +q = p−1b ∈ C. Since q ∈ C therefore equation (2) reduces to +(6) +[aum + bum, un1, un2, . . . , unk] = 0 +Denote c = a + b = � +ij cijeij, for cij ∈ C and 1 ≤ i, j ≤ 2. Suppose c /∈ C then by +Fact (1.3), all the entries of c is non-zero i.e. cij ̸= 0. Choosing u = e22 in equation +(6) and right multiply by e11 we get: +c12e12 = 0 +which implies c12 = 0, a contradiction. Therefore c = a + b ∈ C. +□ +Lemma 1.13. Let R be a prime ring of characteristic different from 2, U be the +Utumi quotient ring of R and C be the extended centroid of R. Let a, b ∈ U, β be +an outer automorphism on U, and m, n1, n2, . . . , nk ≥ 1 are fixed integers such that +(7) +[aum + β(um)b, un1, un2, . . . , unk] = 0 +for all u ∈ R then β is the identity map on R and a + b ∈ C unless b = 0 and +a ∈ C. + +GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS +5 +Proof. Since R and U satisfy the same generalized polynomial identity with auto- +morphisms (see Fact 1.7), it follows that U satisfies +(8) +[aum + β(um)b, un1, un2, . . . , unk] = 0 +We may assume a /∈ C and b ̸= 0 then U satisfies non-trivial generalized polyno- +mial identity. Therefore by ([14],theorem 3), U is dense subring of the ring of linear +transformtion of a vector space V over a division ring D. If β is not Frobenius then +from Fact (1.2), U satisfies +(9) +[aum + zmb, un1, un2, . . . , unk] = 0 +then by [9], we get, a, b ∈ C. In particular from equation (9), U satisfies +(10) +b[zm, un1, un2, . . . , unk] = 0 +then by posner’s theorem [21] there exists a suitable filed F and a positive integer +n such that U and Mn(F) satisfies the same polynomial identity. For i ̸= j, choose +z = eii + eij and u = eii in equation (10), we get, +0 = [eii + eij, eii]k = (−1)keij +a contradiction, where eij is the usual matrix unit with 1 at the (i, j)th entry and +zero elsewhere. +Now we assume that β is Frobenius and dimDV ≥ 2 (because if dimDV = 1 then +U will be commutative). If char(R) = 0 then β(x) = x because β is Frobenius. +This implies that β is inner which is a contradiction. Thus we may assume that +char(R) = p ̸= 0 and β(λ) = λps for all λ ∈ C and for some fixed integer s ≥ 0. +Now replace u by λu in equation (8) then U satisfies +(11) +λm+n1+...+nk[aum + λm(ps−1)β(um)b, un1, un2, . . . , unk] = 0. +In particular choose λm = γ then U satisfies +(12) +[aum + γ(ps−1)β(um)b, un1, un2, . . . , unk] = 0. +From equation (8) and (12) we get +(13) +(γ(ps−1) − 1)[β(um)b, un1, un2, . . . , unk] = 0 +that is U satisfies +(14) +[β(um)b, un1, un2, . . . , unk] = 0. +Again from equation (8) and (14) we have that [aum, un1, un2, . . . , unk] = 0 for all +u ∈ U, this implies that a ∈ C ( see [9]). Thus we may consider b ̸= 0. +Now let e2 = e ∈ soc(R) then by equation (14), +(15) +[β(e)b, e]k = 0. +Right multiply above relation by (1 − e) gives +(16) +eβ(e)b(1 − e) = 0. +In particular choosing the idempotent (1 − e) − (1 − e)xe for all x ∈ U in equation +(16), we have that U satisfies: +(17) +(1 − e)β(1 − e)b(1 − e)xe + (1 − e)β(1 − e)β(x)β(e)b(e + (1 − e)xe) +−(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)β(x)β(e)b(e + (1 − e)xe). +Since β is outer then from Fact (1.2), U satisfies : +(18) +(1 − e)β(1 − e)b(1 − e)xe + (1 − e)β(1 − e)zβ(e)b(e + (1 − e)xe) + +6 +A. PANDEY, B. PRAJAPATI +−(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe). +In particular for x = 0, we get (1 − e)β(1 − e)zβ(e)be = 0, for all z ∈ U. Hence +by primeness of U, we have either (1 − e)β(1 − e) = 0 or β(e)be = 0 for any +e2 = e ∈ Soc(U). +If we consider the case (1 − e)β(1 − e) = 0, then equation (17) reduces to: +−(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe). +In particular U satisfies +(1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe). +Now replace z by ze in above relation we get that +(19) +(1 − e)xeβ(1 − e)zeβ(e)be. +Now since eβ(1−e) ̸= 0, otherwise from (1−e)β(1−e) = 0 and we get β(1−e) = 0, +which is a contradiction. Thus from equation (19), we get eβ(e)be = 0, then by +equation (16) we get that eβ(e)b = 0. Thus, in any case equation (15) implies that +(20) +β(e)be = 0. +for any idempotent element e ∈ U. If we choose the idempotent element ex(1−e)+e, +for all x ∈ U then from equation (20), U satisfies: +� +β +� +ex(1 − e) + β(e) +� +b(ex(1 − e) + e). +Since β(e)be = 0, U satisfies +β(e)β(x)b(ex(1 − e) + e). +Since β is outer then from Fact (1.2), U satisfies: +β(e)yb(ex(1 − e) + e) +for all x, y ∈ U. In particular for x = 0, β(e)ybe = 0 i.e. be = 0 (by primness of U) +for all e2 = e ∈ U. Let M denotes the additive subgroup of U, which is generated +by all the idempotent elements of U, then bM = 0. Moreover, by [18](page 18, +corollary), [U, U] ⊆ M, i.e. b[U, U] = 0 implying b = 0, a contradiction. +□ +Remark: Lemma 1.12 and Lemma 1.13 cover all the cases to prove Proposition +1.11. +Proof of main theorem: From the given hypothesis we have: +(21) +[F(um), un1, un2, . . . , unk] = 0 +for all u ∈ I. Since R, I and U satisfy the same generalized polynomial identities +as well as the same differential identities with automorphism (see Fact 1.6, 1.7). +Hence, +(22) +[F(um), un1, un2, . . . , unk] = 0 +for all u ∈ U, where F(u) = cu + d(u) for some c ∈ U and d is skew derivation +of U with associated automorphism β (see Fact 1.5). We divide the proof into the +following cases: + +GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS +7 +Case 1: If d is inner derivation then d(u) = pu − β(u)p for all u ∈ U and for +some p ∈ U. Then U satisfies +(23) +[(c + p)um − β(um)p, un1, un2, . . . , unk] = 0. +Then by Proposition 1.11 we get the required result. +Case2: If d is outer then U satisfies +(24) +� +cum + +m−1 +� +i=0 +β(ui)d(u)um−i−1, un1, un2, . . . , unk +� += 0. +by [33], we have: +(25) +� +cum + +m−1 +� +i=0 +β(ui)yum−i−1, un1, un2, . . . , unk +� += 0. +In particular U satisfies +(26) +� m−1 +� +i=0 +β(ui)yum−i−1, un1, un2, . . . , unk +� += 0 +Subcase 1: If β is outer derivation then from Fact (1.2), U satisfies: +(27) +� +cum + +m−1 +� +i=0 +ziyum−i−1, un1, un2, . . . , unk +� += 0 +for all u, y, z ∈ U. In particular for z = 0, we have that +(28) +[yum−1, un1, un2, . . . , unk] = 0. +Then by Posner’s theorem, there exists a suitable field F and a positive integer +n such that U and Mn(F) satisfy the same generalized polynomial identity. For +i ̸= j, choose y = eii + eji and u = eii in equation (28), we get +0 = [eii + eji, eii]k = eji +which is a contradiction. +Subcase 2: If β is inner then β(x) = pxp−1, for all x ∈ U and for some p ∈ U. +Hence from equation (26), U satisfies: +(29) +� m−1 +� +i=0 +puip−1yum−i−1, un1, un2, . . . , unk +� += 0. +Since β is not identity, hence p /∈ C, therefore equation (29) is a non-trivial polyno- +mial identity for R. By [14], U is isomorphic to a dense ring of linear transformation +on some vector space V over C. Firstly, we consider the case when dimCV ≥ 3. +Since p /∈ C therefore there exists some v ∈ V such that {p−1v, v} is linearly C- +independent. Since dimCV ≥ 3, there must exists w1 ∈ V such that {p−1v, v, w1} +is linearly C- independent. Since U is dense, therefore by Jacobson density theorem +( see Fact 1.9), there exists y1, y2 ∈ U such that +y1w1 = 0, y2w1 = v, y1p−1v = p−1v, y1v = v. +Therefore by equation (29) we get that +0 = +� m−1 +� +i=0 +pyi +1p−1y2ym−i−1 +1 +, yn1 +1 , yn2 +1 , . . . , ynk +1 +� +w1 = (−1)kv ̸= 0 + +8 +A. PANDEY, B. PRAJAPATI +which is a contradiction. Now if dimCV = 2, i.e. U ∼= M2(C), ring of 2-order +matrix over field C. +p = +�p11 +p12 +p21 +p22 +� +, p−1 = +1 +det(p) +� p22 +−p12 +−p21 +p11 +� +choosing u = e22, y = e21 in equation (29), we get that +(30) +p11p22 = 0 +Similarly, by choosing u = e11, y = e22 in equation (29), we get that +(31) +p11p12 = 0 +Since p is invertible therefore p22 and p12 can not be zero simultaneously, thus from +equation (30) and (31), it follows p11 = 0. This implies p12 ̸= 0, otherwise p will +be singular matrix. +Choose φ(u) = (1 − e12)u(1 + e12) ∈ Aut(U), then +φ +�� m−1 +� +i=0 +puip−1yum−i−1, un1, un2, . . . , unk +�� += 0 +i.e. U satisfies: +(32) +� m−1 +� +i=0 +φ(p)uiφ(p−1)yum−i−1, un1, un2, . . . , unk +� += 0. +Denote φ(p)11 as the (1, 1)th-entry of φ(p), then by the same argument as above +we get 0 = φ(p)11 = p21, which is a contradiction (because p is invertible.). +For the case dimCV = 1, R will be commutative and we have nothing to prove in +this case. +Following is the very natural consequence of our main theorem: +Corollary 1.14. Let R be a prime ring with its Utumi ring of quotients U, F +a nonzero generalized derivation of R and I a non-zero ideal of R. Suppose that +[F(xm), xn]k = 0 for all x ∈ I,where n, k ≥ 1 are fixed integers. Then there exists +β ∈ C such that F(x) = βx for all x ∈ R. +Proof. Choosing n1 = n2 = . . . = nk = n in our main theorem we get the required +result. +□ +Future research: Recently, C.K. Liu in [24] investigated the structure of F and +G if they satisfy the identity [F(xn)xm + xmG(xn), xr]k = 0 for all x ∈ I, where +F, G are non-zero generalized derivation on a prime ring R, I is the non-zero ideal +of R and m, n, k ≥ 1 are fixed integers. In the light of [24] with this article one +can try to find the structure of F and G if they satisfy the identity [F(xn)xm + +xmG(xn), xn1, xn2, . . . , xnk] = 0 for all x ∈ I, where m, n, n1, n2, . . . , nk ≥ 1 are +fixed integers. +Acknowledgement +The authors is highly thankful to the referee(s) for valuable suggestions and +comments. This research is not funded. + +GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS +9 +References +[1] Dhara, Basudeb and De Filippis, Vincenzo, Engel conditions of generalized derivations on +left ideals and Lie ideals in prime rings, Communications in Algebra, 48 (1), 154–167, 2020. +[2] Dhara, Basudeb, Annihilator condition on power values of derivations, Indian Journal of Pure +and Applied Mathematics, 42 (4), 357–369, 2011. +[3] Alba¸s, Emine and Arga¸c, Nurcan and Filippis, Vincenzo De, Generalized derivations with +Engel conditions on one-sided ideals, 36 (6), 2063–2071, 2008. +[4] De Filippis, Vincenzo and Di Vincenzo, Onofrio Mario, Vanishing derivations and centralizers +of generalized derivations on multilinear polynomials, Communications in Algebra, 40 (6), +1918–1932, (2012). +[5] Beidar, Konstant I and Martindale, Wallace S and Mikhalev, Alexander V, Rings with gen- +eralized identities, CRC Press, 1995. +[6] Argac, Nurc and Carini, Luisa and De Filippis, V, An Engel condition with generalized +derivations on Lie ideals, Taiwanese Journal of Mathematics, 419–433, 2008. +[7] Alahmadi, Adel and Ali, Shakir and Khan, Abdul Nadim and Khan, Mohammad Salahuddin, +A characterization of generalized derivations on prime rings, Communications in Algebra, 44 +(8) 3201–3210, 2016. +[8] Kharchenko, Vladislav Kirillovich, Differential identities of prime rings, Algebra and Logic, +17 (2), 155–168, 1978. +[9] Dhara, Basudeb and Ali, Asma and Das, Deepankar, Engel conditions of generalized deriva- +tions on Lie ideals and left sided ideals in prime rings and Banach Algebras, Afrika Matem- +atika, 27 (7), 1391–1401, 2016. +[10] Beidar, KI, Rings with generalized identities. 3., Vestnik Moskovskogo Universiteta Seriya i +Matematika, Mekhanika, 4, 66–73, 1978. +[11] Chuang, Chen Lian, GPIs having coefficients in Utumi quotient rings, Proceedings of the +American Mathematical Society, 103 (3), 723–728, 1988. +[12] Demir, C¸agri and Arga¸c, Nurcan, A result on generalized derivations with Engel conditions +on one-sided ideals, Journal of the Korean Mathematical Society, 47 (3), 483–494, 2010. +[13] Posner, Edward C, Derivations in prime rings, Proceedings of the American Mathematical +Society, 8 (6), 1093–1100, 1957. +[14] Martindale 3rd, Wallace S, Prime rings satisfying a generalized polynomial identity, Journal +of Algebra, 12 (4), 576–584, 1969. +[15] Erickson, Theodore and Martindale, Wallace and Osborn, James, Prime nonassociative alge- +bras, Pacific Journal of Mathematics, 60 (1), 49–63 1975. +[16] Faith, C and Utumi, Y, On a new proof of Litoff’s theorem, Acta Mathematica Academiae +Scientiarum Hungarica, 14 (3-4), 369–371, 1963. +[17] Lee, Tsiu Kwen, Generalized derivations of left faithful rings, Communications in Algebra, +27 (8), 4057–4073, 1999. +[18] Herstein, Israel Nathan, Topics in ring theory, University of Chicago press, 1969. +[19] Di Vincenzo, OM, On the n-th centralizer of a Lie ideal, Boll. 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Structure of rings, + +10 +A. PANDEY, B. PRAJAPATI +[29] Carini, Luisa and De Filippis, Vincenzo and Scudo, Giovanni, Identities with product of +generalized skew derivations on multilinear polynomials, Communications in Algebra, 44 (7), +3122–3138, 2016. +[30] De Filippis, Vincenzo and Di Vincenzo, Onofrio Mario, Generalized Skew Derivations and +Nilpotent Values on Lie Ideals, Algebra Colloquium, 26 (04), 2019. +[31] C. L. Chuang. Differential identities with automorphisms and antiautomorphisms, ii. Journal +of Algebra,160(1):130–171, 1993. +[32] V. Kharchenko.Automorphisms and Derivations of Associative Rings, volume 69. Springer +Science and Business Media, 1991. +[33] C.-L. Chuang and T.-K.Identities with a single skew derivation,Journal of Algebra, 288(1):59 +– 77, 2005. +A. Pandey, School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, +INDIA. +Email address: ashutoshpandey064@gmail.com +B. Prajapati, School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, +INDIA. +Email address: balchand@aud.ac.in + diff --git a/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/load_file.txt b/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d26780faf8ebd9bc80263071adfeada0d580453d --- /dev/null +++ b/DdFRT4oBgHgl3EQfxTg4/content/tmp_files/load_file.txt @@ -0,0 +1,500 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf,len=499 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='13641v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='AC] 31 Jan 2023 GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS ASHUTOSH PANDEY, BALCHAND PRAJAPATI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let R be a prime ring of characteristic different from 2, U be the Utumi quotient ring of R and C be the extended centroid of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let F be a generalized skew derivation on R, I be a non-zero ideal of R and m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers such that [F (um), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ I then there exists λ ∈ C such that F (x) = λx for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Introduction Throughout the article R denotes a prime ring with center Z(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The Utumi quotient ring of R is denoted by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The center of U is called the extended centroid of R and it is denoted by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The definition and construction of U can be found in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The commutator ab − ba of two elements a and b of R is denoted by [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Define [a, b]0 = a and for k ≥ 1 the kth commutator of a and b is defined as [a, b]k = [[a, b], b]k−1 = �k i=0(−1)i�k i � biabk−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Also [a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , ak] = [[a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , ak−1], ak] for all a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , ak ∈ R, and for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' An additive mapping d : R → R is said to be a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' An additive mapping F : R → R is said to be a generalized derivation if there exists a derivation d on R such that F(xy) = F(x)y + xd(y) for all x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In [13] Posner proved that if d is derivation of a prime ring R such that [d(x), x] ∈ Z(R) for all x ∈ R then either d = 0 or R is a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In [25] Lanski generalized the Posner’s result by proving it on Lie ideal L of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' More precisely, Lanski proved that if [d(x), x]k ∈ Z(R) for all x ∈ L and k ≥ 1 then char(R) = 2 and R ⊆ M2(F), for a field F, equivalently R satisfies standard identity s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In 2008, Arga¸c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' in [6], generalized Lanski’s result by replacing derivation d by generalized derivation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' More precisely they proved that [F(x), x]k = 0, for all x ∈ L, then either F(x) = ax with a ∈ C or R satisfies the standard identity s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The study of generalized deriva- tions on Lie ideals and on left ideals are given in [2, 5, 4, 6] where further references can be found out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' More recently in [9] Dhara¸c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' proved the following: Let R be a prime ring with its Utumi ring of quotients U, G a nonzero generalized derivation of R and L a noncentral Lie ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Suppose that [G(xm), xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , xnk] = 0 for all x ∈ L,where m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then one of the fol- lowing holds: (1) there exists β ∈ C such that G(x) = βx for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' (2) R satisfies the standard identity s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In this article we continue this line of investigation concerning the identity [F(um), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ R, where m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' 16N60, 16W25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Key Words and Phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Lie ideals, generalized skew derivations, extended centroid, Utumi quotient ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PANDEY, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PRAJAPATI integers and F is a generalized skew derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' More precisely we shall prove the following: Main Theorem: Let R be a prime ring of characteristic different from 2, U be the Utumi quotient ring of R and C be the extended centroid of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let F be a generalized skew derivation on R, I be a two sided ideal of R and m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers such that [F(um), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ I then there exists λ ∈ C such that F(x) = λx for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' We recall the following facts that are useful to prove our main theorem: Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let f(xi, d(xi), α(xi)) is a generalized polynomial identity for a prime ring R, d is a outer skew derivation and α is outer automorphism of R then R also satisfies the generalized polynomial identity f(xi, yi, zi), where xi, yi, zi are distinct indeterminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' ([33, Theorem 1]) Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' ([32, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='9]) Let R be a prime ring satisfies polynomial identity of the type f(xαi△k j ) = 0, where f(z(i,k) j ) is generalized polynomial identity with coefficient from U, △1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , △n are mutually different correct words from a reduced set of skew derivations commuting with all the corresponding automorphisms and α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , αm are mutually outer automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In this case the identity f(z(i,k) j ) = 0 is valid for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let K be an infinite field and m ≥ 2 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , Pk are non- scalar matrices in Mm(K) then there exists some invertible matrix P ∈ Mm(K) such that each matrix PP1P −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , PPkP −1 has all non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' [4] Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let K be any field and R = Mm(K) be the algebra of all m × m matrices over K with m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then the matrix unit eij is an element of [R, R] for all 1 ≤ i ̸= j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Every generalized skew derivation F of R can be uniquely extended to a generalized derivation of U and its assume the form F(x) = ax + d(x), for some a ∈ U and a skew derivation d on U [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If I is a two-sided ideal of R, then R, I and U satisfy the same differential identities [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If I is a two-sided ideal of R, then R, I and U satisfies the same general- ized polynomial identities with coefficients in U ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Further R, I and U satisfy the same generalized polynomial identities with automorphism in U [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' (Kharchenko [Theorem 2,[8]] Let R be a prime ring, d a non zero deriva- tion on R and I a non zero ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If I satisfies the differential identity f(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , rn, d(r1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , d(rn)) = 0 for all r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , rn ∈ I, then either (i) I satisfies the generalized polynomial identity f(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , rn, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , xn) = 0 or (ii) d is U-inner i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=', for some q ∈ U, d(x) = [q, x] and I satisfies the generalized polynomial identity f(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , rn, [q, r1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , [q, rn]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' � Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='1 (Jacobson density theorem)[5] � Let R be a primitive ring with VR a faithful irreducible R-module and D = End(VR), then for any positive GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS 3 integer n if v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , vn are D-independent in V and w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , wn are arbitrary in V then there exists r ∈ R such that vir = wi for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let X = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='} be a countable set consisting of noncommuting indeterminates x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='. Let C{X} be the free algebra over C on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' We denote T = U ∗C C{X}, the free product of the C-algebras U and C{X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' The elements of T are called the generalized polynomials with coefficients in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let B be a set of C-independent vectors of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then any element f ∈ T can be represented in the form f = � i aini, where ai ∈ C and ni are B-monomials of the form p0u1p1u2p2 · unpn, with p0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , pn ∈ B and u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , un ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Any generalized polynomial f = � i aini is trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=', zero element in T if and only if ai = 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' For further details we refer the reader to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' We begin with the following Proposition: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let R be a prime ring of characteristic different from 2, U be the Utumi quotient ring of R, C be the extended centroid of R and β ∈ Aut(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let a, b ∈ U and m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers such that (1) [aum + β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ R then either β is identity map on R and a, b ∈ C or there exists an invertible element p such that β(x) = pxp−1, for all x ∈ R with p−1b ∈ C and a + b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' We need to prove the following lemmas to prove Proposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='11): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let R be a prime ring of characteristic different from 2, U be the Utumi quotient ring of R and C be the extended centroid of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let a, b ∈ U and m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers such that (2) [aum + pump−1b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ R then p−1b ∈ C and a + b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' First assume that R does not satisfy any non-trivial generalized polynomial identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let T = U ∗ C{u}, the free product of U and C{u}, C-algebra in single indeterminate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then equation (2) is a GPI in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If p−1b /∈ C then p−1b and 1 are linearly independent over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus from Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='10), equation (2) implies un1+n2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='+nkpump−1b = 0 in T implying p−1b = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore we conclude that p−1b ∈ C and hence equation (2) reduces to : (3) [aum + bum, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 again by [9], equation (3) implies that a + b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now we consider the case when equation (2) is a nontrivial polynomial identity for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since R and U satisfy the same generalized polynomial identities (see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore U satisfies equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In case C is infinite the generalized polynomial identity (2) is also satisfied by U⊗C ¯C where ¯C is the algebraic closure of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since both U and U⊗C ¯C are prime and centrally closed [15], we may replace R by U or U⊗C ¯C according as C is infinite or finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus we may assume that R is centrally closed over C which is either finite or algebraically closed such that [aum + pump−1b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' By Martindale’s result [14], R is a primitive ring with non-zero socle H and eHe is a simple central algebra finite 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PANDEY, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PRAJAPATI dimensional over C, for any minimal idempotent element e ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus there exists a vector space V over a division ring D such that R is isomorphic to a dense subring of ring of D-linear transformations of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since C is either finite or algebraically closed, D must coincide with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Assume first that dimCV ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If p−1b /∈ C then there exists v ∈ V such that {p−1bv, v} is linearly C-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since dimCV ≥ 3 there exists w ∈ V such that {p−1bv, v, w} is linearly C-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' By Jacobson’s theorem (see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='9) there exists x ∈ R such that : xv = 0, xp−1bv = p−1bv Then, 0 = [axm + pxmp−1b, xn1, xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , xnk]v = bv, a contradiction, because if bv = 0, then {p−1bv, v, w} will be C-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus {p−1bv, v} is linearly C- dependent therefore p−1b ∈ C and hence equation (2) reduces to [aum + bum, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 which implies a + b ∈ C by [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now if dimCV = 2, then U ∼= M2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Denote p = � ij eijpij, q = p−1b = � ij eijqij ∈ M2(C), for pij, qij ∈ C and 1 ≤ i, j ≤ 2, where eij is the usual matrix unit with 1 at (i, j)th place and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Assume q /∈ C then by Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='3), all the entries in q is non-zero i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' qij ̸= 0 for 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Choosing u = e11 in equation (2) and right multiplying by e22 we get: (4) p11q12 = 0 implying p11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let φ be an automorphism of U then (5) [φ(a)um + φ(p)umφ(p−1b), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 is also an identity of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus φ(p), φ(q)and φ(a) must satisfy equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Denote φ(p) = � ij eijp′ ij, φ(q) = � ij eijq′ ij, for p′ ij, q′ ij ∈ C and 1 ≤ i, j ≤ 2, then from equation (4), we have p′ 11q′ 12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular chossing φ(u) = (1 + e21)u(1 − e21) we get p12q12 = 0, which implies p12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus 1st row of p is zero, which is a contradiction because p is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus q12 = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore q = p−1b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since q ∈ C therefore equation (2) reduces to (6) [aum + bum, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 Denote c = a + b = � ij cijeij, for cij ∈ C and 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Suppose c /∈ C then by Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='3), all the entries of c is non-zero i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' cij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Choosing u = e22 in equation (6) and right multiply by e11 we get: c12e12 = 0 which implies c12 = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore c = a + b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let R be a prime ring of characteristic different from 2, U be the Utumi quotient ring of R and C be the extended centroid of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let a, b ∈ U, β be an outer automorphism on U, and m, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers such that (7) [aum + β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ R then β is the identity map on R and a + b ∈ C unless b = 0 and a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since R and U satisfy the same generalized polynomial identity with auto- morphisms (see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='7), it follows that U satisfies (8) [aum + β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 We may assume a /∈ C and b ̸= 0 then U satisfies non-trivial generalized polyno- mial identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore by ([14],theorem 3), U is dense subring of the ring of linear transformtion of a vector space V over a division ring D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If β is not Frobenius then from Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2), U satisfies (9) [aum + zmb, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 then by [9], we get, a, b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular from equation (9), U satisfies (10) b[zm, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 then by posner’s theorem [21] there exists a suitable filed F and a positive integer n such that U and Mn(F) satisfies the same polynomial identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' For i ̸= j, choose z = eii + eij and u = eii in equation (10), we get, 0 = [eii + eij, eii]k = (−1)keij a contradiction, where eij is the usual matrix unit with 1 at the (i, j)th entry and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now we assume that β is Frobenius and dimDV ≥ 2 (because if dimDV = 1 then U will be commutative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If char(R) = 0 then β(x) = x because β is Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' This implies that β is inner which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus we may assume that char(R) = p ̸= 0 and β(λ) = λps for all λ ∈ C and for some fixed integer s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now replace u by λu in equation (8) then U satisfies (11) λm+n1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='+nk[aum + λm(ps−1)β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular choose λm = γ then U satisfies (12) [aum + γ(ps−1)β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' From equation (8) and (12) we get (13) (γ(ps−1) − 1)[β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 that is U satisfies (14) [β(um)b, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Again from equation (8) and (14) we have that [aum, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ U, this implies that a ∈ C ( see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus we may consider b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now let e2 = e ∈ soc(R) then by equation (14), (15) [β(e)b, e]k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Right multiply above relation by (1 − e) gives (16) eβ(e)b(1 − e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular choosing the idempotent (1 − e) − (1 − e)xe for all x ∈ U in equation (16), we have that U satisfies: (17) (1 − e)β(1 − e)b(1 − e)xe + (1 − e)β(1 − e)β(x)β(e)b(e + (1 − e)xe) −(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)β(x)β(e)b(e + (1 − e)xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since β is outer then from Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2), U satisfies : (18) (1 − e)β(1 − e)b(1 − e)xe + (1 − e)β(1 − e)zβ(e)b(e + (1 − e)xe) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PANDEY, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PRAJAPATI −(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular for x = 0, we get (1 − e)β(1 − e)zβ(e)be = 0, for all z ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Hence by primeness of U, we have either (1 − e)β(1 − e) = 0 or β(e)be = 0 for any e2 = e ∈ Soc(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If we consider the case (1 − e)β(1 − e) = 0, then equation (17) reduces to: −(1 − e)xeβ(1 − e)b(e + (1 − e)xe) + (1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular U satisfies (1 − e)xeβ(1 − e)zβ(e)b(e + (1 − e)xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now replace z by ze in above relation we get that (19) (1 − e)xeβ(1 − e)zeβ(e)be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now since eβ(1−e) ̸= 0, otherwise from (1−e)β(1−e) = 0 and we get β(1−e) = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus from equation (19), we get eβ(e)be = 0, then by equation (16) we get that eβ(e)b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Thus, in any case equation (15) implies that (20) β(e)be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' for any idempotent element e ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' If we choose the idempotent element ex(1−e)+e, for all x ∈ U then from equation (20), U satisfies: � β � ex(1 − e) + β(e) � b(ex(1 − e) + e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since β(e)be = 0, U satisfies β(e)β(x)b(ex(1 − e) + e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since β is outer then from Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2), U satisfies: β(e)yb(ex(1 − e) + e) for all x, y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular for x = 0, β(e)ybe = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' be = 0 (by primness of U) for all e2 = e ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let M denotes the additive subgroup of U, which is generated by all the idempotent elements of U, then bM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Moreover, by [18](page 18, corollary), [U, U] ⊆ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' b[U, U] = 0 implying b = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' □ Remark: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='12 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='13 cover all the cases to prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Proof of main theorem: From the given hypothesis we have: (21) [F(um), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since R, I and U satisfy the same generalized polynomial identities as well as the same differential identities with automorphism (see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Hence, (22) [F(um), un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0 for all u ∈ U, where F(u) = cu + d(u) for some c ∈ U and d is skew derivation of U with associated automorphism β (see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' We divide the proof into the following cases: GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS 7 Case 1: If d is inner derivation then d(u) = pu − β(u)p for all u ∈ U and for some p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then U satisfies (23) [(c + p)um − β(um)p, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='11 we get the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Case2: If d is outer then U satisfies (24) � cum + m−1 � i=0 β(ui)d(u)um−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' by [33], we have: (25) � cum + m−1 � i=0 β(ui)yum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular U satisfies (26) � m−1 � i=0 β(ui)yum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0 Subcase 1: If β is outer derivation then from Fact (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='2), U satisfies: (27) � cum + m−1 � i=0 ziyum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0 for all u, y, z ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In particular for z = 0, we have that (28) [yum−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then by Posner’s theorem, there exists a suitable field F and a positive integer n such that U and Mn(F) satisfy the same generalized polynomial identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' For i ̸= j, choose y = eii + eji and u = eii in equation (28), we get 0 = [eii + eji, eii]k = eji which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Subcase 2: If β is inner then β(x) = pxp−1, for all x ∈ U and for some p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Hence from equation (26), U satisfies: (29) � m−1 � i=0 puip−1yum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since β is not identity, hence p /∈ C, therefore equation (29) is a non-trivial polyno- mial identity for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' By [14], U is isomorphic to a dense ring of linear transformation on some vector space V over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Firstly, we consider the case when dimCV ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since p /∈ C therefore there exists some v ∈ V such that {p−1v, v} is linearly C- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since dimCV ≥ 3, there must exists w1 ∈ V such that {p−1v, v, w1} is linearly C- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Since U is dense, therefore by Jacobson density theorem ( see Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='9), there exists y1, y2 ∈ U such that y1w1 = 0, y2w1 = v, y1p−1v = p−1v, y1v = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Therefore by equation (29) we get that 0 = � m−1 � i=0 pyi 1p−1y2ym−i−1 1 , yn1 1 , yn2 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , ynk 1 � w1 = (−1)kv ̸= 0 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PANDEY, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' PRAJAPATI which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Now if dimCV = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' U ∼= M2(C), ring of 2-order matrix over field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' p = �p11 p12 p21 p22 � , p−1 = 1 det(p) � p22 −p12 −p21 p11 � choosing u = e22, y = e21 in equation (29), we get that (30) p11p22 = 0 Similarly, by choosing u = e11, y = e22 in equation (29), we get that (31) p11p12 = 0 Since p is invertible therefore p22 and p12 can not be zero simultaneously, thus from equation (30) and (31), it follows p11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' This implies p12 ̸= 0, otherwise p will be singular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Choose φ(u) = (1 − e12)u(1 + e12) ∈ Aut(U), then φ �� m−1 � i=0 puip−1yum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk �� = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' U satisfies: (32) � m−1 � i=0 φ(p)uiφ(p−1)yum−i−1, un1, un2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , unk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Denote φ(p)11 as the (1, 1)th-entry of φ(p), then by the same argument as above we get 0 = φ(p)11 = p21, which is a contradiction (because p is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' For the case dimCV = 1, R will be commutative and we have nothing to prove in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Following is the very natural consequence of our main theorem: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Let R be a prime ring with its Utumi ring of quotients U, F a nonzero generalized derivation of R and I a non-zero ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Suppose that [F(xm), xn]k = 0 for all x ∈ I,where n, k ≥ 1 are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Then there exists β ∈ C such that F(x) = βx for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Choosing n1 = n2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' = nk = n in our main theorem we get the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' □ Future research: Recently, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Liu in [24] investigated the structure of F and G if they satisfy the identity [F(xn)xm + xmG(xn), xr]k = 0 for all x ∈ I, where F, G are non-zero generalized derivation on a prime ring R, I is the non-zero ideal of R and m, n, k ≥ 1 are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' In the light of [24] with this article one can try to find the structure of F and G if they satisfy the identity [F(xn)xm + xmG(xn), xn1, xn2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , xnk] = 0 for all x ∈ I, where m, n, n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' , nk ≥ 1 are fixed integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Acknowledgement The authors is highly thankful to the referee(s) for valuable suggestions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' This research is not funded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' GENERALIZED SKEW DERIVATION ON IDEAL WITH ENGEL CONDITIONS 9 References [1] Dhara, Basudeb and De Filippis, Vincenzo, Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Communications in Algebra, 48 (1), 154–167, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' [2] Dhara, Basudeb, Annihilator condition on power values of derivations, Indian Journal of Pure and Applied Mathematics, 42 (4), 357–369, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' [3] Alba¸s, Emine and Arga¸c, Nurcan and Filippis, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Differential identities with automorphisms and antiautomorphisms, ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Journal of Algebra,160(1):130–171, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' [32] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Kharchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='Automorphisms and Derivations of Associative Rings, volume 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Springer Science and Business Media, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Chuang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='Identities with a single skew derivation,Journal of Algebra, 288(1):59 – 77, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Pandey, School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Email address: ashutoshpandey064@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='com B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Prajapati, School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content=' Email address: balchand@aud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFRT4oBgHgl3EQfxTg4/content/2301.13641v1.pdf'} diff --git a/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/2301.01967v1.pdf.txt b/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/2301.01967v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d236ab105afdda340ca91beb1ed7c3ceca59a74 --- /dev/null +++ b/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/2301.01967v1.pdf.txt @@ -0,0 +1,1419 @@ +arXiv:2301.01967v1 [cs.CL] 5 Jan 2023 +A Survey of Code-switching: Linguistic and Social Perspectives for +Language Technologies +A. Seza Do˘gru¨oz +Ghent University, Gent, Belgium +as.dogruoz@ugent.be +Sunayana Sitaram +Microsoft Research India, Bangalore, India +sunayana.sitaram@microsoft.com +Barbara E. Bullock +UT at Austin, Austin, USA +bbullock@austin.utexas.edu +Almeida Jacqueline Toribio +UT at Austin, Austin, USA +toribio@austin.utexas.edu +Abstract +The analysis of data in which multiple lan- +guages are represented has gained popularity +among computational linguists in recent years. +So far, much of this research focuses mainly +on the improvement of computational meth- +ods and largely ignores linguistic and social +aspects of C-S discussed across a wide range +of languages within the long-established liter- +ature in linguistics. To fill this gap, we offer +a survey of code-switching (C-S) covering the +literature in linguistics with a reflection on the +key issues in language technologies. From the +linguistic perspective, we provide an overview +of structural and functional patterns of C-S +focusing on the literature from European and +Indian contexts as highly multilingual areas. +From the language technologies perspective, +we discuss how massive language models fail +to represent diverse C-S types due to lack of +appropriate training data, lack of robust evalu- +ation benchmarks for C-S (across multilingual +situations and types of C-S) and lack of end-to- +end systems that cover sociolinguistic aspects +of C-S as well. Our survey will be a step to- +wards an outcome of mutual benefit for com- +putational scientists and linguists with a shared +interest in multilingualism and C-S. +1 +Introduction +It is common for individuals in multilingual com- +munities to switch between languages in various +ways, in speech and in writing. +In example 1, +a bilingual child alternates between German and +Turkish (in bold) to describe her teacher at school. +Note that the Turkish possessive case marker (- +si) is attached to a German noun (Karakoc¸ and +Herkenrath, 2019). +1. Frau Kummer. Echte Name-si Christa. +Ms. Kummer. Real Name-Poss.3sg Christa. +‘Ms. Kummer. (Her) real name is Christa’ +The goal of this paper is to inform researchers in +computational linguistics (CL) and language tech- +nologies about the linguistic and social aspects of +code-switching (C-S) found in multilingual con- +texts (e.g. Europe and India) and how linguists +describe and model them. Our intent is to increase +clarity and depth in computational investigations +of C-S and to bridge the fields so that they might +be mutually reinforcing. +It is our hope that in- +terested readers can profit from the insights pro- +vided by the studies reported in this survey, for in- +stance, in understanding the factors that guide C-S +outcomes or in making use of existing annotation +schema across multilingual contexts. +2 +Competing models of C-S +For linguists, the specific ways in which languages +are switched matters. The use of a single Spanish +word in an English tweet (ex. 2) is not as syn- +tactically complicated as the integration in ex. 1. +In fact, it may not signal multilingualism at all, +simply borrowing. Many words, particularly an- +glicisms, circulate globally: marketing, feedback, +gay. +2. This is a good baile! +‘This is a good dance party!’ (Solorio and Liu, +2008) +To produce example (2), the speaker needs to +know only one Spanish word. But, to produce ex- +ample (1), the speaker has to know what word or- +der and case marker to use, and which languages +they should be drawn from. +NLP scholars are +not always concerned with the difference between +examples (1) and (2) so that, with some excep- +tions (Bhat et al., 2016), grammatical work in +NLP tends to rely heavily on the notion of a ma- +trix language model advanced by Joshi (1982) and +later adapted by Myers-Scotton (1997) as the Ma- +trix Language Frame (MLF) model. +The MLF + +holds that one language provides the grammati- +cal frame into which words or phrases from an- +other are embedded and its scope of application +is a clause. Thus, it would not apply to the al- +ternational English-Afrikaans C-S in example (3) +as each clause is in a separate language (Dulm, +2007). +3. I love Horlicks maar hier´s niks +‘I love Horlicks but there’s nothing there ’ +Although it dominates computational approaches +to C-S, the MLF is contested on empirical and the- +oretical grounds. The consistent identification of a +matrix language is not always possible, the criteria +for defining it are ambiguous, and its scope is lim- +ited (Meakins, 2012; Bhat et al., 2016; Adamou, +2016; MacSwan, 2000; Auer and Muhamedova, +2005). Bullock et al. (2018) computationally show +that different ways of determining the matrix lan- +guage only reliably converge over sentences with +simple insertions as in example (2). +For many linguists, the MLF is not the only way, +or even an adequate way, to theorize C-S. The +Equivalence Constraint (Poplack, 1980) captures +the fact that C-S tends to occur at points where the +linear structures of the contributing languages co- +incide, as when the languages involved share word +order. Other syntactic theories are built on the dif- +ferences between lexical and functional elements, +including the Government Constraint (DiSciullo +et al., 1986) and the Functional Head Constraint +(Belazi et al., 1994). Incorporating the latter in +NLP experiments has been shown to improve the +accuracy of computational and speech models (Li +and Fung, 2014; Bhat et al., 2016). Functional el- +ements include negative particles and auxiliaries, +which are respectively classified as Adverbs and +Verbs (lexical classes), in some NLP tag sets (Al- +Ghamdi et al., 2016). This means that NLP exper- +iments often use annotations that are too coarse +to be linguistically informative with regard to C- +S. Constraint-free theories (Mahootian and San- +torini, 1996; MacSwan, 2000) hold that nothing +restricts switching apart from the grammatical re- +quirements of the contributing languages. +Test- +ing such theories in NLP experiments would re- +quire syntactically parsed corpora that are rare for +mixed language data (Partanen et al., 2018). In +sum, working together, theoretical and computa- +tional linguists could create better tools for pro- +cessing C-S than those currently available. +3 +Why do speakers code-switch? +In addition to focusing on the linguistic aspects +and constraints on C-S, linguists are also inter- +ested in the social and cognitive motivations for +switching across languages. +What a (multilin- +gual) speaker is trying to achieve by switching +languages can affect its structural outcome. Lin- +guists recognize that pragmatic, interactional, and +socio-indexical functions may condition C-S pat- +terns. +For instance, Mysl´ın and Levy (2015) +demonstrate that Czech-English speakers switch +to English for high-information content words +in prominent prosodic positions when speaking +Czech. Other uses of C-S with structural traces +include signalling an in-group identity through +backflagging (Muysken, 1995) or emblematic tag- +switching (Poplack, 1980). +These are words or +phrases that are used at the edge of clauses (e.g., +Spanish ojal´a or English so). +Other functions, +among these, quoting a speaker, getting the atten- +tion of an interlocutor, or reiterating an utterance +to soften or intensify a message will also be in- +dicated via C-S in predictable linguistic construc- +tions, such as with verbs of ‘saying’, vocative ex- +pressions, and sequential translation equivalents +(Gumperz, 1982; Zentella, 1997). +According to Clyne (1991), there are eight fac- +tors (e.g. +topic, type of interaction, interlocu- +tors, role relationship, communication channel) +that can influence C-S choices. +Lavric (2007) +explains C-S choices in line with politeness the- +ory, focusing on prestige and face-saving moves in +multilingual conversations. Heller (1992) takes a +macro-social view, arguing that French-English C- +S in Quebec may signal a political choice among +both dominant and subordinate groups. +Gardner-Chloros and Edwards (2004) suggest +that social factors influence language choice, with +different generations of speakers from the same +community exhibiting very different C-S patterns. +Similarly Sebba (1998) argues that as speakers +cognitively construct equivalence between mor- +phemes, words, and phrases across their lan- +guages, communities of the same languages may +do this differently. Evidence from computational +studies suggests that C-S is speaker-dependent (Vu +et al., 2013). Gender and identity also play a role +for C-S practices in English and Greek Cypriot +community in London (Finnis, 2014). +From +a computational perspective, Papalexakis et al. +(2014) investigated the factors that influence C-S + +choices (Turkish-Dutch) in computer mediated in- +teraction and how to predict them automatically. +4 +Code-switching, Borrowing, Transfer, +Loan Translation +While C-S implies active alternation between +grammatical systems, borrowing does not. It is dif- +ficult to know if a lone word insertion (e.g. exam- +ple (2)) constitutes a borrowing or a C-S without +considering how the items are integrated into the +grammar of the receiving language (Poplack et al., +1988). When such analyses are done, most lone- +item insertions are analyzable as one-time bor- +rowings, called nonce borrowings (Sankoff et al., +1990). +Similarly, what looks like complex C-S +may not be perceived as switching at all. Auer +(1999) distinguishes a continuum of mixing types: +prototypical C-S is pragmatic and intentional, Lan- +guage Mixing serves no pragmatic purpose, and +Mixed Languages are the single code of a com- +munity. These can look structurally identical, but +the latter can be modeled as a single language +(e.g. languages like Michif Cree (Bakker, 1997) +or Gurinji Kriol (Meakins, 2012)) rather than the +intertwining of two. Bilaniuk (2004) describes the +Surzhyk spoken by urban Russian-Ukrainian bilin- +guals (in Ukraine) as ‘between C-S and Mixed +Language’ since speakers are highly bilingual and +the direction of switching is indeterminate. +Loan translation and transfer involve the words +from only one language but the semantics and +grammatical constructions from the other. In ex- +ample 4, the Turkish verb yapmak,‘ to do’, takes +on the Dutch meaning of doen in Turkish spoken +in the Netherlands (Do˘gru¨oz and Backus, 2009). +4. ˙Ilkokul-u ˙Istanbul-da yap-tı-m. +primary.school-ACC ˙Istanbul-LOC do-past- +1sg. +‘I finished primary school in Istanbul.’ +In transfer, grammatical constructions can be +borrowed from one language to another without +the words being borrowed. Treffers-Daller (2012) +demonstrates the transfer of verb particles from +Germanic languages into French. +In Brussels +French (Belgium), the construction chercher apr`es +‘look after’ (for ‘look for’) is a translation of the +Dutch equivalent and, in Ontario French (Canada), +chercher pour is the translation equivalent of En- +glish ‘look for’. +In reference French (France), +there is normally no particle following the verb. +The degree to which linguistic features like loan +translation and transfer can be found alongside C- +S is unknown. +5 +C-S across Languages: European +Context +The contexts in which people acquire and use mul- +tiple languages in Europe are diverse. Some ac- +quire their languages simultaneously from birth, +while others acquire them sequentially, either natu- +rally or via explicit instruction. Multilingualism is +the norm in many zones where local residents may +speak different languages to accommodate their +interlocutors. Speakers who use local dialects or +minoritized varieties may also be engaged in C-S +when switching between their variety and a domi- +nant one (Mills and Washington, 2015; Blom and +Gumperz, 1972). +C-S in bilingual language acquisition of chil- +dren has been studied across language contact con- +texts in Europe. In Germany, Herkenrath (2012) +and Pfaff (1999) focused on Turkish-German C-S +and Meisel (1994) on German-French C-S of bilin- +gual children. +From a comparative perspective, +Poeste et al. (2019) analyzed C-S among bilingual, +trilingual, and multilingual children growing up in +Spain and Germany. In the Netherlands, Bosma +and Blom (2019) focused on C-S among bilingual +Frisian-Dutch children. In addition to analyzing +C-S in children’s speech, Juan-Garau and Perez- +Vidal (2001) and Lanza (1998) investigated C-S +in the interaction patterns between bilingual chil- +dren and their parents (i.e. Spanish-Catalan and +English-Norwegian respectively). +Within an educational setting, Kleeman (2012) +observed C-S among bilingual (North Sami- +Norwegian) kindergarten children in the North of +Norway. Similarly, Jørgensen (1998) and Crom- +dal (2004) report the use of C-S for resolving dis- +putes among bilingual (Turkish-Danish) children +in Denmark and multilingual (Swedish-English +and/or a Non-Scandinavian Language) children in +Sweden respectively. +C-S does not only take place between standard +languages but between minority languages and +dialects as well. +For example, Themistocleous +(2013) studied C-S between Greek and Cypriot +Greek and Deuchar (2006) focused on the C-S +between Welsh and English in the UK. Berruto +(2005) reports cases of language mixing between +standard Italian and Italoromance dialects in Italy. + +In the Balkans, Kyuchukov (2006) analyzed C-S +between Turkish-Bulgarian and Romani in Bul- +garia. +C-S between dialects and/or standard vs. +minority languages in computer mediated interac- +tion was analyzed by Siebenhaar (2006) among +Swiss-German dialects and by Robert-Tissot and +Morel (2017) through SMS corpora collected +across Germanic (i.e. English and German) and +Romance languages (French, Spanish, Italian) in +Switzerland. +C-S is commonly observable across immigrant +contexts in Europe. In the UK, Georgakopoulou +and Finnis (2009) described the C-S patterns +between English and Cypriot Greek while Issa +(2006) focused on the C-S between English and +Cypriot Turkish communities in London. +Wei +and Milroy (1995) analyzed the C-S between En- +glish and Chinese from a conversational analysis +point of view based on the interactions of bilin- +gual (Chinese-English) families in Northeastern +England. In addition, O˙za´nska-Ponikwia (2016) +investigated the Polish-English C-S in the UK as +well. +C-S among immigrant community mem- +bers have also been widely studied in Germany +(e.g. Turkish-German C-S by Keim (2008) and +C¸ etino˘glu (2017), Russian-German C-S by Khaki- +mov (2016)). In the Netherlands, C-S studies in- +clude Turkish-Dutch C-S by Backus (2010) and +Dutch-Morroccan C-S by Nortier (1990). +In +Belgium, Meeuws and Blommaert (1998) stud- +ied the French-Lingala-Swahili C-S among immi- +grants of Zaire and Treffers-Daller (1994) stud- +ied French-Dutch C-S in Brussels. +In Spain, +Jieanu (2013) describes the Romanian-Spanish C- +S among the Romanian immigrants. In addition +to the C-S analyses within spoken interactions of +immigrant communities across Europe, there are +also studies about C-S within computer mediated +communication as well. +These studies include +Greek-German C-S by Androutsopoulos (2015) +in Germany, Turkish-Dutch C-S by Papalexakis +et al. (2014), Papalexakis and Do˘gru¨oz (2015) and +a comparison of Turkish-Dutch and Moroccan- +Dutch C-S by Dorleijn and Nortier (2009) in the +Netherlands. Similarly, Marley (2011) compared +French-Arabic C-S within computer mediated in- +teraction across Moroccan communities in France +and the UK. +In addition to daily communication, some lin- +guists are also interested in the C-S observed in +historical documents. +While Swain (2002) ex- +plored Latin-Greek C-S by Cicero (Roman States- +man), Dunkel (2000) analyzed C-S in his com- +munication with Atticus (Roman philosopher who +studied in Athens) in the Roman Empire. Argenter +(2001) reports cases of language mixing within the +Catalan Jewish community (in Spain) in the 14th +and 15th centuries and Rothman (2011) highlights +the C-S between Italian, Slavic and Turkish in +the historical documents about Ottoman-Venetian +relations. +In Switzerland, Volk and Clematide +(2014) worked on detecting and annotating C-S +patterns in diachronic and multilingual (English, +French, German, Italian, Romansh and Swiss Ger- +man) Alpine Heritage corpus. +Within the media context, Martin (1998) inves- +tigated English C-S in written French advertising, +and Onysko (2007) investigated the English C-S +in German written media through corpus analyses. +Zhiganova (2016) indicates that German speakers +perceive C-S into English for advertising purposes +with both positive and negative consequences. +Similar to humans, institutions and/or organiza- +tions could also have multilingual communication +with their members and/or audience. For exam- +ple, Wodak et al. (2012) analyzed the C-S and lan- +guage choice at the institutional level for European +Union institutions. +6 +C-S across Languages: Indian Context +According to the 2011 Census (Chandramouli, +2011), 26% of the population of India is bilin- +gual, while 7% is trilingual. There are 121 ma- +jor languages and 1599 other languages in India, +out of which 22 (Assamese, Bangla, Bodo, Do- +gri, Gujarati, Hindi, Kashmiri, Kannada, Konkani, +Maithili, Malayalam, Manipuri, Marathi, Nepali, +Oriya, Punjabi, Tamil, Telugu, Sanskrit, Santali, +Sindhi, Urdu) are scheduled languages with an of- +ficial recognition (almost 97% of the population +speaks one of the scheduled languages). +Most +of the population ( 93%) speak languages from +the Indo-Aryan (Hindi, Bengali, Marathi, Urdu, +Gujarati, Punjabi, Kashmiri, Rajasthani, Sindhi, +Assamese, Maithili, Odia) and Dravidian (Kan- +nada, Malayalam, Telugu, Tamil) language fami- +lies. The census excludes languages with a popu- +lation lower than 10,000 speakers. Given this, it is +probably difficult to find monolingual speakers in +India considering the linguistic diversity and wide- +spread multilingualism. + +Kachru (1978) provides one of the early stud- +ies on the types and functions of C-S in India with +a historical understanding of the multilingual con- +text. In addition to the mutual influences and con- +vergence of Indo-Aryan and Dravidian languages +internally, he mentions Persian and English as out- +side influences on Indian languages. +Similarly, +Sridhar (1978) provides an excellent comparative +overview about the functions of C-S in Kannada +in relation to the Perso-Arabic vs. English influ- +ences. +Kumar (1986) gives examples about the +formal (e.g. within NPs, PPs, VPs) and functional +(i.e. social and stylistic) aspects of Hindi-English +C-S from a theoretical point of view. +More re- +cently, Doley (2013) explains how fish mongers +in a local fish market in Assam adjust and switch +between Assamese, English and local languages +strategically to sell their products to multilingual +clientele. Another observation about C-S in daily +life comes from Boro (2020) who provides exam- +ples of English, Assamese and Bodo (another lan- +guage spoken in the Assam region) C-S and bor- +rowings. In addition to English, Portuguese was +also in contact with the local languages as a result +colonization in South India. For example, Kapp +(1997) explains the Portuguese influence through +borrowings in Dravidian languages (i.e. Kannada +and Telugu) spoken in India. +Instead of automatic data collection and meth- +ods of analyses, the C-S examples for the above- +mentioned studies are (probably) encountered and +collected by the authors themselves in daily life in- +teractions over a period of time with limited means. +Nowadays, these small sets of data would be re- +garded as insignificant in computational areas of +research. +However, ignoring these studies and +data could have serious consequences since cru- +cial information about the social and cultural dy- +namics in a multilingual setting would also be lost. +For example, Nadkarni (1975) proves this point +by explaining how social factors influence the C- +S between Saraswat Brahmin dialect of Konkani +(Indo-Aryan language) and Kannada (Dravidian +language) in the South of India. Both languages +have been in contact with each other for over +four hundred years. Saraswat Brahmins are flu- +ent in both Konkani and Kannada but they do not +speak Konkani with Kannada speakers and they +also do not C-S between Konkani and Kannada. +Nadkarni (1975) attributes this preference to the +high prestige associated with Konkani within the +given social context. Since Kannada (perceived +as less prestigious) is widely spoken in that re- +gion, Konkani speakers learn and speak Kannada +for functional purposes in daily life which does not +involve C-S. However, it is not common for Kan- +nada speakers to learn and speak Konkani (Nad- +karni, 1975). +C-S in India has been investigated through +written media, advertising and film industry as +well. +Si (2011) analyzed Hindi-English C-S in +the scripts of seven Bollywood movies which were +filmed between 1982 and 2004. +Her results in- +dicate a change of direction C-S over the years. +More specifically, Hindi was the dominant lan- +guage with occasional switches to English for the +early productions but English became the domi- +nant language especially for younger generations +in the later productions. +A similar trend has +been observed for Bengali movie scripts as well. +Through analyzing movie scripts (between 1970s +and 2010s), Chatterjee (2016) finds a drastic in- +crease in the use of bilingual verbs (e.g. +reno- +vate koreche “renovation do”) over time and at- +tributes this rise to the increasing popularity of +English in Indian society. Within the immigrant +context, Gardner-Chloros and Charles (2007) fo- +cused on the types and functions of C-S between +Hindi and English across the TV programs (e.g. +highly scripted vs. +loosely scripted programs) +of a British/Asian cable channel in the UK. Al- +though they have come across C-S in a variety +of TV shows, the least amount of C-S was en- +countered in the news broadcasts (i.e. +highly +scripted). In general, they have encountered less +C-S on TV broadcasts in comparison to the natu- +ral speech and attribute this factor to the conscious- +ness of TV personalities about pure language use +(instead of C-S). Similarly, Zipp (2017) analyzed +Gujarati-English C-S within a radio show target- +ing British South Asians living in the US and con- +cluded that C-S was part of identity construction +among youngsters (group identity). Pratapa and +Choudhury (2017) perform a quantitative study of +18 recent Bollywood (Hindi) movies and find that +C-S is used for establishing identity, social dynam- +ics between characters and the socio-cultural con- +text of the movie. +From an advertising point of view, Kathpalia +and Wee Ong (2015) analyzed C-S in Hinglish +(i.e. Hindi, English, Urdu, Sanskrit according to +their definition) billboards about the Amul brand + +in India. After compiling the advertisements on +billboards (1191), they classified the structures +and functions of C-S. Their results indicate more +intrasentential C-S than intersentential ones on +the billboards. +In terms of function, the ad- +vertisers used C-S to indicate figures of speech +(e.g. +puns, associations, contradictory associa- +tions, word-creation and repetitions) to attract the +attention of the target group. +Mohanty (2006) provides an extended overview +of the multilingual education system in India ex- +ploring the types and quality of schools across +a wide spectrum. +In general, high-cost English +Medium (EM) education is valued by upper-class +and affluent families. Although low-cost EM edu- +cation is also available for lower income families, +he questions its impact in comparison to education +in the local languages. +Sridhar (2002) explains +that C-S is commonly practiced among students +in schools across India. In addition, she finds it +unrealistic to ask the students to separate the two +languages harshly. +In immigrant contexts, Mar- +tin et al. (2006) investigates how Gujarati-English +C-S is used among the South Asian students in +educational settings in the UK. Another analy- +sis reveals a shift from Bengali toward English +among the younger generations of the immigrant +Bengali community in the UK (Al-Azami, 2006). +In terms of the C-S patterns, first generation im- +migrants integrate English words while speaking +Bengali whereas English dominates the conver- +sations of younger generations with occasional +switches to Bengali. There are also studies about +Bengali-English C-S in the UK school settings +(Pagett, 2006) and Bangladesh (Obaidullah, 2016) +as well. +However, a systematic comparison be- +tween Bengali-English C-S in India, Bangladesh +and immigrant settings are lacking. +In their study about aphasic patients, Shya- +mala Chengappa and Bhat (2004) report increased +frequency of C-S between Malayalam and English +for aphasic patients in comparison to the control +group. However, there were less differences be- +tween the groups in terms of functions of C-S. +Deepa and Shyamala (2019) find that amount and +types of C-S could be used to differentiate between +healthy and mild dementia patients who are bilin- +gual in Kannada and English. Although both stud- +ies are carried out with limited subjects, they offer +insights about the use of C-S in health settings as +well. +7 +Computational Approaches to C-S +There has been significant interest in building lan- +guage technologies for code-switched languages +over the last few years, spanning a diverse range +of tasks such as Language Identification, Part +of Speech Tagging, Sentiment Analysis and Au- +tomatic Speech Recognition. +In the European +language context, work has mainly focused on +Turkish-Dutch, Frisian-Dutch, Turkish-German +and Ukranian-Russian with some initial attempts +being made in parsing Russian-Komi text. +In +the Indian language context, Hindi-English is the +most widely studied language pair for compu- +tational processing, with some recent work on +Telugu-English, Tamil-English, Bengali-English +and Gujarati-English. Sitaram et al. (2019) pro- +vide a comprehensive survey of research in com- +putational processing of C-S text and speech and +Jose et al. (2020) present a list of datasets available +for C-S research. However, despite significant ef- +forts, language technologies are not yet capable +of processing C-S as seamlessly as monolingual +data. We identify three main limitations of the cur- +rent state of computational processing of C-S: data, +evaluation and user-facing applications. +7.1 +Data +The use of Deep Neural Networks, which require +large amounts of labeled and unlabeled training +data have become the de facto standard for build- +ing speech and NLP systems. Since C-S languages +tend to be low resourced, building Deep Learning- +based models is challenging due to the lack of +large C-S datasets. +Massive multilingual Lan- +guage Models (LMs) such as multilingual BERT +(Devlin et al., 2019) and XLM-R (Conneau et al., +2020) have shown promise in enabling the cover- +age of low-resource languages without any labeled +data by using the zero-shot framework. +These +LMs are typically trained in two phases: a “pre- +training” phase, in which unlabeled data from one +or multiple languages may be used and a “fine- +tuning” phase, in which task-specific labeled data +is used to build a system capable of solving the +task. +Since multilingual LMs are trained on multiple +languages at the same time, it has been suggested +that these models may be capable of processing +C-S text (Johnson et al., 2017), with promising re- +sults initially reported on POS tagging (Pires et al., +2019). Khanuja et al. (2020) found that multilin- + +gual BERT outperforms older task-specific mod- +els on C-S tasks, however, the performance on +C-S is much worse than the performance on the +same tasks in a monolingual setting. Further, these +LMs are either trained primarily on monolingual +datasets such as Wikipedia in the case of mBERT, +or Common Crawl 1 in the case of XLM-R. So, +they are either not exposed to C-S data at all dur- +ing training, or they miss out on several language +pairs, types and functions of C-S that are encoun- +tered in daily life but not available on the web. +Since massive multilingual LMs are now replac- +ing traditional models across many NLP applica- +tions, it is crucial to consider how they can be +trained on C-S data, or made to work for C-S by +incorporating other sources of knowledge. +7.2 +Evaluation +Much of speech and NLP research is now driven +by standard benchmarks that evaluate models +across multiple tasks and languages. +Due to +the shortage of standardized datasets for C-S, un- +til recently, the evaluation of C-S models was +performed over individual tasks and language +pairs. +Khanuja et al. (2020) and Aguilar et al. +(2020) proposed the first evaluation benchmarks +for C-S that span multiple tasks in multiple lan- +guage pairs. The GLUECoS benchmark (Khanuja +et al., 2020) consists of the following C-S tasks +in Spanish-English and Hindi-English: Language +Identification (LID), Part of Speech (POS) tag- +ging, Named Entity Recognition (NER), Senti- +ment Analysis, Question Answering and Natural +Language Inference (NLI). The LINCE bench- +mark (Aguilar et al., 2020) covers Language +Identification, Named Entity Recognition, Part- +of-Speech Tagging, and Sentiment Analysis in +four language pairs: +Spanish-English, Nepali- +English, Hindi-English, and Modern Standard +Arabic-Egyptian Arabic. +Although these benchmarks are important start- +ing points for C-S, it is clear that they do not +represent the entire spectrum of C-S, both from +the point of view of potential applications and +language pairs. +Further, it is important to note +that while state-of-the-art models perform well on +tasks such as LID, POS tagging and NER, they are +only slightly better than chance when it comes to +harder tasks like NLI, showing that current models +are not capable of processing C-S language. More- +1http://www.commoncrawl.org +over, many of the C-S tasks in the benchmarks +above consist of annotated tweets, which only rep- +resent a certain type of C-S. Due to these limita- +tions, we currently do not have an accurate picture +of how well models are able to handle C-S. +7.3 +User-facing applications +Although speech and NLP models for C-S have +been built for various applications, a major limi- +tation of the work done so far in computational +processing of C-S is the lack of end-to-end user- +facing applications that interact directly with users +in multilingual communities. For example, there +is no widely-used spoken dialogue system that +can understand as well as produce code-switched +speech, although some voice assistants may recog- +nize and produce C-S in limited scenarios in some +locales. Although computational implementations +of grammatical models of C-S exist (Bhat et al., +2016), they do not necessarily generate natural C- +S utterances that a bilingual speaker would pro- +duce (Pratapa et al., 2018). Most crucially, current +computational approaches to C-S language tech- +nologies do not usually take into account the lin- +guistic and social factors that influence why and +when speakers/users choose to code-switch. +Bawa et al. (2020) conducted a Wizard-of-Oz +study using a Hindi-English chatbot and found that +not only did bilingual users prefer chatbots that +could code-switch, they also showed a preference +towards bots that mimicked their own C-S patterns. +Rudra et al. (2016) report a study on 430k tweets +from Hindi-English bilingual users and find that +Hindi is preferred for the expression of negative +sentiment. +In a follow-up study, Agarwal et al. +(2017) find that Hindi is the preferred language for +swearing in Hindi-English C-S tweets, and swear- +ing may be a motivating factor for users to switch +to Hindi. The study also finds a gender difference, +with women preferring to swear in English more +often than Hindi. Such studies indicate that mul- +tilingual chatbots and intelligent agents need to +be able to adapt to users’ linguistic styles, while +also being capable of determining when and how +to code-switch. +Due to the paucity of user-facing systems and +standard benchmarks covering only a handful of +simpler NLP tasks, it is likely that we overesti- +mate how well computational models are able to +handle C-S. In sum, language technologies for C- +S seem to be constrained by the lack of availabil- + +ity of diverse C-S training data, evaluation bench- +marks and the absence of user-facing applications. +They need to go beyond pattern recognition and +grammatical constraints of C-S in order to process +and produce C-S the way humans do. Hence, it +is important for the CL community to be aware of +the vast literature around C-S in linguistics, partic- +ularly as we proceed to solving more challenging +tasks. +8 +Conclusion +The goal of this paper was to inform computa- +tional linguists and language technologists about +the linguistic and social aspects C-S studies focus- +ing on the European and Indian multilingual con- +texts. There are some similarities (e.g. themes for +linguistic research in C-S) but also differences be- +tween the two contexts in terms of the social, cul- +tural and historical characteristics. For example, +C-S in immigrant communities has been a com- +mon theme for both multilingual contexts. In Eu- +rope, C-S has been widely studied within the im- +migrant communities who have come through la- +bor immigration in the 1960s. However, there is +a need for more research about the C-S in immi- +grant languages with a more recent history as well +as minority languages and regional dialects. An- +alyzing C-S in the immigration context is even +more complex for Indian languages. +There are +hardly any systematic linguistic comparisons be- +tween the C-S within the same language pairs in +India and immigrant contexts (e.g. C-S between +Hindi-English in India vs. Hindi-English in the +US/UK). There is also a need for more research +about C-S between less known language pairs in +India. However, some of these languages are not +even officially listed (e.g. in census results) since +they have less than 10,000 speakers. In these cases, +collecting and analyzing the multilingual and C-S +data becomes more difficult. +A common flaw that is shared both by linguis- +tics and computational areas of research is to focus +only on the positive evidence and assume that C-S +will occur in all multilingual contexts. However, +there is also a need for negative evidence to fal- +sify this assumption. As illustrated through an ex- +ample from Konkani-Kannada language contact in +India (see section 6), bilingual speakers may pre- +fer not to C-S due to historical, social and cultural +factors operating in that setting. Therefore, devel- +oping an automatic C-S system for a random pair +of languages without an in-depth and systematic +analysis of linguistic and social aspects of C-S in +a particular context would not be very useful for +the targeted users and/or language technologists. +To date, the literature focusing on the social and +linguistic aspects of C-S is less visible for CL re- +searchers. This lack of visibility leads to misunder- +standings and/or incomplete citations of earlier re- +search which would have saved time and resources +for CL research if resolved. One of the reasons +is perhaps the differences in publishing traditions +between humanities and computational areas of re- +search. Conference and workshop proceedings are +commonly accepted means of publication in com- +putational linguistics. Whereas, journal publica- +tions, books and/or chapters are the publication +forms in humanities. However, guidelines about +how to cite publications in humanities are often +missing in computational venues. There are also +differences in terms of length, review cycles and +open access policies between the two fields which +may influence the visibility of research output for +each other. It is perhaps useful to remember that +science advances by standing on the shoulders of +giants (i.e. building upon earlier research). With +our contribution to the conference, we hope to con- +nect the two fields and start a scientific dialogue to +enhance the advancement in both fields. +References +Evangelia Adamou. 2016. 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The study of the perception +of code-switching to English in German advertising. +Procedia: Social and Behavioral Sciences, 236:225– +229. +Lena Zipp. 2017. Code-switching in the media: Iden- +tity negotiations in a Gujarati diaspora radio pro- +gram. International Journal of the Sociology of Lan- +guage, 2017(247):33 – 48. + diff --git a/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/load_file.txt b/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e35878e2b45aa5a25444b010001ddd7d0203cb0 --- /dev/null +++ b/DtA0T4oBgHgl3EQfAv-e/content/tmp_files/load_file.txt @@ -0,0 +1,848 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf,len=847 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='01967v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='CL] 5 Jan 2023 A Survey of Code-switching: Linguistic and Social Perspectives for Language Technologies A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Seza Do˘gru¨oz Ghent University, Gent, Belgium as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='dogruoz@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='be Sunayana Sitaram Microsoft Research India, Bangalore, India sunayana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='sitaram@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='com Barbara E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bullock UT at Austin, Austin, USA bbullock@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='edu Almeida Jacqueline Toribio UT at Austin, Austin, USA toribio@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='edu Abstract The analysis of data in which multiple lan- guages are represented has gained popularity among computational linguists in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' So far, much of this research focuses mainly on the improvement of computational meth- ods and largely ignores linguistic and social aspects of C-S discussed across a wide range of languages within the long-established liter- ature in linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' To fill this gap, we offer a survey of code-switching (C-S) covering the literature in linguistics with a reflection on the key issues in language technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' From the linguistic perspective, we provide an overview of structural and functional patterns of C-S focusing on the literature from European and Indian contexts as highly multilingual areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' From the language technologies perspective, we discuss how massive language models fail to represent diverse C-S types due to lack of appropriate training data, lack of robust evalu- ation benchmarks for C-S (across multilingual situations and types of C-S) and lack of end-to- end systems that cover sociolinguistic aspects of C-S as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Our survey will be a step to- wards an outcome of mutual benefit for com- putational scientists and linguists with a shared interest in multilingualism and C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 1 Introduction It is common for individuals in multilingual com- munities to switch between languages in various ways, in speech and in writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In example 1, a bilingual child alternates between German and Turkish (in bold) to describe her teacher at school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Note that the Turkish possessive case marker (- si) is attached to a German noun (Karakoc¸ and Herkenrath, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Frau Kummer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Echte Name-si Christa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Kummer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Real Name-Poss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='3sg Christa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' ‘Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Kummer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (Her) real name is Christa’ The goal of this paper is to inform researchers in computational linguistics (CL) and language tech- nologies about the linguistic and social aspects of code-switching (C-S) found in multilingual con- texts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Europe and India) and how linguists describe and model them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Our intent is to increase clarity and depth in computational investigations of C-S and to bridge the fields so that they might be mutually reinforcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' It is our hope that in- terested readers can profit from the insights pro- vided by the studies reported in this survey, for in- stance, in understanding the factors that guide C-S outcomes or in making use of existing annotation schema across multilingual contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2 Competing models of C-S For linguists, the specific ways in which languages are switched matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The use of a single Spanish word in an English tweet (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2) is not as syn- tactically complicated as the integration in ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In fact, it may not signal multilingualism at all, simply borrowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Many words, particularly an- glicisms, circulate globally: marketing, feedback, gay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' This is a good baile!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' ‘This is a good dance party!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (Solorio and Liu, 2008) To produce example (2), the speaker needs to know only one Spanish word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' But, to produce ex- ample (1), the speaker has to know what word or- der and case marker to use, and which languages they should be drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' NLP scholars are not always concerned with the difference between examples (1) and (2) so that, with some excep- tions (Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2016), grammatical work in NLP tends to rely heavily on the notion of a ma- trix language model advanced by Joshi (1982) and later adapted by Myers-Scotton (1997) as the Ma- trix Language Frame (MLF) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The MLF holds that one language provides the grammati- cal frame into which words or phrases from an- other are embedded and its scope of application is a clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Thus, it would not apply to the al- ternational English-Afrikaans C-S in example (3) as each clause is in a separate language (Dulm, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' I love Horlicks maar hier´s niks ‘I love Horlicks but there’s nothing there ’ Although it dominates computational approaches to C-S, the MLF is contested on empirical and the- oretical grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The consistent identification of a matrix language is not always possible, the criteria for defining it are ambiguous, and its scope is lim- ited (Meakins, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Adamou, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' MacSwan, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Auer and Muhamedova, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bullock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2018) computationally show that different ways of determining the matrix lan- guage only reliably converge over sentences with simple insertions as in example (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For many linguists, the MLF is not the only way, or even an adequate way, to theorize C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The Equivalence Constraint (Poplack, 1980) captures the fact that C-S tends to occur at points where the linear structures of the contributing languages co- incide, as when the languages involved share word order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Other syntactic theories are built on the dif- ferences between lexical and functional elements, including the Government Constraint (DiSciullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 1986) and the Functional Head Constraint (Belazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Incorporating the latter in NLP experiments has been shown to improve the accuracy of computational and speech models (Li and Fung, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Functional el- ements include negative particles and auxiliaries, which are respectively classified as Adverbs and Verbs (lexical classes), in some NLP tag sets (Al- Ghamdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' This means that NLP exper- iments often use annotations that are too coarse to be linguistically informative with regard to C- S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Constraint-free theories (Mahootian and San- torini, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' MacSwan, 2000) hold that nothing restricts switching apart from the grammatical re- quirements of the contributing languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Test- ing such theories in NLP experiments would re- quire syntactically parsed corpora that are rare for mixed language data (Partanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In sum, working together, theoretical and computa- tional linguists could create better tools for pro- cessing C-S than those currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 3 Why do speakers code-switch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to focusing on the linguistic aspects and constraints on C-S, linguists are also inter- ested in the social and cognitive motivations for switching across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' What a (multilin- gual) speaker is trying to achieve by switching languages can affect its structural outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Lin- guists recognize that pragmatic, interactional, and socio-indexical functions may condition C-S pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For instance, Mysl´ın and Levy (2015) demonstrate that Czech-English speakers switch to English for high-information content words in prominent prosodic positions when speaking Czech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Other uses of C-S with structural traces include signalling an in-group identity through backflagging (Muysken, 1995) or emblematic tag- switching (Poplack, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' These are words or phrases that are used at the edge of clauses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', Spanish ojal´a or English so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Other functions, among these, quoting a speaker, getting the atten- tion of an interlocutor, or reiterating an utterance to soften or intensify a message will also be in- dicated via C-S in predictable linguistic construc- tions, such as with verbs of ‘saying’, vocative ex- pressions, and sequential translation equivalents (Gumperz, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Zentella, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' According to Clyne (1991), there are eight fac- tors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' topic, type of interaction, interlocu- tors, role relationship, communication channel) that can influence C-S choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Lavric (2007) explains C-S choices in line with politeness the- ory, focusing on prestige and face-saving moves in multilingual conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Heller (1992) takes a macro-social view, arguing that French-English C- S in Quebec may signal a political choice among both dominant and subordinate groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Gardner-Chloros and Edwards (2004) suggest that social factors influence language choice, with different generations of speakers from the same community exhibiting very different C-S patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly Sebba (1998) argues that as speakers cognitively construct equivalence between mor- phemes, words, and phrases across their lan- guages, communities of the same languages may do this differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Evidence from computational studies suggests that C-S is speaker-dependent (Vu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Gender and identity also play a role for C-S practices in English and Greek Cypriot community in London (Finnis, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' From a computational perspective, Papalexakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2014) investigated the factors that influence C-S choices (Turkish-Dutch) in computer mediated in- teraction and how to predict them automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 4 Code-switching, Borrowing, Transfer, Loan Translation While C-S implies active alternation between grammatical systems, borrowing does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' It is dif- ficult to know if a lone word insertion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' exam- ple (2)) constitutes a borrowing or a C-S without considering how the items are integrated into the grammar of the receiving language (Poplack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' When such analyses are done, most lone- item insertions are analyzable as one-time bor- rowings, called nonce borrowings (Sankoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly, what looks like complex C-S may not be perceived as switching at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Auer (1999) distinguishes a continuum of mixing types: prototypical C-S is pragmatic and intentional, Lan- guage Mixing serves no pragmatic purpose, and Mixed Languages are the single code of a com- munity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' These can look structurally identical, but the latter can be modeled as a single language (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' languages like Michif Cree (Bakker, 1997) or Gurinji Kriol (Meakins, 2012)) rather than the intertwining of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bilaniuk (2004) describes the Surzhyk spoken by urban Russian-Ukrainian bilin- guals (in Ukraine) as ‘between C-S and Mixed Language’ since speakers are highly bilingual and the direction of switching is indeterminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Loan translation and transfer involve the words from only one language but the semantics and grammatical constructions from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In ex- ample 4, the Turkish verb yapmak,‘ to do’, takes on the Dutch meaning of doen in Turkish spoken in the Netherlands (Do˘gru¨oz and Backus, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' ˙Ilkokul-u ˙Istanbul-da yap-tı-m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='school-ACC ˙Istanbul-LOC do-past- 1sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' ‘I finished primary school in Istanbul.’ In transfer, grammatical constructions can be borrowed from one language to another without the words being borrowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Treffers-Daller (2012) demonstrates the transfer of verb particles from Germanic languages into French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Brussels French (Belgium), the construction chercher apr`es ‘look after’ (for ‘look for’) is a translation of the Dutch equivalent and, in Ontario French (Canada), chercher pour is the translation equivalent of En- glish ‘look for’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In reference French (France), there is normally no particle following the verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The degree to which linguistic features like loan translation and transfer can be found alongside C- S is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 5 C-S across Languages: European Context The contexts in which people acquire and use mul- tiple languages in Europe are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Some ac- quire their languages simultaneously from birth, while others acquire them sequentially, either natu- rally or via explicit instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Multilingualism is the norm in many zones where local residents may speak different languages to accommodate their interlocutors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Speakers who use local dialects or minoritized varieties may also be engaged in C-S when switching between their variety and a domi- nant one (Mills and Washington, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Blom and Gumperz, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S in bilingual language acquisition of chil- dren has been studied across language contact con- texts in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Germany, Herkenrath (2012) and Pfaff (1999) focused on Turkish-German C-S and Meisel (1994) on German-French C-S of bilin- gual children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' From a comparative perspective, Poeste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2019) analyzed C-S among bilingual, trilingual, and multilingual children growing up in Spain and Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the Netherlands, Bosma and Blom (2019) focused on C-S among bilingual Frisian-Dutch children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to analyzing C-S in children’s speech, Juan-Garau and Perez- Vidal (2001) and Lanza (1998) investigated C-S in the interaction patterns between bilingual chil- dren and their parents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Spanish-Catalan and English-Norwegian respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Within an educational setting, Kleeman (2012) observed C-S among bilingual (North Sami- Norwegian) kindergarten children in the North of Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly, Jørgensen (1998) and Crom- dal (2004) report the use of C-S for resolving dis- putes among bilingual (Turkish-Danish) children in Denmark and multilingual (Swedish-English and/or a Non-Scandinavian Language) children in Sweden respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S does not only take place between standard languages but between minority languages and dialects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For example, Themistocleous (2013) studied C-S between Greek and Cypriot Greek and Deuchar (2006) focused on the C-S between Welsh and English in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Berruto (2005) reports cases of language mixing between standard Italian and Italoromance dialects in Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the Balkans, Kyuchukov (2006) analyzed C-S between Turkish-Bulgarian and Romani in Bul- garia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S between dialects and/or standard vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' minority languages in computer mediated interac- tion was analyzed by Siebenhaar (2006) among Swiss-German dialects and by Robert-Tissot and Morel (2017) through SMS corpora collected across Germanic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' English and German) and Romance languages (French, Spanish, Italian) in Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S is commonly observable across immigrant contexts in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the UK, Georgakopoulou and Finnis (2009) described the C-S patterns between English and Cypriot Greek while Issa (2006) focused on the C-S between English and Cypriot Turkish communities in London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Wei and Milroy (1995) analyzed the C-S between En- glish and Chinese from a conversational analysis point of view based on the interactions of bilin- gual (Chinese-English) families in Northeastern England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition, O˙za´nska-Ponikwia (2016) investigated the Polish-English C-S in the UK as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S among immigrant community mem- bers have also been widely studied in Germany (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Turkish-German C-S by Keim (2008) and C¸ etino˘glu (2017), Russian-German C-S by Khaki- mov (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the Netherlands, C-S studies in- clude Turkish-Dutch C-S by Backus (2010) and Dutch-Morroccan C-S by Nortier (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Belgium, Meeuws and Blommaert (1998) stud- ied the French-Lingala-Swahili C-S among immi- grants of Zaire and Treffers-Daller (1994) stud- ied French-Dutch C-S in Brussels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Spain, Jieanu (2013) describes the Romanian-Spanish C- S among the Romanian immigrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to the C-S analyses within spoken interactions of immigrant communities across Europe, there are also studies about C-S within computer mediated communication as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' These studies include Greek-German C-S by Androutsopoulos (2015) in Germany, Turkish-Dutch C-S by Papalexakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2014), Papalexakis and Do˘gru¨oz (2015) and a comparison of Turkish-Dutch and Moroccan- Dutch C-S by Dorleijn and Nortier (2009) in the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly, Marley (2011) compared French-Arabic C-S within computer mediated in- teraction across Moroccan communities in France and the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to daily communication, some lin- guists are also interested in the C-S observed in historical documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' While Swain (2002) ex- plored Latin-Greek C-S by Cicero (Roman States- man), Dunkel (2000) analyzed C-S in his com- munication with Atticus (Roman philosopher who studied in Athens) in the Roman Empire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Argenter (2001) reports cases of language mixing within the Catalan Jewish community (in Spain) in the 14th and 15th centuries and Rothman (2011) highlights the C-S between Italian, Slavic and Turkish in the historical documents about Ottoman-Venetian relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Switzerland, Volk and Clematide (2014) worked on detecting and annotating C-S patterns in diachronic and multilingual (English, French, German, Italian, Romansh and Swiss Ger- man) Alpine Heritage corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Within the media context, Martin (1998) inves- tigated English C-S in written French advertising, and Onysko (2007) investigated the English C-S in German written media through corpus analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Zhiganova (2016) indicates that German speakers perceive C-S into English for advertising purposes with both positive and negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similar to humans, institutions and/or organiza- tions could also have multilingual communication with their members and/or audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For exam- ple, Wodak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2012) analyzed the C-S and lan- guage choice at the institutional level for European Union institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 6 C-S across Languages: Indian Context According to the 2011 Census (Chandramouli, 2011), 26% of the population of India is bilin- gual, while 7% is trilingual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There are 121 ma- jor languages and 1599 other languages in India, out of which 22 (Assamese, Bangla, Bodo, Do- gri, Gujarati, Hindi, Kashmiri, Kannada, Konkani, Maithili, Malayalam, Manipuri, Marathi, Nepali, Oriya, Punjabi, Tamil, Telugu, Sanskrit, Santali, Sindhi, Urdu) are scheduled languages with an of- ficial recognition (almost 97% of the population speaks one of the scheduled languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Most of the population ( 93%) speak languages from the Indo-Aryan (Hindi, Bengali, Marathi, Urdu, Gujarati, Punjabi, Kashmiri, Rajasthani, Sindhi, Assamese, Maithili, Odia) and Dravidian (Kan- nada, Malayalam, Telugu, Tamil) language fami- lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The census excludes languages with a popu- lation lower than 10,000 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Given this, it is probably difficult to find monolingual speakers in India considering the linguistic diversity and wide- spread multilingualism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Kachru (1978) provides one of the early stud- ies on the types and functions of C-S in India with a historical understanding of the multilingual con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to the mutual influences and con- vergence of Indo-Aryan and Dravidian languages internally, he mentions Persian and English as out- side influences on Indian languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly, Sridhar (1978) provides an excellent comparative overview about the functions of C-S in Kannada in relation to the Perso-Arabic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' English influ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Kumar (1986) gives examples about the formal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' within NPs, PPs, VPs) and functional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' social and stylistic) aspects of Hindi-English C-S from a theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' More re- cently, Doley (2013) explains how fish mongers in a local fish market in Assam adjust and switch between Assamese, English and local languages strategically to sell their products to multilingual clientele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Another observation about C-S in daily life comes from Boro (2020) who provides exam- ples of English, Assamese and Bodo (another lan- guage spoken in the Assam region) C-S and bor- rowings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition to English, Portuguese was also in contact with the local languages as a result colonization in South India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For example, Kapp (1997) explains the Portuguese influence through borrowings in Dravidian languages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Kannada and Telugu) spoken in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Instead of automatic data collection and meth- ods of analyses, the C-S examples for the above- mentioned studies are (probably) encountered and collected by the authors themselves in daily life in- teractions over a period of time with limited means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Nowadays, these small sets of data would be re- garded as insignificant in computational areas of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, ignoring these studies and data could have serious consequences since cru- cial information about the social and cultural dy- namics in a multilingual setting would also be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For example, Nadkarni (1975) proves this point by explaining how social factors influence the C- S between Saraswat Brahmin dialect of Konkani (Indo-Aryan language) and Kannada (Dravidian language) in the South of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Both languages have been in contact with each other for over four hundred years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Saraswat Brahmins are flu- ent in both Konkani and Kannada but they do not speak Konkani with Kannada speakers and they also do not C-S between Konkani and Kannada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Nadkarni (1975) attributes this preference to the high prestige associated with Konkani within the given social context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Since Kannada (perceived as less prestigious) is widely spoken in that re- gion, Konkani speakers learn and speak Kannada for functional purposes in daily life which does not involve C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, it is not common for Kan- nada speakers to learn and speak Konkani (Nad- karni, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S in India has been investigated through written media, advertising and film industry as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Si (2011) analyzed Hindi-English C-S in the scripts of seven Bollywood movies which were filmed between 1982 and 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Her results in- dicate a change of direction C-S over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' More specifically, Hindi was the dominant lan- guage with occasional switches to English for the early productions but English became the domi- nant language especially for younger generations in the later productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' A similar trend has been observed for Bengali movie scripts as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Through analyzing movie scripts (between 1970s and 2010s), Chatterjee (2016) finds a drastic in- crease in the use of bilingual verbs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' reno- vate koreche “renovation do”) over time and at- tributes this rise to the increasing popularity of English in Indian society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Within the immigrant context, Gardner-Chloros and Charles (2007) fo- cused on the types and functions of C-S between Hindi and English across the TV programs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' highly scripted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' loosely scripted programs) of a British/Asian cable channel in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Al- though they have come across C-S in a variety of TV shows, the least amount of C-S was en- countered in the news broadcasts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' highly scripted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In general, they have encountered less C-S on TV broadcasts in comparison to the natu- ral speech and attribute this factor to the conscious- ness of TV personalities about pure language use (instead of C-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Similarly, Zipp (2017) analyzed Gujarati-English C-S within a radio show target- ing British South Asians living in the US and con- cluded that C-S was part of identity construction among youngsters (group identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Pratapa and Choudhury (2017) perform a quantitative study of 18 recent Bollywood (Hindi) movies and find that C-S is used for establishing identity, social dynam- ics between characters and the socio-cultural con- text of the movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' From an advertising point of view, Kathpalia and Wee Ong (2015) analyzed C-S in Hinglish (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Hindi, English, Urdu, Sanskrit according to their definition) billboards about the Amul brand in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' After compiling the advertisements on billboards (1191), they classified the structures and functions of C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Their results indicate more intrasentential C-S than intersentential ones on the billboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In terms of function, the ad- vertisers used C-S to indicate figures of speech (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' puns, associations, contradictory associa- tions, word-creation and repetitions) to attract the attention of the target group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Mohanty (2006) provides an extended overview of the multilingual education system in India ex- ploring the types and quality of schools across a wide spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In general, high-cost English Medium (EM) education is valued by upper-class and affluent families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Although low-cost EM edu- cation is also available for lower income families, he questions its impact in comparison to education in the local languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Sridhar (2002) explains that C-S is commonly practiced among students in schools across India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In addition, she finds it unrealistic to ask the students to separate the two languages harshly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In immigrant contexts, Mar- tin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2006) investigates how Gujarati-English C-S is used among the South Asian students in educational settings in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Another analy- sis reveals a shift from Bengali toward English among the younger generations of the immigrant Bengali community in the UK (Al-Azami, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In terms of the C-S patterns, first generation im- migrants integrate English words while speaking Bengali whereas English dominates the conver- sations of younger generations with occasional switches to Bengali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There are also studies about Bengali-English C-S in the UK school settings (Pagett, 2006) and Bangladesh (Obaidullah, 2016) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, a systematic comparison be- tween Bengali-English C-S in India, Bangladesh and immigrant settings are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In their study about aphasic patients, Shya- mala Chengappa and Bhat (2004) report increased frequency of C-S between Malayalam and English for aphasic patients in comparison to the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, there were less differences be- tween the groups in terms of functions of C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Deepa and Shyamala (2019) find that amount and types of C-S could be used to differentiate between healthy and mild dementia patients who are bilin- gual in Kannada and English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Although both stud- ies are carried out with limited subjects, they offer insights about the use of C-S in health settings as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 7 Computational Approaches to C-S There has been significant interest in building lan- guage technologies for code-switched languages over the last few years, spanning a diverse range of tasks such as Language Identification, Part of Speech Tagging, Sentiment Analysis and Au- tomatic Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the European language context, work has mainly focused on Turkish-Dutch, Frisian-Dutch, Turkish-German and Ukranian-Russian with some initial attempts being made in parsing Russian-Komi text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In the Indian language context, Hindi-English is the most widely studied language pair for compu- tational processing, with some recent work on Telugu-English, Tamil-English, Bengali-English and Gujarati-English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Sitaram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2019) pro- vide a comprehensive survey of research in com- putational processing of C-S text and speech and Jose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2020) present a list of datasets available for C-S research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, despite significant ef- forts, language technologies are not yet capable of processing C-S as seamlessly as monolingual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' We identify three main limitations of the cur- rent state of computational processing of C-S: data, evaluation and user-facing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='1 Data The use of Deep Neural Networks, which require large amounts of labeled and unlabeled training data have become the de facto standard for build- ing speech and NLP systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Since C-S languages tend to be low resourced, building Deep Learning- based models is challenging due to the lack of large C-S datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Massive multilingual Lan- guage Models (LMs) such as multilingual BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2019) and XLM-R (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2020) have shown promise in enabling the cover- age of low-resource languages without any labeled data by using the zero-shot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' These LMs are typically trained in two phases: a “pre- training” phase, in which unlabeled data from one or multiple languages may be used and a “fine- tuning” phase, in which task-specific labeled data is used to build a system capable of solving the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Since multilingual LMs are trained on multiple languages at the same time, it has been suggested that these models may be capable of processing C-S text (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2017), with promising re- sults initially reported on POS tagging (Pires et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Khanuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2020) found that multilin- gual BERT outperforms older task-specific mod- els on C-S tasks, however, the performance on C-S is much worse than the performance on the same tasks in a monolingual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Further, these LMs are either trained primarily on monolingual datasets such as Wikipedia in the case of mBERT, or Common Crawl 1 in the case of XLM-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' So, they are either not exposed to C-S data at all dur- ing training, or they miss out on several language pairs, types and functions of C-S that are encoun- tered in daily life but not available on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Since massive multilingual LMs are now replac- ing traditional models across many NLP applica- tions, it is crucial to consider how they can be trained on C-S data, or made to work for C-S by incorporating other sources of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='2 Evaluation Much of speech and NLP research is now driven by standard benchmarks that evaluate models across multiple tasks and languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Due to the shortage of standardized datasets for C-S, un- til recently, the evaluation of C-S models was performed over individual tasks and language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Khanuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2020) and Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2020) proposed the first evaluation benchmarks for C-S that span multiple tasks in multiple lan- guage pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The GLUECoS benchmark (Khanuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2020) consists of the following C-S tasks in Spanish-English and Hindi-English: Language Identification (LID), Part of Speech (POS) tag- ging, Named Entity Recognition (NER), Senti- ment Analysis, Question Answering and Natural Language Inference (NLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The LINCE bench- mark (Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2020) covers Language Identification, Named Entity Recognition, Part- of-Speech Tagging, and Sentiment Analysis in four language pairs: Spanish-English, Nepali- English, Hindi-English, and Modern Standard Arabic-Egyptian Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Although these benchmarks are important start- ing points for C-S, it is clear that they do not represent the entire spectrum of C-S, both from the point of view of potential applications and language pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Further, it is important to note that while state-of-the-art models perform well on tasks such as LID, POS tagging and NER, they are only slightly better than chance when it comes to harder tasks like NLI, showing that current models are not capable of processing C-S language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' More- 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='org over, many of the C-S tasks in the benchmarks above consist of annotated tweets, which only rep- resent a certain type of C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Due to these limita- tions, we currently do not have an accurate picture of how well models are able to handle C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='3 User-facing applications Although speech and NLP models for C-S have been built for various applications, a major limi- tation of the work done so far in computational processing of C-S is the lack of end-to-end user- facing applications that interact directly with users in multilingual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For example, there is no widely-used spoken dialogue system that can understand as well as produce code-switched speech, although some voice assistants may recog- nize and produce C-S in limited scenarios in some locales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Although computational implementations of grammatical models of C-S exist (Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2016), they do not necessarily generate natural C- S utterances that a bilingual speaker would pro- duce (Pratapa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Most crucially, current computational approaches to C-S language tech- nologies do not usually take into account the lin- guistic and social factors that influence why and when speakers/users choose to code-switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Bawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2020) conducted a Wizard-of-Oz study using a Hindi-English chatbot and found that not only did bilingual users prefer chatbots that could code-switch, they also showed a preference towards bots that mimicked their own C-S patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Rudra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2016) report a study on 430k tweets from Hindi-English bilingual users and find that Hindi is preferred for the expression of negative sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In a follow-up study, Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' (2017) find that Hindi is the preferred language for swearing in Hindi-English C-S tweets, and swear- ing may be a motivating factor for users to switch to Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The study also finds a gender difference, with women preferring to swear in English more often than Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Such studies indicate that mul- tilingual chatbots and intelligent agents need to be able to adapt to users’ linguistic styles, while also being capable of determining when and how to code-switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Due to the paucity of user-facing systems and standard benchmarks covering only a handful of simpler NLP tasks, it is likely that we overesti- mate how well computational models are able to handle C-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In sum, language technologies for C- S seem to be constrained by the lack of availabil- ity of diverse C-S training data, evaluation bench- marks and the absence of user-facing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' They need to go beyond pattern recognition and grammatical constraints of C-S in order to process and produce C-S the way humans do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Hence, it is important for the CL community to be aware of the vast literature around C-S in linguistics, partic- ularly as we proceed to solving more challenging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 8 Conclusion The goal of this paper was to inform computa- tional linguists and language technologists about the linguistic and social aspects C-S studies focus- ing on the European and Indian multilingual con- texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There are some similarities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' themes for linguistic research in C-S) but also differences be- tween the two contexts in terms of the social, cul- tural and historical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' For example, C-S in immigrant communities has been a com- mon theme for both multilingual contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In Eu- rope, C-S has been widely studied within the im- migrant communities who have come through la- bor immigration in the 1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, there is a need for more research about the C-S in immi- grant languages with a more recent history as well as minority languages and regional dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' An- alyzing C-S in the immigration context is even more complex for Indian languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There are hardly any systematic linguistic comparisons be- tween the C-S within the same language pairs in India and immigrant contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' C-S between Hindi-English in India vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Hindi-English in the US/UK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There is also a need for more research about C-S between less known language pairs in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, some of these languages are not even officially listed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' in census results) since they have less than 10,000 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' In these cases, collecting and analyzing the multilingual and C-S data becomes more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' A common flaw that is shared both by linguis- tics and computational areas of research is to focus only on the positive evidence and assume that C-S will occur in all multilingual contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, there is also a need for negative evidence to fal- sify this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' As illustrated through an ex- ample from Konkani-Kannada language contact in India (see section 6), bilingual speakers may pre- fer not to C-S due to historical, social and cultural factors operating in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Therefore, devel- oping an automatic C-S system for a random pair of languages without an in-depth and systematic analysis of linguistic and social aspects of C-S in a particular context would not be very useful for the targeted users and/or language technologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' To date, the literature focusing on the social and linguistic aspects of C-S is less visible for CL re- searchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' This lack of visibility leads to misunder- standings and/or incomplete citations of earlier re- search which would have saved time and resources for CL research if resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' One of the reasons is perhaps the differences in publishing traditions between humanities and computational areas of re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Conference and workshop proceedings are commonly accepted means of publication in com- putational linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Whereas, journal publica- tions, books and/or chapters are the publication forms in humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' However, guidelines about how to cite publications in humanities are often missing in computational venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' There are also differences in terms of length, review cycles and open access policies between the two fields which may influence the visibility of research output for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' It is perhaps useful to remember that science advances by standing on the shoulders of giants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' building upon earlier research).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' With our contribution to the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Li Wei and Lesley Milroy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Conversational code-switching in a Chinese community in Britain: A sequential analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Journal of Pragmatics, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Ruth Wodak, Michał Krzy˙zanowski, and Bernhard Forchtner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The interplay of language ideolo- gies and contextual cues in multilingual interactions: Language choice and code-switching in European Union institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Language in Society, 41:157– 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Ana Celia Zentella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Growing up bilingual : Puerto Rican children in New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Blackwell, Malden, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Anna Zhiganova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' The study of the perception of code-switching to English in German advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Procedia: Social and Behavioral Sciences, 236:225– 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Lena Zipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' Code-switching in the media: Iden- tity negotiations in a Gujarati diaspora radio pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} +page_content=' International Journal of the Sociology of Lan- guage, 2017(247):33 – 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtA0T4oBgHgl3EQfAv-e/content/2301.01967v1.pdf'} diff --git a/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/2301.01040v1.pdf.txt b/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/2301.01040v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..226ad40fdcf5157f72a404cc4bcbf9c547faa829 --- /dev/null +++ b/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/2301.01040v1.pdf.txt @@ -0,0 +1,751 @@ +Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction +using a circularly polarized electric field +Amrutha S V,∗ Anupama Sebastian, Puthiyapurayil Sibeesh, Shreyas Punacha, and T K Shajahan† +Department of Physics +National Institute of Technology Karnataka +(Dated: January 4, 2023) +We present the first experimental study of unpinning a spiral wave of excitation using a circularly +polarized electric field. The experiments are conducted in the Belousov-Zhabotinsky(BZ) reaction, +and the system is modeled using the Oregenator model. The mechanism of unpinning in the BZ +reaction differs from that in the physiological medium. We show that the wave unpins when the +electric force opposes the propagation of the spiral wave. We developed an analytical relation of +the unpinning phase with the initial phase, the pacing ratio, and the field strength and verified the +same. +The Belousov-Zhabotinsky (BZ) reaction has served as +the prototype of a large class of systems that display ex- +citation waves, including the waves of action potentials +seen in the heart [1], brain [2], retina [3], and waves of +communication in the social amoeba dictyostelium dis- +coideum [4, 5]. +Excitation waves in these systems ex- +hibit strikingly similar spatio-temporal patterns such as +expanding target waves or rotating spiral waves [6–8]. +Recently there has been a renewed interest in the pat- +tern formation in the BZ reaction because of the active +nature of the chemical waves: their wavefronts are electri- +cally charged [9, 10] and resultant changes in the surface +tension on the droplets of BZ reagents in an oily medium +can propel the droplets [11, 12]. +A characteristic feature of excitation waves is their ten- +dency to pin to heterogeneities in the medium [13–16]. +A pinned rotating wave requires a carefully administered +stimulus to remove it from the heterogeneity [17]. This is +especially pertinent in cardiac tissue since stable pinned +rotating waves can be life-threatening [17, 18]. +Several groups have proposed methods for controlling +such pinned waves using either pulsed electric field [1, 19] +or, more recently, circularly polarized electric field [20– +22]. Numerical studies have shown that circularly polar- +ized electric fields (CPEF) are more efficient in control- +ling cardiac excitation waves [20, 22, 23]. In particular, +CPEF requires less energy and is more efficient in con- +trolling pinned rotating waves [20, 21]. Our systematic +investigations on the mechanism of CPEF indicated that +the spiral wave could be unpinned if the frequency of the +CPEF is more than a cut-off frequency [23]. +It is observed that chemical waves are also prone to +pinning [13], and they can also be unpinned using elec- +tric field [9, 24]. However, there is an essential distinction +between the chemical wave and the waves in physiological +tissue. In the latter, the electric force does not affect the +excitation wave directly, instead, they unpin by inducing +secondary excitations from the heterogeneities [25]. In +∗ Copyrights Reserved +† shajahantk@nitk.edu.in +the chemical medium, on the other hand, the wavefront +contains charged ions such as Br− and Fe3+, which can +be moved by the applied electric field, , i.e., the elec- +tric field in a chemical medium exerts an advective force +directly on the wavefront [9, 26, 27]. +Such an electric +force on the wave is not reported in the physiological tis- +sue. It is also observed that the chemical wave unpins +as it moves away from the anode, and not when moving +towards it [9]. +So far, there have not been any experimental reports of +unpinning spiral waves using CPEF, either in the chem- +ical medium or the cardiac tissue. However, CPEF is re- +alized in BZ medium to control spiral turbulence [28]. In +this paper, we report the first experimental studies of spi- +ral wave unpinning using CPEF in an excitable medium. +However, the mechanism of how CPEF acts on a chemical +wave is different from that of the cardiac excitation wave. +In particular, we find no cut-off frequency for CPEF to +unpin a chemical wave. We vary the pacing ratio, ini- +tial spiral phase, and field strength. We deduced that +the wave unpins when the component of the electric field +vector along the direction of the spiral equals or exceeds +a critical field strength. Based on this, we predict the +unpinning angle as a function of the initial position of +the spiral wave, the frequency, and the strength of the +electric field. We show that our analytical formulation +agrees with experimental data and numerical results. +In this paper, we focus on the unpinning of an anti- +clockwise (ACW) rotating spiral using a CPEF rotating +in the same direction. We conducted our studies in the +ferroin-catalyzed BZ reaction in a petri-dish, as described +in detail in Ref. [9]. Briefly, we start with the following +initial reagents: [H2SO4] = 0.16 M, [NaBrO3] = 40 mM, +[Malonic acid] = 40 mM, and [Ferroin] = 0.5 mM. The +reaction mixture is embedded in 1.4 % w/v of agar gel +to avoid any hydrodynamic perturbations. +The single +reaction layer of thickness 3 × 10−3 m is taken in a glass +petri dish of diameter 0.1 m. A circular excitation wave is +created at the center of the reaction medium by inserting +a silver wire. By disrupting the motion of the circular +wavefront, a pair of counter-rotating spirals are created. +To generate a pinned spiral wave, a glass bead of diameter +1.2 mm is carefully placed at the tip of one of the spirals. +arXiv:2301.01040v1 [nlin.PS] 3 Jan 2023 + +2 +FIG. 1. (a) Schematic diagram of the experimental system: The positions of two pairs of field electrodes with respect +to the glass bead are shown schematically (not to scale). Unpinning of an anti-clockwise rotating spiral using CPEF: +(b) An ACW rotating spiral pinned to a spherical bead of diameter 1.2 mm in the experiment. The natural period of pinned +spiral tip Ts = 297 s. (c) An applied CPEF of strength E0 ≃ 1.38 V/cm, and period TE = 125 s unpins the spiral tip from +the obstacle. (d) An ACW rotating spiral pinned to an obstacle of diameter 1.0 s.u in the simulation with Ts = 1.77 t.u is +subjected to a CPEF of strength E0 ≃ 0.6 and period TE = 1.18 t.u. The unpinned spiral tip drifts away from the obstacle at +t = 1.27 t.u. The arrows show the direction of the applied CPEF. +The pinning of the spiral tip to the obstacle is confirmed +after 1-2 rotations. An anticlockwise circularly polarized +electric field (CPEF) is applied using two pairs of copper +electrodes as in Fig. 1(a). Images of the reaction medium +are recorded using a CCD camera at every 30s interval +for 1 − 2 hours. +To model this experiment, we use a two-dimensional +Oregonator model. The model equations are given by +∂u +∂t = 1 +ϵ (u(1−u)−fv(u − q) +u + q +)+Du∇2u+Mu( ⃗E·∇u) (1) +∂v +∂t = u − v + Dv∇2v + Mv( ⃗E · ∇v). +(2) +Here, u is the activator variable, and v is the in- +hibitor variable (corresponding to the rescaled concen- +trations of [HBrO2] and the catalyst, respectively). ⃗E = +E0cos( 2πt +T )ˆx + E0sin( 2πt +T )ˆy is the circularly polarized +electric field. +The electric field is added as an advec- +tion term for the variables u and v. An obstacle is added +to this domain by setting the diffusion coefficient of the +activator to a very low value. Details of the model and +the simulations are given in Ref. [9]. +The rotating chemical wave in the BZ reaction medium +can get anchored into the boundary of the glass bead and +form a very stable pinned wave, as shown in Fig. 1(b). A +similar situation occurs in the numerical simulation of the +model equations, where the spiral wave can get anchored +to the obstacle in the domain. It is known that this wave +can be unpinned with an electric field [9, 24]. Here we +employ the circularly polarized electric field (CPEF) us- +ing two cross-electrodes (see Fig 1.(a)). The CPEF can +unpin the wave if the amplitude of the electric field equals +or exceeds a certain threshold value (Eth), as shown in +Fig. 1(c). An arrow indicates the instantaneous direc- +tion of the electric field. Similar unpinning is also seen +in the simulations [Fig. 1(d)]. To understand the unpin- +ning process, we measure the location at which the wave +unpins from the obstacle. +We can quantify the spiral +location by the phase of the spiral tip on the obstacle +boundary. The phase is the angle of the spiral tip, mea- +sured in degrees from the +x-axis along the anticlock- +wise direction with the obstacle center as the origin. The +phase of the spiral when we start the CPEF is denoted +by φ0 and the phase when the spiral unpins from the +boundary is denoted by φu [Fig. 2]. The instantaneous +direction of the electric field is denoted by the angle θE. +The direction of the spiral is along the tangent at the +obstacle, and this direction is denoted by ˆrt. We define +the pacing ratio, p, as the ratio of the frequency of the +CPEF (ωcp) to that of the spiral (ωs), i.e., p = ωcp/ωs. +We have varied p from 0.25 to 3. + +t=0s +t=78.5s +t=178.5s +t=300s +(a) +(b) +(c) +t=0s +t=110s +t=176s +t=375s +t=0 +t=0.57 +(p) +t=0.3 +FIG. 2. Schematic diagram showing the phase measurements: +φ0 and φu are the phase of the spiral tip at t = 0 and at the +time of unpinning respectively. θE denotes the phase of the +electric field ⃗E and ˆrt is the tangential vector of spiral rotation +on the obstacle boundary. +All phases are measured in the +anticlockwise direction from the +x axis, with the obstacle +center as the origin. The tail of the resultant field vector ⃗E +marked with a + sign is mentioned as the anode and the head +with a − sign is the cathode. +Our observations can be summarised as follows: (1) +The chemical wave can be unpinned with CPEF for all +pacing ratios (between 0.25 to 3), provided the strength +of the electric field is equal or above a threshold. +0 +1 +2 +3 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +1200 +1300 +theory +0 +1 +2 +3 +(a) Experiment +(b) Simulation +Pacing Ratio (p) +( +u- +0) in degrees +0 = 45 +0 = 135 +0 = 225 +0 = 315 +FIG. 3. Unpinning at E = Eth: For spirals with different +φ0, the phase difference (φu- φ0) is plotted (solid curve) with +the pacing ratio, p in (a) experiments and (b) simulations. +The solid theory lines represent the phases where the unpin- +ning condition is satisfied for the first time (Eq.4). The dashed +lines at the top correspond to the phases where the spiral un- +pins when the unpinning condition is met a second time in its +subsequent rotations. Most of the cases with φ0 = 3150 show +a delayed unpinning. +There is no cut-off frequency and both overdrive pac- +ing (p > 1) and underdrive pacing (p < 1) are equally +effective. (2) The spiral unpinning phase φu varies lin- +early with φ0. It increases for overdrive pacing and de- +creases for underdrive pacing (Fig. 4). (3) (φu- φ0) varies +with the pacing ratio, p, as in Fig. 3. (4) Unpinning is +not guaranteed within one rotation of the spiral. In a +few cases, where either the relative rotation of the spiral- +field pair varies quickly (extremely overdrive or under- +drive pacing), or φ0 lies close to the expected φu (i.e., +(φu - φ0) ≈ 0), the spiral misses unpinning at the first +expected phase. Here, the unpinning may happen later at +a phase where the unpinning condition is satisfied again +(dashed lines in Fig. 3). +(5) It takes several rotations +for the chemical wave to unpin as p approaches 1 (when +the spiral and the CPEF are rotating with the same fre- +quency). For resonant pacing (p = 1), the wave cannot +be unpinned except for a small range of initial conditions +(φ0). This range increases with the strength of the elec- +tric field (Fig. 5). +0 +100 +200 +300 +0 +200 +400 +600 +800 +1000 +0 +100 +200 +300 +0 +200 +400 +600 +800 +1000 +(a) p=0.5 +(b) p=1.5 +( +u- +0) in degrees +Initial Phase ( +0) +experiment +simulation +theory +FIG. 4. Unpinning at E = Eth: (a) The spiral phase differ- +ence (φu − φ0) is plotted against φ0 for p = 0.5 (underdrive +pacing). (b) same as (a) but for p = 1.5 (overdrive pacing). +In both cases (φu − φ0) varies linearly with φ0. The dashed +line indicates the unpinning during the subsequent rotations +of the electric field. To plot the theory line for φ0 ≥ 2700, we +have added ∓2π to Eq.3 and Eq.4 respectively. Otherwise, +the lines keep on decreasing or increasing linearly for under- +drive and overdrive pacing respectively. Circles and triangles +represent the experiment and simulation data respectively. +These results can be analyzed in light of our recent +work with the DC electric fields [9]. +We found that +the electric field exerts a retarding force on the chemical +wavefront, which is maximum when the field direction is +along the direction of the wavefront. For a CPEF with +field strength E = Eth this condition is satisfied when +⃗E. ˆrt = 0. +From this, we can estimate the unpinning +angle as, +φu = pφ0 + 90 +p − 1 +; p > 1 +(3) +φu = 270 − pφ0 +1 − p +; p < 1 +(4) +When E = Eth, the wave can be unpinned only when +θE − φ0 = 90. The theoretical solid curves in Figs. 4 and + +E(t = tunpinning +E(t = 0) ++4 +3 are based on the above equation. +For a field strength greater than Eth, the wave must +unpin when the component of ⃗E along ˆrt reaches the crit- +ical threshold, i.e., ⃗E. ˆrt ≥ Eth. This condition gives an +upper-bound and lower-bound for possible spiral unpin- +ning phases φu. +For overdrive pacing with p > 1, the unpinning phase +window is given by +pφ0 + sin−1( Eth +E ) +p − 1 +≤ φu ≤ pφ0 + π − sin−1( Eth +E ) +p − 1 +(5) +with a width ∆φu = π−2 sin−1( +Eth +E ) +p−1 +. In most of the cases, +the unpinning condition (Eq. 5) is satisfied at the lower +bound of this range. However, unpinning is possible at +any point inside the window (refer Fig.1 in the supple- +mentary material). +For underdrive pacing i.e, for p < 1, the unpinning +phase window is +π + sin−1( Eth +E ) − pφ0 +1 − p +≤ φu ≤ 2π − pφ0 − sin−1( Eth +E ) +1 − p +(6) +The width of this window is ∆φu = π−2 sin−1( +Eth +E ) +1−p +. De- +pending on the strength of the electric field, the width +of the window increases. The window reduces to a point +when E = Eth. +For p = 1, unpinning happens only if the following +condition is satisfied. +π + sin−1(Eth +E ) ≤ φ0 ≤ 2π − sin−1(Eth +E ) +(7) +Thus for p = 1 the range of initial phases that lead to un- +pinning increases with the field strength, E. Figure. 5(a) +shows the initial spiral phases that lead to successful un- +pinning as a function of Eth +E . +FIG. 5. Unpinning of spiral wave with pacing ratio, p = 1 for +different field strength: π + sin−1( Eth +E ) and 2π − sin−1( Eth +E ) +are the lower and upper limit of the range of possible φ0-values +which gives successful unpinning for p = 1. The shaded re- +gion corresponds to the cases of successful unpinning. Circles +and diamonds represent the experiment and simulation data +respectively. +In summary, we have presented the first experimen- +tal studies using a circularly polarized electric field to +unpin an excitation wave. We observed unpinning with +overdrive, underdrive and resonant pacing. Because of +the charge on the chemical wavefront, the mechanism of +chemical wave unpinning differs from that in other ex- +citable media. The wave unpins when the electric field +component along the tangential direction of spiral prop- +agation is equal or more than the critical threshold; i.e., +when the electric force opposite to the instantaneous +spiral propagation is above the threshold value. Based +on this condition, we are able to predict the unpinning +phase, and the same has been verified in simulations and +in experiments. +The unpinning of a rotating chemical wave presents a +unique physical situation. In our studies, the unpinning +happens when the anode ‘catches’ the spiral from behind +while chasing it. If it fails to halt the wave and overtakes +it, it has to come back again to act on it. i.e., the wave +can only be unpinned while propagating away from the +anode. A similar kind of asymmetry in the chemical wave +behavior in an external electric field has been observed +in previous studies. The speed of chemical wave propa- +gation decreases as it propagates towards the anode and +it decreases as it rotates away from it. Depending on the +velocity, the size of a free spiral core varies; as the spiral +accelerates, the core-size decreases, and as it decelerates +the core-size increases [27]. As a result, a drift of the +spiral tip occurs in the medium. The drift occurs with a +parallel component which is always directed towards the +anode and a chirality-dependent perpendicular compo- +nent [29]. The phenomenon of spiral drift is addressed in +numerous experimental and computational studies with + +experiment +simulation +360 +2π - sin +No unpinning +E +270 +Eth +π + sin + 180 +E +No unpinning +90 +0 +0.6 +0.7 +0.8 +0.9 +1.0 +Eth +E5 +dc, ac, and polarized electric fields [29–31]. Most of the +important field effects in the BZ reaction could be ex- +plained with the electromigration of Br− and Fe3+ ions. +Only a few studies investigated the effect of an electric +field on a pinned spiral wave in the BZ reaction. In a +unidirectional field, the spiral always unpins as it rotates +away from the anode [9, 24]. In light of previous results, +we can assume that the wave can only be unpinned when +it is retarded by the electric field, and not when it is accel- +erated by it. During retardation, the core size increases, +and the spiral can only pin weakly to obstacles smaller +than the spiral core [14, 32]. This could be the reason +for the asymmetric nature of the unpinning. As a proto- +type model, the BZ reaction is expected to show all the +qualitative features observed in other excitable systems. +On the contrary, our studies show that the chemical ex- +citation waves interact uniquely with an external electric +field. +ACKNOWLEDGMENTS +We thank Beneesh P B, Deepu Vijayasenan, Ajith K +M, K V Gangadharan, and Muhammed Mansoor C B +for discussions. +Experiments were conducted using a +grant (ECR/2016/000983) from Science and Engineering +Research Board, Department of Science and Technology +(SERB-DST), India. +AUTHOR CONTRIBUTIONS +S.V.A and T.K.S conceived the study. +S.V.A per- +formed the experiments, and P.S. built the experimen- +tal setup. S.P and A.S performed numerical simulations. +S.V.A, A.S, and T.K.S analysed the data. A.S and T.K.S +developed the theory. S.V.A and T.K.S wrote the paper. +All authors helped to edit the paper. +[1] S. Luther, F. H. Fenton, B. G. Kornreich, A. 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The solid bottom line +represents the lower limit of the range of possible φu-values given by the relation φu = (pφ0 + +48.95)/(p−1) for over-drive pacing and φu = (π −pφ0 +48.59)/(1−p) for under-drive pacing. The +upper limit of the range of possible φu-values, given by the relation φu = (pφ0 + π − 48.95)/(p − 1) +for over-drive pacing and φu = (2π−pφ0−48.59)/(1−p) for under-drive pacing, are represented by +the top dashed line. For φ0 = 3150, the above equations must be added with 2π to get the positive +phase values. Circles and triangles represent the experiment and simulation data respectively. +1 +arXiv:2301.01040v1 [nlin.PS] 3 Jan 2023 + +experiment +simulation +LowerLimit +800 +Φo=45 +=135 +Upper Limit +700 +009 +600 +500 +500 +400 +400 +300 +300 +degrees +200 +200 +100 +100 +0 +.s +0.5 +0.0 +0.5 +1.0 +¥3.0 +0.0 +1.0 +1.5 +1.5 +2.0 +2.5 +¥2.02.5 +3.0 +(o +700 +700 +)=225 +d)Φo=315 +一 +600 +600 +nd) +500 +500 +400 +400 +300 +300 +200 +200 +100 +100 +0 +0 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +Pacing Ratio(P)Fig.1 shows the unpinning phase window at E > Eth for different initial phases of the +spiral. Here, the solid lines correspond to the lower limit, and the dashed lines correspond +to the upper limit of the window according to the equations ?? and ??. The unpinning +always happens at a phase within this range. The width of the window varies with the field +strength. +II. +COMPARISON BETWEEN PINNING OBSTACLES OF DIFFERENT GEOM- +ETRY +The results of spiral unpinning from spherical beads are presented in the paper. For +comparison, we have performed similar experiments using cylindrical rods. The experimental +setup is the same as in figue.??a. A cylindrical glass rod of length ≈ 4 mm is inserted +vertically into the medium. +FIG. 2. Comparison of unpinning of spiral pinned to spherical bead and cylindrical rod: φu is +plotted against the pacing ratio,p where p > 1. φ0 = 450 and E = 1.38 V/cm. The diameter of +the obstacles are same and equals 1.2 mm. +Figure.3 shows the variation of unpinning phase with the pacing ratio for a cylindrical +obstacle of radius 1.2 mm. +The unpinning phases for a spherical bead have also been +shown, and in both cases, unpinning occurs at phases that are consistent with the theoretical +predictions. +2 + +bead +350 +rod +Equation +300 + 250 +200 +150 +100 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +Pacing Ratio (p)III. +COMPARISON BETWEEN NUMERICAL MODELS +In this letter, we have used a two-variable reduction of the original three-variable Oreg- +onator model. Here we compare the unpinning studies using both two and three-variable +models. +The three-variable Oregonator model consists of the following equations [? ]. +∂u +∂t = 1 +ϵ(qw − uw + u(1 − u)) + Du∇2u +(1) +∂v +∂t = u − v + Dv∇2v + Mv( ⃗E · ∇v) +(2) +∂w +∂t = 1 +ϵ′(−qw − uw + fv) + Dw∇2w + Mw( ⃗E · ∇w) +(3) +The variables u, v and w represent the re-scaled dimensionless concentrations of HBrO2, +Fe3+, and Br− respectively. The model parameters are q = 0.002, f = 1.4, ϵ = 0.01 as in +Numerical methods in the manuscript along with an additional parameter, ϵ′ = 0.0001. For +both variables v and w, the electric field ⃗E is added as an advection term. However, the +variable u is unaffected in the presence of an electric field as it corresponds to the charge-less +species HBrO2. The values of the ionic mobilities are Mu = 0, Mv = -2, and Mv = 1. The +simulation details can be obtained from our recent paper [? ]. +FIG. 3. Comparison of spiral unpinning obtained in two and three-variable Oregonator models: φu +is plotted against the pacing ratio,p where p > 1. φ0 = 450 and E = 0.6. The obstacle diameter is +1.0 s.u. +3 + +2-variable +350 +3-variable +Equation +300 +250 +200 +150 +100 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +Pacing Ratio (p)Using the three-variable model, we measured the unpinning phase of an ACW spiral +pinned to an obstacle of radius, r=1.0 s.u in an electric field of strength Eth = 0.6. The +unpinning is done for a fixed initial phase with overdrive pacing. The results are in good +agreement with those obtained from the two-variable Oregonator model and from the theory. +4 + diff --git a/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/load_file.txt b/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..199f51b9b15d70f34a460379c94590cde70a4d0e --- /dev/null +++ b/FdAzT4oBgHgl3EQfHPsU/content/tmp_files/load_file.txt @@ -0,0 +1,683 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf,len=682 +page_content='Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction using a circularly polarized electric field Amrutha S V,∗ Anupama Sebastian, Puthiyapurayil Sibeesh, Shreyas Punacha, and T K Shajahan† Department of Physics National Institute of Technology Karnataka (Dated: January 4, 2023) We present the first experimental study of unpinning a spiral wave of excitation using a circularly polarized electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The experiments are conducted in the Belousov-Zhabotinsky(BZ) reaction, and the system is modeled using the Oregenator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The mechanism of unpinning in the BZ reaction differs from that in the physiological medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We show that the wave unpins when the electric force opposes the propagation of the spiral wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We developed an analytical relation of the unpinning phase with the initial phase, the pacing ratio, and the field strength and verified the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The Belousov-Zhabotinsky (BZ) reaction has served as the prototype of a large class of systems that display ex- citation waves, including the waves of action potentials seen in the heart [1], brain [2], retina [3], and waves of communication in the social amoeba dictyostelium dis- coideum [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Excitation waves in these systems ex- hibit strikingly similar spatio-temporal patterns such as expanding target waves or rotating spiral waves [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Recently there has been a renewed interest in the pat- tern formation in the BZ reaction because of the active nature of the chemical waves: their wavefronts are electri- cally charged [9, 10] and resultant changes in the surface tension on the droplets of BZ reagents in an oily medium can propel the droplets [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A characteristic feature of excitation waves is their ten- dency to pin to heterogeneities in the medium [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A pinned rotating wave requires a carefully administered stimulus to remove it from the heterogeneity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' This is especially pertinent in cardiac tissue since stable pinned rotating waves can be life-threatening [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Several groups have proposed methods for controlling such pinned waves using either pulsed electric field [1, 19] or, more recently, circularly polarized electric field [20– 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Numerical studies have shown that circularly polar- ized electric fields (CPEF) are more efficient in control- ling cardiac excitation waves [20, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In particular, CPEF requires less energy and is more efficient in con- trolling pinned rotating waves [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Our systematic investigations on the mechanism of CPEF indicated that the spiral wave could be unpinned if the frequency of the CPEF is more than a cut-off frequency [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' It is observed that chemical waves are also prone to pinning [13], and they can also be unpinned using elec- tric field [9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' However, there is an essential distinction between the chemical wave and the waves in physiological tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In the latter, the electric force does not affect the excitation wave directly, instead, they unpin by inducing secondary excitations from the heterogeneities [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In ∗ Copyrights Reserved † shajahantk@nitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='in the chemical medium, on the other hand, the wavefront contains charged ions such as Br− and Fe3+, which can be moved by the applied electric field, , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', the elec- tric field in a chemical medium exerts an advective force directly on the wavefront [9, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Such an electric force on the wave is not reported in the physiological tis- sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' It is also observed that the chemical wave unpins as it moves away from the anode, and not when moving towards it [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' So far, there have not been any experimental reports of unpinning spiral waves using CPEF, either in the chem- ical medium or the cardiac tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' However, CPEF is re- alized in BZ medium to control spiral turbulence [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In this paper, we report the first experimental studies of spi- ral wave unpinning using CPEF in an excitable medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' However, the mechanism of how CPEF acts on a chemical wave is different from that of the cardiac excitation wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In particular, we find no cut-off frequency for CPEF to unpin a chemical wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We vary the pacing ratio, ini- tial spiral phase, and field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We deduced that the wave unpins when the component of the electric field vector along the direction of the spiral equals or exceeds a critical field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Based on this, we predict the unpinning angle as a function of the initial position of the spiral wave, the frequency, and the strength of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We show that our analytical formulation agrees with experimental data and numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In this paper, we focus on the unpinning of an anti- clockwise (ACW) rotating spiral using a CPEF rotating in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We conducted our studies in the ferroin-catalyzed BZ reaction in a petri-dish, as described in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Briefly, we start with the following initial reagents: [H2SO4] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='16 M, [NaBrO3] = 40 mM, [Malonic acid] = 40 mM, and [Ferroin] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The reaction mixture is embedded in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='4 % w/v of agar gel to avoid any hydrodynamic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The single reaction layer of thickness 3 × 10−3 m is taken in a glass petri dish of diameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A circular excitation wave is created at the center of the reaction medium by inserting a silver wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' By disrupting the motion of the circular wavefront, a pair of counter-rotating spirals are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' To generate a pinned spiral wave, a glass bead of diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='2 mm is carefully placed at the tip of one of the spirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='01040v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='PS] 3 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (a) Schematic diagram of the experimental system: The positions of two pairs of field electrodes with respect to the glass bead are shown schematically (not to scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Unpinning of an anti-clockwise rotating spiral using CPEF: (b) An ACW rotating spiral pinned to a spherical bead of diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='2 mm in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The natural period of pinned spiral tip Ts = 297 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (c) An applied CPEF of strength E0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='38 V/cm, and period TE = 125 s unpins the spiral tip from the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (d) An ACW rotating spiral pinned to an obstacle of diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u in the simulation with Ts = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='77 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u is subjected to a CPEF of strength E0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='6 and period TE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='18 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The unpinned spiral tip drifts away from the obstacle at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='27 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The arrows show the direction of the applied CPEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The pinning of the spiral tip to the obstacle is confirmed after 1-2 rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' An anticlockwise circularly polarized electric field (CPEF) is applied using two pairs of copper electrodes as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Images of the reaction medium are recorded using a CCD camera at every 30s interval for 1 − 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' To model this experiment, we use a two-dimensional Oregonator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The model equations are given by ∂u ∂t = 1 ϵ (u(1−u)−fv(u − q) u + q )+Du∇2u+Mu( ⃗E·∇u) (1) ∂v ∂t = u − v + Dv∇2v + Mv( ⃗E · ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (2) Here, u is the activator variable, and v is the in- hibitor variable (corresponding to the rescaled concen- trations of [HBrO2] and the catalyst, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ⃗E = E0cos( 2πt T )ˆx + E0sin( 2πt T )ˆy is the circularly polarized electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The electric field is added as an advec- tion term for the variables u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' An obstacle is added to this domain by setting the diffusion coefficient of the activator to a very low value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Details of the model and the simulations are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The rotating chemical wave in the BZ reaction medium can get anchored into the boundary of the glass bead and form a very stable pinned wave, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A similar situation occurs in the numerical simulation of the model equations, where the spiral wave can get anchored to the obstacle in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' It is known that this wave can be unpinned with an electric field [9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Here we employ the circularly polarized electric field (CPEF) us- ing two cross-electrodes (see Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The CPEF can unpin the wave if the amplitude of the electric field equals or exceeds a certain threshold value (Eth), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' An arrow indicates the instantaneous direc- tion of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Similar unpinning is also seen in the simulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' To understand the unpin- ning process, we measure the location at which the wave unpins from the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We can quantify the spiral location by the phase of the spiral tip on the obstacle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The phase is the angle of the spiral tip, mea- sured in degrees from the +x-axis along the anticlock- wise direction with the obstacle center as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The phase of the spiral when we start the CPEF is denoted by φ0 and the phase when the spiral unpins from the boundary is denoted by φu [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The instantaneous direction of the electric field is denoted by the angle θE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The direction of the spiral is along the tangent at the obstacle, and this direction is denoted by ˆrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We define the pacing ratio, p, as the ratio of the frequency of the CPEF (ωcp) to that of the spiral (ωs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', p = ωcp/ωs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We have varied p from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='25 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' t=0s t=78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5s t=178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5s t=300s (a) (b) (c) t=0s t=110s t=176s t=375s t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='57 (p) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Schematic diagram showing the phase measurements: φ0 and φu are the phase of the spiral tip at t = 0 and at the time of unpinning respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' θE denotes the phase of the electric field ⃗E and ˆrt is the tangential vector of spiral rotation on the obstacle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' All phases are measured in the anticlockwise direction from the +x axis, with the obstacle center as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The tail of the resultant field vector ⃗E marked with a + sign is mentioned as the anode and the head with a − sign is the cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Our observations can be summarised as follows: (1) The chemical wave can be unpinned with CPEF for all pacing ratios (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='25 to 3), provided the strength of the electric field is equal or above a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 0 1 2 3 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 theory 0 1 2 3 (a) Experiment (b) Simulation Pacing Ratio (p) ( u- 0) in degrees 0 = 45 0 = 135 0 = 225 0 = 315 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Unpinning at E = Eth: For spirals with different φ0, the phase difference (φu- φ0) is plotted (solid curve) with the pacing ratio, p in (a) experiments and (b) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The solid theory lines represent the phases where the unpin- ning condition is satisfied for the first time (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The dashed lines at the top correspond to the phases where the spiral un- pins when the unpinning condition is met a second time in its subsequent rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Most of the cases with φ0 = 3150 show a delayed unpinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' There is no cut-off frequency and both overdrive pac- ing (p > 1) and underdrive pacing (p < 1) are equally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (2) The spiral unpinning phase φu varies lin- early with φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' It increases for overdrive pacing and de- creases for underdrive pacing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (3) (φu- φ0) varies with the pacing ratio, p, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (4) Unpinning is not guaranteed within one rotation of the spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In a few cases, where either the relative rotation of the spiral- field pair varies quickly (extremely overdrive or under- drive pacing), or φ0 lies close to the expected φu (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', (φu - φ0) ≈ 0), the spiral misses unpinning at the first expected phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Here, the unpinning may happen later at a phase where the unpinning condition is satisfied again (dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (5) It takes several rotations for the chemical wave to unpin as p approaches 1 (when the spiral and the CPEF are rotating with the same fre- quency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For resonant pacing (p = 1), the wave cannot be unpinned except for a small range of initial conditions (φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' This range increases with the strength of the elec- tric field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 0 100 200 300 0 200 400 600 800 1000 0 100 200 300 0 200 400 600 800 1000 (a) p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 (b) p=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 ( u- 0) in degrees Initial Phase ( 0) experiment simulation theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Unpinning at E = Eth: (a) The spiral phase differ- ence (φu − φ0) is plotted against φ0 for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 (underdrive pacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' (b) same as (a) but for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 (overdrive pacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In both cases (φu − φ0) varies linearly with φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The dashed line indicates the unpinning during the subsequent rotations of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' To plot the theory line for φ0 ≥ 2700, we have added ∓2π to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='3 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Otherwise, the lines keep on decreasing or increasing linearly for under- drive and overdrive pacing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Circles and triangles represent the experiment and simulation data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' These results can be analyzed in light of our recent work with the DC electric fields [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We found that the electric field exerts a retarding force on the chemical wavefront, which is maximum when the field direction is along the direction of the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For a CPEF with field strength E = Eth this condition is satisfied when ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ˆrt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' From this, we can estimate the unpinning angle as, φu = pφ0 + 90 p − 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' p > 1 (3) φu = 270 − pφ0 1 − p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' p < 1 (4) When E = Eth, the wave can be unpinned only when θE − φ0 = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The theoretical solid curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 4 and E(t = tunpinning E(t = 0) +4 3 are based on the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For a field strength greater than Eth, the wave must unpin when the component of ⃗E along ˆrt reaches the crit- ical threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ˆrt ≥ Eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' This condition gives an upper-bound and lower-bound for possible spiral unpin- ning phases φu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For overdrive pacing with p > 1, the unpinning phase window is given by pφ0 + sin−1( Eth E ) p − 1 ≤ φu ≤ pφ0 + π − sin−1( Eth E ) p − 1 (5) with a width ∆φu = π−2 sin−1( Eth E ) p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In most of the cases, the unpinning condition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 5) is satisfied at the lower bound of this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' However, unpinning is possible at any point inside the window (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='1 in the supple- mentary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For underdrive pacing i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e, for p < 1, the unpinning phase window is π + sin−1( Eth E ) − pφ0 1 − p ≤ φu ≤ 2π − pφ0 − sin−1( Eth E ) 1 − p (6) The width of this window is ∆φu = π−2 sin−1( Eth E ) 1−p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' De- pending on the strength of the electric field, the width of the window increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The window reduces to a point when E = Eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For p = 1, unpinning happens only if the following condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' π + sin−1(Eth E ) ≤ φ0 ≤ 2π − sin−1(Eth E ) (7) Thus for p = 1 the range of initial phases that lead to un- pinning increases with the field strength, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 5(a) shows the initial spiral phases that lead to successful un- pinning as a function of Eth E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Unpinning of spiral wave with pacing ratio, p = 1 for different field strength: π + sin−1( Eth E ) and 2π − sin−1( Eth E ) are the lower and upper limit of the range of possible φ0-values which gives successful unpinning for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The shaded re- gion corresponds to the cases of successful unpinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Circles and diamonds represent the experiment and simulation data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In summary, we have presented the first experimen- tal studies using a circularly polarized electric field to unpin an excitation wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' We observed unpinning with overdrive, underdrive and resonant pacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Because of the charge on the chemical wavefront, the mechanism of chemical wave unpinning differs from that in other ex- citable media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The wave unpins when the electric field component along the tangential direction of spiral prop- agation is equal or more than the critical threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', when the electric force opposite to the instantaneous spiral propagation is above the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Based on this condition, we are able to predict the unpinning phase, and the same has been verified in simulations and in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The unpinning of a rotating chemical wave presents a unique physical situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In our studies, the unpinning happens when the anode ‘catches’ the spiral from behind while chasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' If it fails to halt the wave and overtakes it, it has to come back again to act on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=', the wave can only be unpinned while propagating away from the anode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A similar kind of asymmetry in the chemical wave behavior in an external electric field has been observed in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The speed of chemical wave propa- gation decreases as it propagates towards the anode and it decreases as it rotates away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Depending on the velocity, the size of a free spiral core varies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' as the spiral accelerates, the core-size decreases, and as it decelerates the core-size increases [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' As a result, a drift of the spiral tip occurs in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The drift occurs with a parallel component which is always directed towards the anode and a chirality-dependent perpendicular compo- nent [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The phenomenon of spiral drift is addressed in numerous experimental and computational studies with experiment simulation 360 2π - sin No unpinning E 270 Eth π + sin 180 E No unpinning 90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 Eth E5 dc, ac, and polarized electric fields [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Most of the important field effects in the BZ reaction could be ex- plained with the electromigration of Br− and Fe3+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Only a few studies investigated the effect of an electric field on a pinned spiral wave in the BZ reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In a unidirectional field, the spiral always unpins as it rotates away from the anode [9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' In light of previous results, we can assume that the wave can only be unpinned when it is retarded by the electric field, and not when it is accel- erated by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' During retardation, the core size increases, and the spiral can only pin weakly to obstacles smaller than the spiral core [14, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' This could be the reason for the asymmetric nature of the unpinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' As a proto- type model, the BZ reaction is expected to show all the qualitative features observed in other excitable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' On the contrary, our studies show that the chemical ex- citation waves interact uniquely with an external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Beneesh P B, Deepu Vijayasenan, Ajith K M, K V Gangadharan, and Muhammed Mansoor C B for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Experiments were conducted using a grant (ECR/2016/000983) from Science and Engineering Research Board, Department of Science and Technology (SERB-DST), India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='A and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S conceived the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='A per- formed the experiments, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' built the experimen- tal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='P and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S performed numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='A, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S analysed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S developed the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='A and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='S wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' All authors helped to edit 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Wu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Gao, Chaos 32, 093145 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Punacha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' N K, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Shajahan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' E 102, 032411 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Sutthiopad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Luengviriya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Porjai, B.' metadata={'source': 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[25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Pumir, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Nikolski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' H¨orning, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Isomura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Agladze, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Yoshikawa, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [29] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Schmidt and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' M¨uller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' E 55, 4390 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} 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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 124, 014505 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Gao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Zheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Pan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Zheng, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Zhang, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 7, 1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Pumir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Sinha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Sridhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Argentina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' H¨orning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Filippi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Cherubini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Luther, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Krinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' E 81, 010901 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' SUPPLEMENTARY MATERIALS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' UNPINNING FOR E > Eth FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Unpinning at E > Eth (sin−1( Eth E ) = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='950): Spiral waves with different φ0 are unpinned in a CPEF with both under-drive (p < 1) and over-drive pacing (p > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The solid bottom line represents the lower limit of the range of possible φu-values given by the relation φu = (pφ0 + 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='95)/(p−1) for over-drive pacing and φu = (π −pφ0 +48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='59)/(1−p) for under-drive pacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The upper limit of the range of possible φu-values, given by the relation φu = (pφ0 + π − 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='95)/(p − 1) for over-drive pacing and φu = (2π−pφ0−48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='59)/(1−p) for under-drive pacing, are represented by the top dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For φ0 = 3150, the above equations must be added with 2π to get the positive phase values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Circles and triangles represent the experiment and simulation data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='01040v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='PS] 3 Jan 2023 experiment simulation LowerLimit 800 Φo=45 =135 Upper Limit 700 009 600 500 500 400 400 300 300 degrees 200 200 100 100 0 .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 (o 700 700 )=225 d)Φo=315 一 600 600 nd) 500 500 400 400 300 300 200 200 100 100 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 Pacing Ratio(P)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='1 shows the unpinning phase window at E > Eth for different initial phases of the spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Here, the solid lines correspond to the lower limit, and the dashed lines correspond to the upper limit of the window according to the equations ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' and ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='. The unpinning always happens at a phase within this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The width of the window varies with the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' COMPARISON BETWEEN PINNING OBSTACLES OF DIFFERENT GEOM- ETRY The results of spiral unpinning from spherical beads are presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For comparison, we have performed similar experiments using cylindrical rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The experimental setup is the same as in figue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' A cylindrical glass rod of length ≈ 4 mm is inserted vertically into the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Comparison of unpinning of spiral pinned to spherical bead and cylindrical rod: φu is plotted against the pacing ratio,p where p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' φ0 = 450 and E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='38 V/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The diameter of the obstacles are same and equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='3 shows the variation of unpinning phase with the pacing ratio for a cylindrical obstacle of radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The unpinning phases for a spherical bead have also been shown, and in both cases, unpinning occurs at phases that are consistent with the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 2 bead 350 rod Equation 300 250 200 150 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='00 Pacing Ratio (p)III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' COMPARISON BETWEEN NUMERICAL MODELS In this letter, we have used a two-variable reduction of the original three-variable Oreg- onator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Here we compare the unpinning studies using both two and three-variable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The three-variable Oregonator model consists of the following equations [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ∂u ∂t = 1 ϵ(qw − uw + u(1 − u)) + Du∇2u (1) ∂v ∂t = u − v + Dv∇2v + Mv( ⃗E · ∇v) (2) ∂w ∂t = 1 ϵ′(−qw − uw + fv) + Dw∇2w + Mw( ⃗E · ∇w) (3) The variables u, v and w represent the re-scaled dimensionless concentrations of HBrO2, Fe3+, and Br− respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The model parameters are q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='002, f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='4, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='01 as in Numerical methods in the manuscript along with an additional parameter, ϵ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' For both variables v and w, the electric field ⃗E is added as an advection term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' However, the variable u is unaffected in the presence of an electric field as it corresponds to the charge-less species HBrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The values of the ionic mobilities are Mu = 0, Mv = -2, and Mv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The simulation details can be obtained from our recent paper [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' Comparison of spiral unpinning obtained in two and three-variable Oregonator models: φu is plotted against the pacing ratio,p where p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' φ0 = 450 and E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The obstacle diameter is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 3 2-variable 350 3-variable Equation 300 250 200 150 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='00 Pacing Ratio (p)Using the three-variable model, we measured the unpinning phase of an ACW spiral pinned to an obstacle of radius, r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='u in an electric field of strength Eth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The unpinning is done for a fixed initial phase with overdrive pacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' The results are in good agreement with those obtained from the two-variable Oregonator model and from the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfHPsU/content/2301.01040v1.pdf'} diff --git a/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/2301.11437v1.pdf.txt b/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/2301.11437v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0807b9c649f8727c51af839aeae81b201414ea6c --- /dev/null +++ b/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/2301.11437v1.pdf.txt @@ -0,0 +1,2537 @@ +arXiv:2301.11437v1 [math.NT] 26 Jan 2023 +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION +FIELDS +ANDREW YAO +Abstract. Let K be a global function field. Using Haar measures, we compute the +densities of the Kodiara types and Tamagawa numbers of elliptic curves over a completion +of K. Also, we prove results about the number of iterations of Tate’s algorithm that are +completed when the algorithm is used on elliptic curves over a completion of K. +1. Introduction +Let p be a prime and q = pn for a positive integer n. Let K be a finite extension +of Fq(t). Define MK to be the set of places of K. Suppose P ∈ MK. Let KP be the +completion of K at P and RP be the valuation ring of KP. Suppose E is an elliptic curve +over K with equation +E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 +such that a1, a2, a3, a4, and a6 are elements of K. E has a long Weierstrass form, and if +a1 = a2 = a3 = 0, E has a short Weierstrass form. We study densities for elliptic curves +over K that have a long Weierstrass form. +As an elliptic curve over KP, E has a Kodaira type, which describes its geometry. +Particularly, E has a Tamagawa number cP = [E(KP) : E0(KP)] over KP. A method +to determine the Kodaira type and Tamagawa number of an elliptic curve over KP is +Tate’s algorithm ([6], [7]). The description of Tate’s algorithm in [6] is used in this paper +to compute local densities. Often, steps from this description of Tate’s algorithm are +referred to. +The papers [2] and [3] discuss densities of Kodaira types and Tamagawa products for +elliptic curves over Q. In these papers, the densities at the nonarchimedean places of +Q are considered. In [2] and [3], the densities are for elliptic curves in long and short +Weierstrass form, respectively. +Moreover, [1] discusses densities of Kodaira types and +Tamagawa products for elliptic curves over number fields in short Weierstrass form. Note +that some of the methods for computing local densities with Tate’s algorithm used in +Section 4, Section 5, and Section 6 of this paper are similar to methods used in [1], [2], +and [3]. +Local densities over KP can be obtained using the Haar measure. Let N be a positive +integer. Note that KN +P as an additive group is locally compact, and because of this, Haar’s +theorem can be used on KN +P . Particularly, suppose µP is the Haar measure on KN +P such +that µP(RN +P ) = 1. +Let GP be the set of curves y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 over KP such that +a1, a2, a3, a4, a6 ∈ RP. Because the discriminant of an elliptic curve must be nonzero, not +1 + +2 +ANDREW YAO +all elements of GP are elliptic curves. Also, note that GP can be considered to be R5 +P. +The local densities for GP are obtained from the Haar measure on R5 +P. +Definition 1.1. For an elliptic curve E ∈ GP, let NP(E) be the number of iterations of +Tate’s algorithm that are completed when the algorithm is used on E. +Suppose T is the set of Kodaira types. Let r be an element of T and n be a positive +integer. Define δK(r, n; P) to be the Haar measure of the set of elliptic curves E over KP +with coefficients in RP such that E has Kodaira type r and the Tamagawa number of E +is n. For k ≥ 0, define δK(r, n, k; P) to be the Haar measure of the set of elliptic curves E +over KP with coefficients in RP such that E has Kodaira type r, the Tamagawa number +of E is n, and NP(E) = k. +In this paper, we often consider the number of iterations that Tate’s algorithm completes +when the algorithm is used on an elliptic curve over KP. Note that in order to study this +topic, Proposition 2.4 is useful. Next, we give an important result of the paper. +Theorem 1.2. For a Kodaira type r, positive integer n, and nonnegative integer k, +δK(r, n, k; P) = +1 +Q10k +P +δK(r, n, 0; P). +We prove Theorem 1.2 by considering the cases p ≥ 5, p = 3, and p = 2. Note that the +general method used to prove the theorem is to use translations. The proof of this result +is given in Section 7.1. +Organization. The paper is organized as follows. In Section 2, we introduce elliptic +curves and Tate’s algorithm. Next, in Section 3, for a nonempty finite subset S of MK +and a positive integer N, we discuss how to obtain global densities for ON +K,S. Afterwards, +in Section 4, Section 5, and Section 6, we compute the local densities if the characteristic +p of K is at least 5, equal to 2, and equal to 3, respectively. Finally, in Section 7, we +prove additional results about local and global densities. +Notation. Suppose P is a place of K. Let the degree of P be [RP/πPRP : Fq]. Also, +let QP = |RP/πPRP|. Moreover, let πP be a uniformizer of P in K. Denote vP to be +the valuation vπP over KP; note that vP is also a valuation over K because K ⊂ KP. +Additionally, for a nonnegative integer k, let LP,k be a set of representatives of the cosets +of RP/πk +PRP such that 0 ∈ LP,k. +Suppose S is a finite nonempty subset of MK. We let OK,S be the set of x ∈ K such +that if P ∈ SC = MK\S, vP(x) ≥ 0. Also, let WS be the set of curves y2 + a1xy + a3y = +x3 + a2x2 + a4x + a6 such that a1, a2, a3, a4, a6 ∈ OK,S. +For d ≥ 1, let Td be the number of places of P with degree d. The zeta function of K is +ζK(s) = +∞ +� +d=1 +� +1 − 1 +qds +�−Td +. +Suppose D is a divisor of K. Define L(D) as the set of x ∈ K such that x = 0 or x ̸= 0 +and (x) + D ≥ 0. +Acknowledgements. This research was done in MIT SPUR. The author would like +to thank Hao Peng for providing useful guidance. Also, the author would like to thank +Zhiyu Zhang for suggesting the problem. Additionally, the author would like to thank +David Jerison and Ankur Moitra for giving advice about the project. + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +3 +2. Elliptic Curves and Global Densities +Suppose P is a place of K. Let E be an elliptic curve over KP. There exist a1, a2, a3, a4, a6 ∈ +KP such that E has equation +E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6. +Suppose a1, a2, a3, a4, a6 ∈ KP satisfy this condition. Additionally, define +b2(E) = a2 +1 + 4a2, b4(E) = a1a3 + 2a4, b6(E) = a2 +3 + 4a6, +b8(E) = a2 +1a6 + 4a2a6 − a1a3a4 + a2a2 +3 − a2 +4. +Also, the discriminant of E is +∆(E) = −b2(E)2b8(E) − 8b4(E)3 − 27b6(E)2 + 9b2(E)b4(E)b6(E). +Definition 2.1 ([7]). Elliptic curves E and F over KP are equivalent if there exists +l, m, n, u ∈ KP such that u ̸= 0 and the equation for F can be obtained from the equation +for E by first replacing x with u2x + n and y with u3y + lu2x + m and then dividing by +u6. +Definition 2.2 ([7]). An elliptic curve E over KP is minimal if the equation for E has +coefficients in RP and if there does not exist an elliptic curve F over KP such that the +equation for F has coefficients in RP, F is equivalent to E, and vP(∆(F)) < vP(∆(E)). +The following proposition generalizes Theorem 3.2 of [7] to nonminimal equivalent el- +liptic curves. +Note that this proposition is used later in the paper to compute local +densities. +Proposition 2.3. Let E and F be elliptic curves over KP that have equations with co- +efficients in RP. Assume that E and F are equivalent and satisfy vP(∆(E)) = vP(∆(F)). +Then, there exists l, m, n, u ∈ RP such that vP(u) = 0 and the equation of F can be ob- +tained from the equation of E by first replacing x with u2x+n and y with u3y +lu2x+m +and then dividing by u6. +Proof. The proof of Theorem 3.2 of [7] can be used to prove this proposition. +■ +Proposition 2.4. Let k be a nonnegative integer. Suppose E is an elliptic curve over +KP with equation +E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 +and assume that a1, a2, a3, a4, a6 ∈ RP. For l, m, n ∈ KP, let E′(l, m, n) be the elliptic +curve that is E with x replaced by x + n and y replaced by y + lx + m. NP(E) ≥ k if and +only if there exists l, m, n ∈ RP such that if E′(l, m, n) has equation +E′(l, m, n) : y2 + a′ +1xy + a′ +3y = x3 + a′ +2x2 + a′ +4x + a′ +6, +a′ +i ∈ πki +P RP for i ∈ {1, 2, 3, 4, 6}. +Proof. Suppose l, m, n exist. Let l, m, n satisfy the condition. From Tate’s algorithm, we +have that NP(E) = NP(E′(l, m, n)) ≥ k. +Next, we prove that if NP(E) ≥ k, l, m, and n exist using induction on k. The base +case k = 0 is clear. Let a be a nonnegative integer and assume the result is true for k = a. +We prove the result is true for k = a + 1. Assume NP(E) ≥ a + 1. Because NP(E) ≥ a, + +4 +ANDREW YAO +l, m, n ∈ RP exist such that if x is replaced with x + n and y is replaced with y + lx + m, +the resulting curve E′(l, m, n) : y2+a′ +1xy+a′ +3y = x3+a′ +2x2+a′ +4x+a′ +6 has a′ +i ≡ 0 (mod πia +P ) +for i ∈ {1, 2, 3, 4, 6}. Suppose l, m, n ∈ RP satisfy this condition. Suppose that the curve +that is obtained after Tate’s algorithm is used for a iterations on E′(l, m, n) is +F : y2 + a′ +1 +πa +P +xy + a′ +3 +π3a +P +y = x3 + a′ +2 +π2a +P +x2 + a′ +4 +π4a +P +x + a′ +6 +π6a +P +. +We have that F is E with x replaced with π2a +P x + n and y replaced with π3a +P y + lπ2a +P x + m +divided by π6a +P . +Because NP(E′(l, m, n)) = NP(E) ≥ a+ 1, F will complete at least one more iteration. +During this iteration, suppose x is replaced with x+n′ and y is replaced with y +l′x+m′. +We have that the resulting elliptic curve +F ′ : y2 + a′′ +1xy + a′′ +3y = x3 + a′′ +2x2 + a′′ +4x + a′′ +6 +has a′′ +i ≡ 0 (mod πi +P) for i ∈ {1, 2, 3, 4, 6}. Moreover, F ′ is E with x replaced with +π2a +P x + n + n′π2a +P +and y replaced with +π3a +P y + (l + l′πa +P)π2a +P x + m + m′π3a +P + ln′π2a +P +divided by π6a +P . The equation of +E′(l + l′πa +P, m + m′π3a +P + ln′π2a +P , n + n′π2a +P ) +is +y2 + πa +Pa′′ +1xy + π3a +P a′′ +3y = x3 + π2a +P a′′ +2x2 + π4a +P a′′ +4x + π6a +P a′′ +6, +and πai +P a′′ +i ∈ π(a+1)i +P +RP for i ∈ {1, 2, 3, 4, 6}. +This completes the induction. +We are +done. +■ +Note that Tate’s algorithm cannot be used on a curve in GP with discriminant 0. +However, this is not considered in the calculations of local densities later in the paper. +Suppose r ∈ T, n is a positive integer, and k is a nonnegative integer. The set U of +elliptic curves E ∈ GP with Kodaira type r, Tamagawa number n, and M(E) = k is an +open subset of GP, because if E ∈ U, if multiples of πM +P are added to the coefficients of +E for sufficiently positive large integers M, the resulting curve will be an element of U. +Particularly, the set of elliptic curves is an open subset of GP. In the next proposition, we +prove that the Haar measure of this set is 1; note that it follows that the Haar measure +of the set of curves in GP with discriminant 0 is 0. +Proposition 2.5. The Haar measure of the set of elliptic curves is 1. +Proof. Let M be a positive integer. For E : y2 +a1xy +a3y = x3 +a2x2 +a4x+a6, we see +that the number of solutions for ai, i ∈ {1, 2, 3, 4, 6} modulo πM +P to ∆(E) ≡ 0 (mod πM +P ) +is O(Q4M +P ). Therefore, the Haar measure of the set of elliptic curves with discriminant +equal to 0 is at most O(Q4M +P +) +Q5M +P += O( +1 +QM +P ). The result follows from taking M → ∞. +■ + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +5 +3. Global Densities +Next, global densities are established. Definitions and theorems from [4] are used in +this section. +Let S be a finite nonempty subset of MK. Also, suppose N is a positive integer. Let +Div(S) be the set of divisors +� +P ∈S +nPP +such that for P ∈ S, nP is a nonnegative integer, and there exists P ∈ S such that nP > 0. +Suppose U ⊂ ON +K,S. The upper density of U at S is +dS(U) = lim sup +D∈Div(S) +|U ∩ L(D)N| +|L(D)|N +, +and the lower density of U at S is +dS(U) = lim inf +D∈Div(S) +|U ∩ L(D)N| +|L(D)|N +. +If dS(U) = dS(U), the density dS(U) of U at S exists, and equals dS(U) = dS(U). +Theorem 3.1 ([4], Theorem 2.1). For P ∈ SC, let UP ⊂ KN +P be a measurable set such +that µP(∂UP ) = 0. For a positive integer M, let VM be the set of x ∈ ON +K,S such that +x ∈ UP for some P ∈ SC with degree at least M. Suppose limM→∞ dS(VM) = 0. Let +P : ON +K,S → 2SC, P(a) = {P ∈ SC : a ∈ UP}. Then: +(1) � +P ∈SC µP(UP) is convergent. +(2) For T ⊂ 2SC, ν(T) := dS(P−1(T)) exists. Also, ν defines a measure on 2SC. +(3) ν is concentrated at finite subsets of SC, and for a finite set T of places in SC, +ν(T) = +� +P ∈T +µP(UP) +� +P ∈SC\T +(1 − µP(UP)). +Theorem 3.2 ([4], Theorem 2.2). Let f and g be polynomials in OK,S[x1, . . . , xd] that +are relatively prime. For M ≥ 1, let VM be the set of x ∈ ON +K,S such that f(x) ≡ g(x) ≡ 0 +(mod πP) for some P ∈ SC with degree at least M. Then, limM→∞ dS(VM) = 0. +In this paper, we consider global densities for elliptic curves over K with coefficients +in OK,S in long Weierstrass form. We see that WS can be considered to be O5 +K,S, and +particularly, the global density definitions from above for O5 +K,S can be used on WS. Similar +methods are used in [2] for elliptic curves over Q with coefficients in Z. Note that an elliptic +curve must have a nonzero discriminant, meaning that not all curves in WS are elliptic +curves. However, for D ∈ Div(S), the number of curves in WS with discriminant 0 that +are elements of L(D)5, where WS is considered to be O5 +K,S, is O(|L(D)|4). Particularly, if +proportions over elliptic curves in WS is considered rather than the proportions over WS, +the density is not changed. +Proposition 3.3 is about the global density of nonminimal elliptic curves. Note that the +lemma is used to prove Theorem 7.2. + +6 +ANDREW YAO +Proposition 3.3. For a positive integer M, let VM be the set of elliptic curves E ∈ WS +such that there exists P ∈ SC with degree at least M such that NP(E) ≥ 1. Then, +limM→∞ dS(VM) = 0. +Proof. We prove this with casework on the characteristic p of K. +Suppose that E is +an elliptic curve in GP with equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 for +a1, a2, a3, a4, a6 ∈ RP such that NP(E) ≥ 1. +Assume p ≥ 5. We have that E can be translated to the curve +y2 = x3 + +� +−b2(E)2 +48 ++ b4(E) +2 +� +x − b2(E)3 +864 +− b2(E)b4(E) +24 ++ b6(E) +4 +. +Because NP(E) ≥ 1, using Proposition 2.4, −b2(E)2 +48 ++ b4(E) +2 +≡ 0 (mod πP) and −b2(E)3 +864 +− +b2(E)b4(E) +24 ++ b6(E) +4 +≡ 0 (mod πP). Then, Theorem 3.2 with +f(x1, x2, x3, x4, x6) = −(x2 +1 + 4x2)2 +48 ++ x1x3 + 2x4 +2 +and +g(x1, x2, x3, x4, x6) = −(x2 +1 + 4x2)3 +864 +− (x2 +1 + 4x2)(x1x3 + 2x4) +24 ++ x2 +3 + 4x6 +4 +proves this proposition for p ≥ 5. +Next, assume p = 3. We have that E can be translated to the curve +y2 = x3 + b2(E) +4 +x2 + b4(E) +2 +x + b6(E) +4 +Using Proposition 2.4, +b2(E) +4 +≡ 0 (mod πP) from the coefficient of x2. +Additionally, +∆(E) ≡ 0 (mod πP). Next, Theorem 3.2 with +f(x1, x2, x3, x4, x6) = −(x2 +1 + x2)2(x2 +1x6 + x2x6 − x1x3x4 + x2x2 +3 − x2 +4) + (x1x3 + 2x4)3 +and +g(x1, x2, x3, x4, x6) = x2 +1 + x2 +proves this proposition for p = 3. +Suppose p = 2. Using Proposition 2.4, a1 ≡ 0 (mod πP) from the coefficient of xy. +Also, ∆(E) ≡ 0 (mod πP). Therefore, Theorem 3.2 with +f(x1, x2, x3, x4, x6) = x4 +1(x2 +1x6 + x1x3x4 + x2x2 +3 + x2 +4) + x4 +3 + x3 +1x3 +3 +and +g(x1, x2, x3, x4, x6) = x1 +proves this proposition for p = 2. +■ +4. Local Densities for p ≥ 5 +4.1. Setup. Suppose that the characteristic of K is p ≥ 5. Let P be a place of K. We +compute the local densities over KP of Kodaira types r and Tamagawa numbers n for +elliptic curves in GP. Let G(1) +P +be the set of curves +y2 = x3 + a4x + a6 + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +7 +over KP such that a4, a6 ∈ RP. Note that G(1) +P +can be considered to be R2 +P. Define +ϕ : GP → G(1) +P +as the function such that if E is a curve in GP, ϕ(E) is the curve in G(1) +P +with equation +ϕ(E) : y2 = x3 + +� +−b2(E)2 +48 ++ b4(E) +2 +� +x − b2(E)3 +864 +− b2(E)b4(E) +24 ++ b6(E) +4 +. +If E is an elliptic curve, ϕ(E) is an elliptic curve equivalent to E. +Lemma 4.1. If U is an open subset of G(1) +P , µP(ϕ−1(U)) = µP(U). +Proof. Let V be the set of y2 = x3 + a′ +4x + a′ +6 with a′ +4 ∈ r4 + πn4 +P RP and a′ +6 ∈ r6 + πn6 +P RP. +It suffices to prove that µP(ϕ−1(V )) = µP(V ) = +1 +Qn4+n6 because all open subsets of +G(1) +P +can be written as a disjoint countable union of sets with the form of V . Suppose +E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 ∈ GP. ϕ(E) ∈ V if and only if +−b2(E)2 +48 ++ b4(E) +2 +∈ r4 + πn4 +P RP +and +−b2(E)3 +864 +− b2(E)b4(E) +24 ++ b6(E) +4 +∈ r6 + πn6 +P RP. +Assume that ϕ(E) ∈ V . Let M = max(n4, n6). First, select a1, a2, and a3 modulo +πM +P . Each has QM +P possible residues. Afterwards, a4 will have QM−n4 +P +residues modulo +πM +P ; select the residue for a4. +Finally, a6 has QM−n6 +P +residues modulo πM +P ; select the +residue for a6. We see that if each of a1, a2, a3, a4, a6 are taken modulo πM +P , the number of +combinations of residues is Q5M−n4−n6 +P +. Also, because ai is modulo πM +P for i ∈ {1, 2, 3, 4, 6}, +each combination of residues has a Haar measure of +1 +Q5M +P . We are done. +■ +4.2. Multiple Iterations. Let k be a nonnegative integer. +Suppose Sk is the set of +elliptic curves E ∈ G(1) +P +such that NP(E) ≥ k. +Suppose E is an elliptic curve in G(1) +P +with equation E : y2 = x3 + a4x + a6. Assume +E ∈ Sk. Then, using Proposition 2.4, l, m, n ∈ RP exist such that +� +y + l +πk +P +x + m +π3k +P +�2 +− +� +x + n +π2k +P +�3 +− a4 +π4k +P +� +x + n +π2k +P +� +− a6 +π6k +P +∈ RP[x, y]. +The coefficient of xy is +2l +πk +P , giving that vP(l) ≥ k, and the coefficient of y is +2m +π3k +P , giving +that vP(m) ≥ 3k. Also, the coefficient of x2 is 3n−l2 +π2k +P +, giving that vP(n) ≥ 2k. From this, +we have that vP(a4) ≥ 4k and vP(a6) ≥ 6k. +Define the function φk : Sk → S0, y2 = x3 + a4x + a6 �→ y2 = x3 + +a4 +π4k +P x + +a6 +π6k +P . Note +that Sk ⊂ S0 ⊂ G(1) +P . From Proposition 2.5 and Lemma 4.1, µP(S0) = 1. Next, we show +how we can use φk to compute densities for Sk. +Lemma 4.2. If U is an open subset of G(1) +P , µP(φ−1 +k (U)) = +1 +Q10k +P µP(U). + +8 +ANDREW YAO +Proof. Suppose r4, r6 ∈ RP. Also, suppose n4 and n6 are nonnegative integers. Let V be +the set of elliptic curves y2 = x3 + a′ +4x + a′ +6 with a′ +4 ∈ r4 + πn4 +P RP and a′ +6 ∈ r6 + πn6 +P RP. +Because µP(S0) = 1, µP(V ) = +1 +Qn4+n6 +P +. To prove the lemma, it suffices to prove that +µP(φ−1 +k (V )) = +1 +Q10k +P +µP(V ) = +1 +Qn4+n6+10k +P +. +Suppose E : y2 = x3 + a4x + a6 ∈ G(1) +P +is an elliptic curve. We prove that E ∈ Sk +and φk(E) ∈ V if and only if +a4 +π4k +P +∈ r4 + πn4 +P RP and +a6 +π6k +P +∈ r6 + πn6 +P RP. If φk(E) ∈ V , +then +a4 +π4k +P +∈ r4 + πn4 +P RP and +a6 +π6k +P +∈ r6 + πn6 +P RP. +Assume that +a4 +π4k +P +∈ r4 + πn4 +P RP and +a6 +π6k +P ∈ r6 + πn6 +P RP. From Tate’s algorithm, we have that E ∈ Sk. Then, it is true that +φk(E) ∈ V . +Assume that E ∈ Sk and φk(E) ∈ V . This is true if and only if a4 ∈ π4k +P r4 + πn4+4k +P +R +and a6 ∈ π6k +P r6 + πn6+6k +P +R. Moreover, because µP(S0) = 1, the density of curves y2 = +x3 + a4x + a6 with discriminant 0 such that a4 ∈ π4k +P r4 + πn4+4k +P +and a6 ∈ π6k +P r6 + πn6+6k +P +is 0. Because of this, µP(φ−1 +k (V )) = +1 +Qn4+n6+10k +P +, completing the proof. +■ +4.3. Density Calculations. Note that the density of a set of curves in G(1) +P +is the Haar +measure of the set. In this subsection, we compute the density of the set of minimal +elliptic curves with a given Kodaira type and Tamagawa number over G(1) +P . This can be +extended to nonminimal elliptic curves using Theorem 1.2. Moreover, in this subsection, +we use that the set of curves in G(1) +P +that have a discriminant equal to 0 has a Haar +measure of 0. +Suppose the discriminant is not divisible by πP. We compute the density for this set +by considering a4 and a6 modulo πP. Suppose a4 ∈ r4 + πPRP and a6 ∈ r6 + πPRP. We +find the number of pairs (r4, r6) in L2 +P,1 such that +� r4 +3 +�3 + +� r6 +2 +�2 ≡ 0 (mod πP). If r4 = 0, +r6 has 1 choice, and if −r4 +3 is a square modulo πP, r6 has 2 choices. Otherwise, r6 has 0 +choices. We see that the number of pairs (r4, r6) is QP. Therefore, where each pair (r4, r6) +has a density of +1 +Q2 +P , the density of the discriminant not being divisible by πP is QP −1 +QP . +For this case, Tate’s algorithm ends in step 1 and we get that δK(I0, 1, 0; P) = QP −1 +QP . +Next, assume that the discriminant is divisible by πP. +Furthermore, assume that +a4, a6 ̸≡ 0 (mod πP). Because there are QP − 1 pairs (r4, r6) in L2 +P,1 for this case, the +total density is QP −1 +Q2 +P . Let α be the element of LP,1 such that a4 ≡ −3α2 (mod πP) and +a6 ≡ 2α3 (mod πP). The singular point is (α, 0) and in step 2, x is replaced with x + n +where n = α. Because α ̸≡ 0 (mod πP), Tate’s algorithm ends in step 2. The quadratic +considered in step 2 is T 2 − 3α. We see that for QP −1 +2 +values of α, this quadratic has +roots in RP/πPRP and c = vP(∆(E)). Otherwise, c = 1 if vP(∆(E)) is odd and c = 2 if +vP(∆(E)) is even. +Let N be a positive integer. Suppose a4 ∈ r4 + πN +P RP and a6 ∈ r6 + πN +P RP. We find +the number of pairs (r4, r6) in L2 +P,1 such that +� r4 +3 +�3 + +�r6 +2 +�2 ≡ 0 (mod πN +P ) and r4, r6 ̸= 0. +Because there are +QN +P −QN−1 +P +2 +nonzero residues that are squares modulo πM +P , we have that + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +9 +the number of pairs (r4, r6) is QN +P − QN−1 +P +. Therefore, the density of vP(∆(E)) ≥ N for +a4, a6 ̸≡ 0 (mod πP) is QP −1 +QN+1 +P +. +Suppose N is a positive integer. The density of vP(∆(E)) = N is +QP −1 +QN+1 +P +− QP −1 +QN+2 +P += +(QP −1)2 +QN+2 +P +. We therefore have that δK(I1, 1, 0; P) = (QP −1)2 +Q3 +P +, δK(I2, 2, 0; P) = (QP −1)2 +Q4 +P +, and +δK(IN, N, 0; P) = δK +� +IN, 2 +�N +2 +� +− N + 2, 0; P +� += (QP − 1)2 +2QN+2 +P +for N ≥ 3. +If vP(a4), vP(a6) ≥ 1, the singular point modulo πP from step 2 of Tate’s algorithm is +(0, 0). The total density for this case is +1 +Q2 +P . If vP(a6) = 1, the algorithm ends in step 3. +For this, we get δK(II, 1, 0; P) = QP −1 +Q3 +P . +Assume that vP(a6) ≥ 2. The total density for this case is +1 +Q3 +P . If vP(a4) = 1, the +algorithm ends in step 4, and we get that δK(III, 2, 0; P) = QP −1 +Q4 +P . +Next, suppose vP(a4) ≥ 2. The total density for this case is +1 +Q4 +P . If vP(a6) = 2, the +algorithm ends in step 5. From this, we have that δK(IV, 1, 0; P) = δK(IV, 3, 0; P) = QP −1 +2Q5 +P . +Suppose vP(a6) ≥ 3. The total density for this case is +1 +Q5 +P . In step 6, the polynomial +P(T) ∈ (RP/πPRP)[T] has coefficient of T 2 equal to 0. From adding multiples of π2 +P +to a4, the choices for the coefficient of T are LP,1. Also, from adding multiples of π3 +P +to a6, the choices for the constant term are LP,1. Then, we have that each polynomial +P(T) ∈ (RP/πPRP)[T] with coefficient of T 2 equal to 0 corresponds to a density of +1 +Q7 +P in +G(1) +P . +Assume P(T) has distinct roots in RP/πPRP. The total number of P(T) for this case is +Q2 +P −QP; therefore, the total density for this case is QP −1 +Q6 +P . We have that Tate’s algorithm +ends in step 6 here. The number of P(T) with 0, 1, and 3 roots in RP/πPRP is Q2 +P −1 +3 +, +Q2 +P −QP +2 +, and Q2 +P −3QP +2 +6 +, respectively. With this, δK(I∗ +0, 1, 0; P) = Q2 +P −1 +3Q7 +P , δK(I∗ +0, 2, 0; P) = +QP −1 +2Q6 +P , and δK(I∗ +0, 4, 0; P) = Q2 +P −3QP +2 +6Q7 +P +. +Next, assume that P(T) has a double root and a simple root in RP/πPRP. Then, +Tate’s algorithm enters the subprocedure in step 7. For this case, the total number of +P(T) is QP − 1 and the total density is therefore QP −1 +Q7 +P . In Section 4.4, we compute that +δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = (QP −1)2 +2QN+7 +P +for all positive integers N. +Assume P(T) has a triple root in RP/πPRP. For this case, the total number of P(T) +is 1 and the total density is therefore +1 +Q7 +P . +Because the coefficient of T 2 in P(T) is +0, the triple root is 0. +If vP(a6) = 4, the algorithm ends in step 8. +For this case, +δK(IV ∗, 1, 0; P) = δK(IV ∗, 3, 0; P) = QP −1 +2Q8 +P . +Next, assume that vP(a6) ≥ 5. The total density for this case is +1 +Q8 +P . If vP(a4) = 3, the +algorithm ends in step 9. We then have that δK(III∗, 2, 0; P) = QP −1 +Q9 +P . + +10 +ANDREW YAO +Suppose vP(a4) ≥ 4. The total density for this case is +1 +Q9 +P . If vP(a6) = 5, the algorithm +ends in step 10. Therefore, δK(II∗, 1, 0; P) = QP −1 +Q10 +P . +With density +1 +Q10 +P , we have that vP(a4) ≥ 4 and vP(a6) ≥ 6, meaning that the curve is +not minimal. That is, the curve will complete iteration 1 and continue iteration 2. Note +that the density of nonminimal curves calculated from the algorithm matches Lemma 4.2. +4.4. Subprocedure Density Calculations. Next, we study the densities for the sub- +procedure in step 7 of Tate’s algorithm. +We compute the subprocedure densities by +studying the translation of x in Tate’s algorithm. In the step 7 subprocedure, because +the coefficient of y is initially 0, there will be no translations of y. +Let X be the set of elliptic curves E ∈ G(1) +P +such that NP(E) = 0 and Tate’s algorithm +enters the step 7 subprocedure when used on E. For E ∈ X, let L(E) be the number of +iterations of the step 7 subprocedure that are completed when Tate’s algorithm is used +on E. For a nonnegative integer N, let XN be the set of E ∈ X such that L(E) ≥ N. +Suppose N is an even nonnegative integer. Iteration N of the step 7 subprocedure is +completed if and only if n ∈ RP exists such that vP(n) = 1, vP(a4 + 3n2) ≥ N+6 +2 , and +vP(n3 + 3na4 + a6) ≥ N + 4. Suppose n = n1 satisfies this condition. Suppose n = n2 +also satisfies this condition. We then have that n2 +1 ≡ n2 +2 (mod π +N+6 +2 +P +). This gives that +n1 is equivalent to n2 or −n2 modulo π +N+4 +2 +P +. However, because n3 +1 + n1a4 ≡ n3 +2 + n2a4 +(mod πN+4 +P +), we have that vP(n1 − n2) ≥ N+4 +2 . Moreover, if vP(n1 − n2) ≥ N+4 +2 , n = n2 +also satisfies the condition. +Next, suppose N is an odd nonnegative integer. Iteration N of the subprocedure is +completed if and only if n ∈ RP exists such that vP(n) = 1, vP(a4 + 3n2 +1) ≥ N+5 +2 , and +vP(n3 + na4 + a6) ≥ N + 4. Similarly, we have that if n = n1 satisfies the condition, +n = n2 satisfies the condition if and only if vP(n1 − n2) ≥ N+3 +2 . +Suppose N is a nonnegative integer. Suppose n is an element of LP,⌊ N+4 +2 ⌋ such that +vP(n) = 1. Let Yn,N be the set of curves x3 + 3nx2 + a′ +4x + a′ +6 such that vP(a′ +4) ≥ +� N+6 +2 +� +and vP(a′ +6) ≥ N + 4. Note that Yn,N can be considered to be an open subset of R2 +P. +For E ∈ XN, let nN(E) be the unique value of n ∈ LP,⌊ N+4 +2 ⌋ such that vP(n) = 1, +vP(a4 + 3n2) ≥ +� N+6 +2 +� +, and vP(n3 + na4 + a6) ≥ N + 4. Let θN be the function such that +if E : y2 = x3 + a4x + a6 is an element of XN, +θN(E) : y2 = (x + nN(E))3 + a4(x + nN(E)) + a6 += x3 + 3nN(E)x2 + (a4 + 3nN(E)2)x + nN(E)a4 + a6 + nN(E)3. +Lemma 4.3. If U is an open subset of Yn,N, µP(θ−1 +N (U)) = µP(U). +Proof. Suppose r4, r6 ∈ RP. Also, suppose n4 and n6 are nonnegative integers. Assume +that vP(r4), n4 ≥ ⌊N+4 +2 ⌋ and vP(r6), n6 ≥ N + 4. Let V ⊂ Yn,N be the set of E′ : y2 = +x3 + 3nx2 + a′ +4x + a′ +6 such that a′ +4 ∈ r4 + πn4 +P RP and a′ +6 ∈ r6 + πn6 +P RP. It suffices to prove +that µP(θ−1 +N (V )) = µP(V ). Suppose E : y2 = x3 + a4x + a6 is an elliptic curve. +We prove that that E ∈ XN and θN(E) ∈ V if and only if +a4 + 3n2 ∈ r4 + πn4 +P RP, na4 + a6 + n3 ∈ r6 + πn6 +P RP. + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +11 +Assume that E ∈ XN and θN(E) ∈ V . Because θN(E) ∈ V , we have that nN(E) = n. +Therefore, a4 + 3n2 ∈ r4 + πn4 +P RP and na4 + a6 + n3 ∈ r6 + πn6 +P RP. Next, assume that +a4 + 3n2 ∈ r4 + πn4 +P and na4 + a6 + n3 ∈ r6 + πn6 +P RP. Because vP(a4 + 3n2) ≥ +� N+6 +2 +� +and +vP(na4 + a6 + n3) ≥ N + 4, E ∈ XN. We then have that θN(E) ∈ V . +Let M = max(n4, n6). Modulo πM +P , there are QM−n4 +P +choices for the residue of a4. After +choosing a4 modulo πM +P , there are QM−n6 +P +choices for the residue of a6 modulo πM +P . Each +of these combinations of residues modulo πM +P for a4 and a6 has a density of +1 +Q2M +P +in G(1) +P . +The Haar measure of the Q2M−n4−n6 +P +combinations is +1 +Qn4+n6 +P +. Because the set of curves in +G(1) +P +with discriminant 0 has a Haar measure of 0, +µP(θ−1 +N (V )) = +1 +Qn4+n6 +P += µP(V ). +This finishes the proof. +■ +Let N be a positive integer. We compute the density of I∗ +N. Let n be an element of +LP,⌊ N+3 +2 ⌋ such that vP(n) = 1. We have that the Haar measure of the set of E ∈ Yn,N−1 +that do not complete iteration N is +QP −1 +Q⌊ N+5 +2 ⌋+N+4 +P +. With Lemma 4.3, because there are +(QP − 1)Q⌊ N−1 +2 ⌋ +P +values of n, the density of I∗ +N is (QP −1)2 +QN+7 +P +. From adding multiples of πN+4 +P +to a6, c = 2 and c = 4 have equal density. Therefore, +δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = (QP − 1)2 +2QN+7 +P +. +5. Local Densities for p = 3 +5.1. Setup. Suppose that the characteristic of K is p = 3. Let P be a place of K and +G(2) +P +be the set of curves +y2 = x3 + a2x2 + a4x + a6 +over KP such that a2, a4, a6 ∈ RP. Note that G(2) +P +can be considered to be R3 +P. Define +ϕ : GP → G(2) +P +as the function such that if E is a curve in GP, ϕ(E) is the curve in G(2) +P +with equation +y2 = x3 + b2(E) +4 +x2 + b4(E) +2 +x + b6(E) +4 +. +Note that if E is an elliptic curve, E and ϕ(E) are equivalent. +Lemma 5.1. If U is an open subset of G(2) +P , µP(ϕ−1(U)) = µP(U). +Proof. This can be proved using a method similar to the proof of Lemma 4.1. +■ +5.2. Multiple Iterations. Let k be a nonnegative integer. +Suppose Sk is the set of +elliptic curves E ∈ G(2) +P +such that NP(E) ≥ k. +Suppose E ∈ Sk has equation E : y2 = x3 + a2x2 + a4x + a6. From Proposition 2.4, +l, m, n ∈ RP exist such that +� +y + l +πk +P +x + m +π3k +P +�2 += +� +x + n +π2k +P +�3 ++ a2 +π2k +P +� +x + n +π2k +P +�2 ++ a4 +π4k +P +� +x + n +π2k +P +� ++ a6 +π6k +P + +12 +ANDREW YAO +has coefficients in RP. From the coefficient of xy, vP(l) ≥ k, and from the coefficient of +y, vP(m) ≥ 3k. Therefore, we have that +y2 = +� +x + n +π2k +P +�3 ++ a2 +π2k +P +� +x + n +π2k +P +�2 ++ a4 +π4k +P +� +x + n +π2k +P +� ++ a6 +π6k +P +has coefficients in RP. Note that vP(a2) ≥ 2k also. +For an elliptic curve E ∈ G(2) +P +with equation E : y2 = x3 + a2x2 + a4x + a6, let Ak(E) +be the set of n ∈ RP such that +y2 = x3 + a2 +π2k +P +x2 + 2na2 + a4 +π4k +P +x + n2a2 + na4 + a6 + n3 +π6k +P +has coefficients in RP. The next proposition is useful for computing local densities for +multiple iterations. +Proposition 5.2. Let E be an elliptic curve in G(2) +P . E ∈ Sk if and only if a unique +element n ∈ LP,k exists such that n ∈ Ak(E). +Proof. Assume a unique element n ∈ LP,k exists such that n ∈ Ak(E). Then, Ak(E) is +nonempty, and using Proposition 2.4, E ∈ Sk. +Next, assume E ∈ Sk. From Proposition 2.4, we have that Ak(E) is nonempty. Let the +equation of E be E : y2 = x3 + a2x2 + a4x + a6 for a2, a4, a6 ∈ RP. +Suppose n ∈ Ak(E). From replacing x with x+n′ for n′ ∈ RP, we have that n+n′π2k +P ∈ +Ak(E). Therefore, n ∈ LP,k exists such that n ∈ Ak(E). +Next, we prove uniqueness. Assume n1, n2 ∈ Ak(E) ∩ LP,k. Let +F : y2 = x3 + a2 +π2k +P +x2 + a4 +π4k +P +x + a6 +π6k +P +. +For 1 ≤ i ≤ 2, let Fi be F with x replaced by x + +ni +π2k +P . Note that F1, F2 ∈ G(2) +P . +From the coefficients of x in F1 and F2, +2n1a2 + a4 ≡ 2n2a2 + a4 ≡ 0 +(mod π4k +P ). +Also, from the constant terms of F1 and F2, +n2 +1a2 + n1a4 + n3 +1 ≡ n2 +2a2 + n2a4 + n3 +2 +(mod π6k +P ). +For the sake of contradiction, assume that vP(n1 − n2) < 2k. Let a = vP(n1 − n2). Note +that +vP(n3 +1 − n3 +2) = vP((n1 − n2)3) = 3a. +We have that +n2 +1a2 + n1a4 − n2 +2a2 − n2a4 = (n1 − n2)(n1a2 + n2a2 + a4). +Because a4 ≡ n1a2 ≡ n2a2 (mod π4k +P ), +n1a2 + n2a2 + a4 ≡ 3a4 ≡ 0 +(mod π4k +P ). +From this, +vP(n2 +1a2 + n1a4 − n2 +2a2 − n2a4) = vP((n1 − n2)(n1a2 + n2a2 + a4)) +≥ a + 4k +> 3a. + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +13 +Since vP(n3 +1 − n3 +2) = 3a, +vP(n2 +1a2 + n1a4 + n3 +1 − n2 +2a2 − n2a4 − n3 +2) = 3a < 6k, +which is a contradiction. Therefore, vP(n1 − n2) ≥ 2k and n1 = n2. +■ +Using Proposition 5.2, for E ∈ Sk, let n(E) be the unique n ∈ LP,2k such that the +n ∈ Ak(E). Define φk : Sk → S0 to be the function such that if E ∈ Sk has equation +E : y2 = x3 + a2x2 + a4x + a6, φk(E) ∈ S0 has equation +φk(E) : y2 = x3 + a2 +π2k +P +x2 + 2n(E)a2 + a4 +π4k +P +x + n(E)2a2 + n(E)a4 + a6 + n(E)3 +π6k +P +. +Note that Sk ⊂ S0 ⊂ G(2) +P . Also, using Proposition 2.5 and Lemma 5.1, µP(S0) = 1. +For n ∈ LP,2k, suppose Sk,n is the set of E ∈ Sk such that n(E) = n, and let φk,n be φk +restricted to Sk,n. +Lemma 5.3. Suppose n ∈ LP,k. If U is an open subset of G(2) +P , µP(φ−1 +k,n(U)) = +1 +Q12k +P µP(U). +Proof. Suppose r2, r4, r6 ∈ RP. Also, suppose n2, n4, and n6 are nonnegative integers. +Let V be the set of y2 = x3 + a′ +2x2 + a′ +4x+ a′ +6 such that a′ +2 ∈ r2 + πn2 +P RP, a′ +4 ∈ r4 + πn4 +P RP, +and a′ +6 ∈ r6 + πn6 +P RP. Suppose E : y2 = x3 + a2x2 + a4x + a6 ∈ G(2) +P . E ∈ Sk,n and +φk,n(E) ∈ V if and only if +a2 +π2k +P +∈ r2 + πn2 +P RP, 2na2 + a4 +π4k +P +∈ r4 + πn4 +P RP, n2a2 + na4 + a6 + n3 +π6k +P +∈ r6 + πn6 +P RP. +Assume that E ∈ Sk,n and φk,n(E) ∈ V . Let M = max(n2+2k, n4+4k, n6+6k). There +are QM−n2−2k +P +ways to pick a2 modulo πM +P . Afterwards, a4 will have QM−n4−4k +P +choices +for the residue modulo πM +P ; pick a4 modulo πM +P . Next, a6 has QM−n6−6k +P +choices for the +residue modulo πM +P . Select the residue for a6. The number of combinations of residues is +Q3M−n2−n4−n6−12k +P +and each combination of residues has a Haar measure of Q−3M +P +. Also, +because µP(S0) = 1, the set of curves with discriminant 0 counted in these combinations +of residues has a Haar measure 0. Therefore, µP(φ−1 +k,n(V )) = +1 +Qn2+n4+n6+12k +P +. With this, +µP(φ−1 +k,n(U)) = +1 +Q12k +P µP(U) for all open subsets U of G(2) +P . +■ +Lemma 5.4. If U is an open subset of G(2) +P , µP(φ−1 +k (U)) = +1 +Q10k +P µP(U). +Proof. Let U be an open subset of G(2) +P +We have that φ−1 +k (U) = � +n∈LP,2k φ−1 +k,n(U). Using +Lemma 5.3, +µP(φ−1 +k (U)) = +� +n∈LP,2k +µP(φ−1 +k,n(U)) = +� +n∈LP,2k +1 +Q12k +P +µP(U) = +1 +Q10k +P +µP(U), +completing the proof. +■ + +14 +ANDREW YAO +5.3. Density Calculations for vP(a2) = 0. Suppose vP(a2) = 0. The density for this +case over G(2) +P +is QP −1 +QP . The discriminant is −a3 +2a6 + a2 +2a2 +4 − a3 +4. +From adding multiples of πP to a6, the set of curves with discriminant not divisible by +πP has density (QP −1)2 +Q2 +P +. Then, we add (QP −1)2 +Q2 +P +to δK(I0, 1, 0; P). +Assume the discriminant is divisible by πP . The algorithm ends in step 2. Because +vP(a2) = 0, the coefficient of a6 in the discriminant is not divisible by πP. Then, we see +that for N ≥ 0, the density over G(2) +P +of curves such that vP(a2) = 0 and vP(∆(E)) = N +is (QP −1)2 +QN+2 +P +. If a2 ≡ r2 (mod πP) for r2 ∈ LP,1 such that r2 ̸= 0, T 2 + a2 is irreducible +over RP/πPRP for QP −1 +2 +values of r2. Using step 2 of Tate’s algorithm, we have that +δK(I1, 1, 0; P) = (QP −1)2 +Q3 +P +, δK(I2, 2, 0; P) = (QP −1)2 +Q4 +P +, and +δK(IN, N, 0; P) = δK +� +IN, 2 +�N +2 +� +− N + 2, 0; P +� += (QP − 1)2 +2QN+2 +P +for N ≥ 3. +5.4. Density Calculations for vP(a2) ≥ 1. Next, suppose vP(a2) ≥ 1. The density for +this is +1 +QP and modulo πP, the discriminant is −a3 +4. +Assume the discriminant is not divisible by πP. This occurs if and only if a4 is not +divisible by πP, and the density of this case is QP −1 +Q2 +P . Adding this density to δK(I0, 1, 0; P) +gives that δK(I0, 1, 0; P) = QP −1 +QP . +Next, assume the discriminant is divisible by πP. The total density for these cases will +be +1 +Q2 +P . Suppose α1 is an element of LP,1 such that a6 + α3 +1 ≡ 0 (mod πP). A singular +point is (α1, 0). We have that x is replaced with x + n where n = α1. The resulting curve +has equation +y2 = (x + n)3 + a2(x + n)2 + a4(x + n) + a6. +We have that n2a2 + na4 + a6 + n3 is not divisible by π2 +P with density QP −1 +Q3 +P +by adding +multiples of πP to a6. For this case, δK(II, 1, 0; P) = QP −1 +Q3 +P . +Assume n2a2 + na4 + a6 + n3 is divisible by π2 +P. The total density for this case is +1 +Q3 +P . +The density of vP(2na2 + a4) = 1 is QP −1 +Q4 +P +from replacing a4 with a4 + πPd and a6 with +a6 − α1πPd for d ∈ LP,1. If vP(2na2 + a4) = 1, the algorithm ends in step 4. We then +have that δK(III, 2, 0; P) = QP −1 +Q4 +P . +Assume 2na2 + a4 is divisible by π2 +P. The total density for this case is +1 +Q4 +P . We have +that vP(n2a2 + na4 + a6 + n3) = 2 with density QP −1 +Q5 +P +from adding multiples of π2 +P to a6. +If this is true, the algorithm ends in step 5. Afterwards, we have that δK(IV, 1, 0; P) = +δK(IV, 3, 0; P) = QP −1 +2Q5 +P . +Suppose vP(n2a2 + na4 + a6 + n3) ≥ 3. +The total density for this case is +1 +Q5 +P . +In +step 6, there is no translation. Suppose a2 is replaced by a2 + d1πP, a4 is replaced with +a4 − 2α1d1πP, and a6 is replaced with a6 + α2 +1d1πP for d1 ∈ LP,1. Note that the previous +parts of the algorithm will not be changed. However, this changes the coefficient of x2 +from a2 to a2 + d1πP , which changes the coefficient of T 2 of P(T) in step 6. Next, replace + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +15 +a4 with a4 + d2π2 +P and a6 with a6 − α1d2π2 +P for d2 ∈ πP. Similarly, this does not change +the previous parts of the algorithm. However, d2π2 +P will be added to the coefficient of x, +which adds d2 to the coefficient of T of P(T). Afterwards, replace a6 with a6 + d3π3 +P for +d3 ∈ LP,1. This adds d3 to the constant term P(T). With this, the choices for P(T) are +the monic polynomials with degree 3 in (RP/πPRP)[T]; each choice for P(T) corresponds +to a density of +1 +Q8 +P . Moreover, the number of P(T) with a double root and triple root are +QP(QP − 1) and QP, respectively. +Assume P(T) has distinct roots. We have that the algorithm ends in step 6, with +δK(I∗ +0, 1, 0; P) = +Q2 +P −1 +3Q7 +P , δK(I∗ +0, 2, 0; P) = QP −1 +2Q6 +P , and δK(I∗ +0, 4, 0; P) = +Q2 +P −3QP +2 +6Q7 +P +. +Assume P(T) has a double root. For this case, Tate’s algorithm ends in step 7 and the +total density is QP −1 +Q7 +P . In Section 5.5, we compute that δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = +(QP −1)2 +2QN+7 +P +for all positive integers N. +Next, assume P(T) has a triple root. The density for this case is +1 +Q7 +P . Let α2 be the +element of LP,1 such that +n2a2 + na4 + a6 + n3 ≡ −π3 +P α3 +2 +(mod π4 +P). +Then, for the translation in step 8, we let n = α1+α2πP. Suppose vP(n2a2+na4+a6+n3) = +4. This occurs with density QP −1 +Q8 +P +by adding multiples of π4 +P to a6. In this case, Tate’s +algorithm ends in step 8, and δK(IV ∗, 1, 0; P) = δK(IV ∗, 3, 0; P) = QP −1 +2Q8 +P . +Assume vP(n2a2 + na4 + a6 + n3) ≥ 5. The total density for this case is +1 +Q8 +P . Consider +replacing a4 with a4 + dπ3 +P and a6 with a6 − (α1 + α2πP)dπ3 +P for d ∈ LP,1. This does not +change previous parts of the algorithm but adds dπ3 +P to the coefficient of x. Therefore, +vP(2na2 + a4) = 3 with density QP −1 +Q9 +P . For this, we have that Tate’s algorithm ends in +step 9 and δK(III∗, 2, 0; P) = QP −1 +Q9 +P . +Suppose vP(2na2+a4) ≥ 4. The total density of this case is +1 +Q9 +P . From adding multiples +of π6 +P to a6, vP(n3 +a2n2 +a4n+a6) = 5 with density QP −1 +Q10 +P . Also, if vP(n3 +a2n2 +a4n+ +a6) = 5, the algorithm ends in step 10. This gives that δK(II∗, 1, 0; P) = QP −1 +Q10 +P . +5.5. Subprocedure Density Calculations. Let X be the set of elliptic curves E ∈ G(2) +P +such that NP(E) = 0 and Tate’s algorithm enters the step 7 subprocedure when used on +E. For E ∈ X, let L(E) be the number of iterations of the step 7 subprocedure that are +completed when Tate’s algorithm is used on E. For a nonnegative integer N, let XN be +the set of E ∈ X such that L(E) ≥ N. +Suppose N is an even nonnegative integer. Iteration N of the step 7 subprocedure is +completed if and only if n ∈ RP exists such that vP(a2) = 1, vP(2na2 + a4) ≥ N+6 +2 , and +vP(n3 + n2a2 + na4 + a6) ≥ N + 4. Assume n = n1 satisfies the condition. Suppose +n = n2 satisfies the condition also. Because vP(a2) = 1, vP(n1 − n2) ≥ +N+4 +2 . Next, +assume vP(n1 − n2) ≥ N+4 +2 . We show that n = n2 also satisfies the condition. Clearly, +vP(2n2a2 + a4) ≥ N+6 +2 . Moreover, we have that +n2 +2a2 + n2a4 = n2 +1a2 + n1a4 + 1 +2(n2 − n1)((2n1a2 + a4) + (2n2a2 + a4)). + +16 +ANDREW YAO +Therefore, vP(n3 +2 +n2 +2a2 +n2a4 +a6) ≥ N +4. We have that n = n2 satisfies the condition +if and only if vP(n1 − n2) ≥ N+4 +2 . +Next, suppose N is an odd nonnegative integer. Iteration N of the step 7 subprocedure +is completed if and only if n ∈ RP exists such that vP(n2a2 + na4 + a6 + n3) ≥ N + 4 +and vP(2na2 + a4) ≥ N+5 +2 . Assume n = n1 satisfies the condition. Similarly to when N is +even, we have that n = n2 also satisfies the condition if and only if vP(n1 − n2) ≥ N+3 +2 . +Suppose N is a nonnegative integer. Let YN be the set of curves y2 = x3+a′ +2x2+a′ +4x+a′ +6 +with vP(a′ +2) = 1, vP(a′ +4) ≥ +� N+6 +2 +� +, and vP(a′ +6) ≥ N + 4. For E ∈ XN, let nN(E) be the +unique value of n in LP,⌊ N+4 +2 ⌋ from above. Suppose θN(E), with θN : XN → YN, is the +curve +θN(E) : y2 = (x + nN(E))3 + a2(x + nN(E))2 + a4(x + nN(E)) + a6 += x3 + a2x2 + (2nN(E)a2 + a4)x + nN(E)2a2 + nN(E)a4 + a6. +Lemma 5.5. If U is an open subset of YN, µP(θ−1 +N (U)) = Q⌊ N+4 +2 ⌋ +P +µP(U). +Proof. Suppose n ∈ LP,⌊ N+4 +2 ⌋. Let XN,n be the set of E ∈ XN with nN(E) = n and θN,n +be θN restricted to XN,n. Suppose U is an open subset of YN. Using a method similar to +the proof of Lemma 4.3, we have that +µP(θ−1 +N,n(U)) = µP(U). +Because there are Q⌊ N+4 +2 ⌋ +P +values of n, the result follows. +■ +Suppose N is a positive integer. Using Lemma 5.5, we can compute the density of the +curves E with NP(E) = 0 that have type I∗ +N and Tamagawa number 2 or 4. The Haar +measure of the curves in YN−1 that end in iteration N is +(QP −1)2 +Q +N+6+⌊ N+5 +2 ⌋ +P +. With Lemma 4.1, +we have that δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = (QP −1)2 +2QN+7 +P +. +6. Local Densities for p = 2 +6.1. Setup. Assume that the characteristic of K is p = 2. Let P be a place of K and +G(3) +P +be the set of curves +y2 + a1xy + a3y = x3 + a4x + a6 +over KP such that a1, a3, a4, a6 ∈ RP. +Note that G(3) +P +can be considered to be R4 +P. +Define ϕ : GP → G(3) +P +as the function such that if E is the curve in GP with equation +E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, ϕ(E) is the curve in G(3) +P +with equation +ϕ(E) : y2 + a1xy + +� +a3 − a1a2 +3 +� +y = x3 + +� +a4 − a2 +2 +3 +� +x + 2a3 +2 +27 − a2a4 +3 ++ a6. +Note that if E is an elliptic curve, E and ϕ(E) are equivalent. +Lemma 6.1. If U is an open subset of G(3) +P , µP(ϕ−1(U)) = µP(U). +Proof. This can be proved using a method similar to the proof of Lemma 4.1. +■ + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +17 +6.2. Multiple Iterations. Let k be a nonnegative integer. +Suppose Sk is the set of +elliptic curves E ∈ G(3) +P +such that NP(E) ≥ k. +For an elliptic curve E ∈ G(3) +P +with equation E : y2 + a1xy + a3y = x3 + a4x + a6, let +Ak(E) be the set of (l, m, n) ∈ R3 +P such that if X = x + +n +π2k +P and Y = y + +l +πk +P x + +m +π3k +P , +� +y + l +πk +P +x + m +π3k +P +�2 ++ a1 +πk +P +� +x + l2 + a1l +π2k +P +� � +y + l +πk +P +x + m +π3k +P +� ++ a3 +π3k +P +� +y + l +πk +P +x + m +π3k +P +� +− +� +x + l2 + a1l +π2k +P +�3 +− a4 +π4k +P +� +x + l2 + a1l +π2k +P +� +− a6 +π6k +P +∈ RP[x, y]. +Proposition 6.2. Let E be an elliptic curve in G(3) +P . E ∈ Sk if and only if a unique pair +(l, m) ∈ LP,k × LP,3k exists such that (l, m, l2 + a1l) ∈ Ak(E). +Proof. Suppose a unique pair (l, m) satisfying the conditions exists. Because Ak(E) is +nonempty, E ∈ Sk from Proposition 2.4. +Assume E ∈ Sk. Then, using Proposition 2.4, Ak(E) is nonempty. Let the equation of +E be E : y2 + a1xy + a3y = x3 + a4x + a6 for a1, a3, a4, a6 ∈ RP. +From replacing y with y + l′x for l′ ∈ RP, if (l, m, n) ∈ Ak(E), (l + l′πk +P , m, n) ∈ Ak(E). +Therefore, there exist l ∈ LP,k and m, n ∈ RP such that (l, m, n) ∈ Ak(E). Moreover, +if (l, m, n) ∈ Ak(E), l2 + a1l + n ≡ 0 (mod π2k +P ). +With this, from replacing x with +x+ l2+a1l+n +π2k +P +, if (l, m, n) ∈ Ak(E), (l, m+l(l2 +a1l +n), l2 +a1l) ∈ Ak(E). Therefore, there +exist l ∈ LP,k and m ∈ RP such that (l, m, l2 + a1l) ∈ Ak(E). Next, from replacing y with +y + m′ for m′ ∈ RP, there exists l ∈ LP,k and m ∈ LP,3k such that (l, m, l2 + a1l) ∈ Ak(E). +Next, we prove that (l, m) is unique. Assume that (l1, m1), (l2, m2) ∈ LP,k × LP,3k and +(l1, m1, l2 +1 + a1l1), (l2, m2, l2 +2 + a1l2) ∈ Ak(E). We prove that (l1, m1) = (l2, m2). +Let F be the curve +F : y2 + a1 +πk +P +xy + a3 +π3k +P +y = x3 + a4 +πk +P +x + a6 +π6k +P +. +For 1 ≤ i ≤ 2, let Fi be F with x replaced by x + l2 +i +a1li +π2k +P +and y replaced by y + li +πk +P x + mi +π3k +P . +Note that Fi ∈ G(3) +P +because (li, mi, l2 +i + aili) ∈ Ak(E) for 1 ≤ i ≤ 2. From this, a1 ≡ 0 +(mod πk +P). +Suppose a1 ̸= 0. We have that F1 and F2 are equivalent and vP(∆(F1)) = vP(∆(F2)). +Then, using Proposition 2.3, let τ be a translation from the equation of F1 to the equation +of F2 that replaces x with u2x + n′ and y with u3y + l′u2x + m′, where u, l′, m′, n′ ∈ RP +and vP(u) = 0. +The coefficient of xy after τ is applied to the equation of F1 is +a1 +uπk +P . However, the +coefficient of xy in F2 is a1 +πk +P . Therefore, u = 1 and a1 ≡ 0 (mod πk +P). +Next, the coefficient of y after τ is applied to the equation of F1 +a1l2 +1 + a2 +1l1 + a3 + π2k +P a1n′ +π3k +P +. +However, the coefficient of y in F2 is +a1l2 +2 + a2 +1l2 + a3 +π3k +P +. + +18 +ANDREW YAO +Therefore, +l2 +1 + a1l1 + π2k +P n′ = l2 +2 + a1l2. +Because a1 ≡ 0 (mod πk +P), we have that l1 ≡ l2 (mod πk +P). Therefore, l1 = l2. From this, +n′ = 0. +The coefficient of x2 after τ is applied to the equation of F1 is +n′ + (l′)2 + a1l′ +πk +P +. +This equals the coefficient of x2 in F2, which is 0. Because n′ = 0, we have that l′ = 0 or +l′ = a1 +πk +P . +From setting the coefficient of x after τ is applied to the equation of F1 equal to the +coefficient of x in F2, +a1 +πk +P +· +� m1 +π3k +P ++ m′ +� ++ a1(l2 +1 + a1l1) + a3 +π3k +P +· l′ = a1 +πk +P +· m2 +π3k +P +. +Suppose l′ = 0. Then m1 +π3k +P + m′ = m2 +π3k +P . It follows that m1 ≡ m2 (mod π3k +P ) and m1 = m2. +Suppose l′ = a1 +πk +P . We have that +m1 +π3k +P ++ m′ + a1(l2 +1 + a1l1) + a3 +π3k +P += m2 +π3k +P +. +However, using that the coefficient of y in F2 is an element of RP, +a1(l2 +1 + a1l1) + a3 ≡ a1(l2 +2 + a1l2) + a3 ≡ 0 +(mod π3k +P ). +Therefore, m1 ≡ m2 (mod π3k +P ) and m1 = m2. +Assume a1 = 0. From the coefficient of y in F2, we have that a3 ≡ 0 (mod π3k +P ). Also, +from the coefficients of x in F1 and F2, l4 +1 + a3l1 ≡ l4 +2 + a3l2 (mod π4k +P ). This gives that +l1 = l2. Afterwards, from the constant terms of F1 and F2, m2 +1 + a3m1 ≡ m2 +2 + a3m2 +(mod π6 +P). From this, we obtain that m1 = m2. +■ +Using Proposition 6.2, for E ∈ Sk, let the unique pair (l, m) ∈ LP,k × LP,3k such that +(l, m, l2 +a1l) ∈ Ak(E) be (l(E), m(E)). Define φk : Sk → S0 to be the function such that +if E ∈ Sk has equation E : y2 + a1xy + a3y = x3 + a4x + a6, φk(E) has equation +φk(E) : y2 + a1 +πk +P +xy + a1(l(E)2 + a1l(E)) + a3 +π3k +P += x3+ +l(E)2(l(E)2 + a1l(E)) + a1m(E) + a3l(E) + a4 +π4k +P +x+ +(a1m(E) + a4 + a2 +1l(E)2 + l(E)4)(l(E)2 + a1l(E)) + a3m(E) + a6 + m(E)2 +π6k +P +. + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +19 +The equation for φk(E) is equivalent to +� +y + l(E) +πk +P +x + m(E) +π3k +P +�2 ++ a1 +πk +P +� +x + l(E)2 + a1l(E) +π2k +P +� � +y + l(E) +πk +P +x + m(E) +π3k +P +� ++ +a3 +π3k +P +� +y + l(E) +πk +P +x + m(E) +π3k +P +� += +� +x + l(E)2 + a1l(E) +π2k +P +�3 ++ a4 +π4k +P +� +x + l(E)2 + a1l(E) +π2k +P +� ++ a6 +π6k +P +. +Note that S0 ⊂ G(3) +P , and from Proposition 2.5 and Lemma 6.1, µP(S0) = 1. For l ∈ LP,k +and m ∈ LP,3k, let Sk,l,m be the set of E ∈ Sk such that l(E) = l and m(E) = m. Assume +that φk,l,m is φk restricted to Sk,l,m. +Lemma 6.3. Suppose l ∈ LP,k and m ∈ LP,3k. If U is an open subset of G(3) +P , µP(φ−1 +k,l,m(U)) = +1 +Q14k +P µP(U). +Proof. This can be proved with a method that is similar to the proof of Lemma 5.3. +■ +Lemma 6.4. If U is an open subset of G(3) +P , µP(φ−1 +k (U)) = +1 +Q10k +P µP(U). +Proof. Let U be an open subset of G(3) +P . We have that φ−1 +k (U) = � +l∈LP,k,m∈LP,3k φ−1 +k,l,m(U). +Using Lemma 6.3, +µP(φ−1 +k (U)) = +� +l∈LP,k +� +m∈LP,3k +µP(φ−1 +k,l,m(U)) = +� +l∈LP,k +� +m∈LP,3k +1 +Q14k +P +µP(U) = +1 +Q10k +P +µP(U), +completing the proof. +■ +6.3. Density Calculations for vP(a1) = 0. Suppose that vP(a1) = 0. This case has +density QP −1 +QP . The discriminant is +a4 +1(a2 +1a6 + a1a3a4 + a2 +4) + a4 +3 + a3 +1a3 +3. +Note that by considering a6 modulo πP, the discriminant is not divisible by πP with +density (QP −1)2 +Q2 +P +. For this case, the algorithm ends in step 1. Then, we add (QP −1)2 +Q2 +P +to +δK(I0, 1, 0; P). +Assume the discriminant is divisible by πP. Let (α1, α2) be the singular point modulo +πP; it can be proven that α1, α2 ∈ RP. Also, α1 ≡ −a3 +a1 (mod πP). In step 2, replace x +by x + n and y by y + m with n = α1 and m = α2. Afterwards, the coefficient of xy is a1, +which is not divisible by πP. The algorithm then ends in step 2. +We see that the discriminant is linear in a6. Therefore, we have that vP(a1) = 0 and +vP(∆(E)) = N with density (QP −1)2 +QN+2 +P +for N ≥ 0. Note that the polynomial considered +in step 2 is T 2 + a1T + α1. +Suppose a1 ≡ r1 (mod πP) and a3 ≡ r3 (mod πP) for +r1, r3 ∈ LP,1 such that r1 ̸= 0. Given r1, T 2 + a1T + α1 is irreducible over RP/πPRP for +QP +2 +values of r3. Afterwards, using step 2 of Tate’s algorithm, we get that in this case, + +20 +ANDREW YAO +δK(I1, 1, 0; P) = (QP −1)2 +Q3 +P +, δK(I2, 2, 0; P) = (QP −1)2 +Q4 +P +, and +δK(IN, N, 0; P) = δK +� +IN, 2 +�N +2 +� +− N + 2, 0; P +� += (QP − 1)2 +2QN+2 +P +for N ≥ 3. +6.4. Density Calculations for vP(a1) ≥ 1. In this subsection, we assume that vP(a1) ≥ +1. The density for this is +1 +QP , and the discriminant modulo πP is a4 +3. +Suppose vP(a3) = 0. The density for this case is QP −1 +Q2 +P . Here, the discriminant is not +divisible by πP. Tate’s algorithm then ends in step 1, and we add QP −1 +Q2 +P +to δK(I0, 1, 0; P). +Following this, we obtain that δK(I0, 1, 0; P) = QP −1 +QP . +Next, assume vP(a3) ≥ 1. The total density for this is +1 +Q2 +P . The singular point modulo +πP is (x, y) = (α1, α2) for α1, α2 ∈ LP,1 such that a4 ≡ α2 +1 (mod πP) and a6 ≡ α2 +2 +(mod πP). We replace x with x + n and y with y + m, where n = α1 and m = α2. The +curve is +(y + m)2 + a1(x + n)(y + m) + a3(y + m) = (x + n)3 + a4(x + n) + a6. +If π2 +P does not divide mna1 +ma3 +na4+a6+m2 +n3, the algorithm ends in step 3. By +adding multiples of πP to a6, this occurs with density QP −1 +Q3 +P . We have that δK(II, 1, 0; P) = +QP −1 +Q3 +P . +Assume π2 +P divides mna1 + ma3 + na4 + a6 + m2 + n3. The total density for this case +is +1 +Q3 +P . We have that +b8 = n(na1 + a3)2 + (ma1 + a4 + n2)2. +If b8 is not divisible by π3 +P, the algorithm ends in step 4. By adding multiples of πP to a4, +we have that δK(III, 2, 0; P) = QP −1 +Q4 +P . +Assume b8 is divisible by π3 +P. The total density for this case is +1 +Q4 +P . If vP(na1 + a3) = 1, +the algorithm ends in step 5. Assume a4 ≡ 0 (mod πP). Then, replace a3 with a3 + dπP +and a4 with a4 + βdπP for β, d ∈ LP,1 such that β2 ≡ α1 (mod πP). This will not affect +previous parts of the algorithm; particularly, this will not change b8 modulo π3 +P. However, +na1 + a3 will be increased by dπP. Therefore, we have that vP(na1 + a3) = 1 with density +QP −1 +Q5 +P . From this, δK(IV, 1, 0; P) = δK(IV, 3, 0; P) = QP −1 +2Q5 +P . +Assume vP(na1 + a3) ≥ 2. +The total density for this case is +1 +Q5 +P . +Let α3 be the +element of LP,1 such that n ≡ α2 +3 (mod πP). Also, let α4 be the element of LP,1 such that +mna1 + ma3 + na4 + a6 + m2 + n3 ≡ α2 +4π2 +P (mod π3 +P). After the transformation in step 6, +the equation of the curve is +(y + lx + m)2 + a1(x + n)(y + lx + m) + a3(y + lx + m) += (x + n)3 + a4(x + n) + a6, +where n = α1, l = α3, and m = α2 + α4πP. Suppose that in step 6, the polynomial +P(T) ∈ (RP/πPRP)[T] is P(T) = T 3 + w2T 2 + w1T + w0. +Suppose a4 ≡ 0 (mod πP). Because 0 ∈ LP,1, we have that n = l = 0. This means +that w2 = 0. Then, we can replace a4 with a4 + d1π2 +P for d1 ∈ LP,1, and the previous + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +21 +parts of the algorithm will not be changed. With this, the choices for w1 modulo πP are +the elements of LP,1. Following this, from replacing a6 with a6 + d2π3 +P for d2 ∈ LP,1, the +choices for w0 modulo πP are the elements of LP,1. We have that the number of P(T) +with a double root and no roots in RP/πPRP are QP − 1 and 1, respectively. Moreover, +we have that the number of P(T) with 3 distinct roots in RP/πPRP and 0 roots, 1 root, +and 3 roots in RP/πPRP are Q2 +P −1 +3 +, Q2 +P −QP +2 +, and Q2 +P −3QP +2 +6 +, respectively. +Suppose a4 ̸≡ 0 (mod πP). Consider the translation of replacing a1 with a1 + d1πP, a3 +with a3 + α1d1πP, a4 with a4 + (α2 + α4πP)d1πP, and a6 with a6 + α1(α2 + α4πP)d1πP for +d1 ∈ LP,1. After this, the parts of the algorithm before step 6 do not change. In step 6, +w0 and w1 do not change. However, w2 increases by α3d1. Because α3 ̸= 0, the choices for +w2 are the elements of LP,1. Next, replace a6 with a6 + d2π3 +P for d2 ∈ LP,1. With this, the +choices for w0 are also the elements of LP,1. The number of P(T) with a double root and +no roots in RP/πPRP are the same as above. Also, the number of P(T) with 3 distinct +roots in RP/πPRP and 0 roots, 1 root, and 3 roots in RP/πPRP are the same as above. +Suppose P(T) has distinct roots. For this case, the total density is QP −1 +Q6 +P +and Tate’s +algorithm ends in step 6. We see that δK(I∗ +0, 1, 0; P) = Q2 +P −1 +3Q7 +P , δK(I∗ +0, 2, 0; P) = QP −1 +2Q6 +P , and +δK(I∗ +0, 4, 0; P) = +Q2 +P −3QP +2 +6Q7 +P +. +Assume that P(T) has a double root and a simple root. +For this case, the total +density is QP −1 +Q7 +P +and Tate’s algorithm ends in step 7. In Section 6.5, we compute that +δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = (QP −1)2 +2QN+7 +P +for all positive integers N. +Next, suppose P(T) has a triple root. For this case, the density is +1 +Q7 +P , and the root of +P(T) is √w1 modulo πP. If a4 ≡ 0 (mod πP), the triple root is 0 modulo πP. Let α5 be +an element of LP,1 such that +(m + ln)a1 + la3 + a4 + n2 ≡ α2 +5π2 +P +(mod π3 +P). +Then, the translation in step 8 sets n to be n = α1 + α5πP. +Suppose a4 ≡ 0 (mod πP). Replace a3 with a3 + dπ2 +P and a6 with a6 + (α2 + α4πP)dπ2 +P +for some d ∈ LP,1. Then, note that the previous parts of the algorithm, including P(T), +are unchanged. However, the coefficient of y increases by dπ2 +P. We have that for one value +of d, the coefficient of y is divisible by π3 +P. Next, suppose a4 ̸≡ 0 (mod πP). Replace a1 +with a1 + dπ2 +P and a4 with a4 + (α2 + α4πP)dπ2 +P for some d ∈ LP,1. The previous parts of +the algorithm, including P(T), are unchanged. However, the coefficient of y increases by +(α1 + α5πP)dπ2 +P. Similarly, we have that for one value of d, the coefficient of y is divisible +by π3 +P. From this, we get that the coefficient of y is not divisible by π3 +P and the algorithm +ends in step 8 with density QP −1 +Q8 +P . We then have that δK(IV ∗, 1, 0; P) = δK(IV ∗, 3, 0; P) = +QP −1 +2Q8 +P . +Assume the coefficient of y is divisible by π3 +P. The total density of this case is +1 +Q8 +P . Let +α6 be the element of LP,1 such that +mna1 + ma3 + na4 + a6 + m2 + n3 ≡ α2 +6π4 +P +(mod π5 +P). +Then, m is set to m = α2 + α4πP + α6π2 +P in step 9. If π4 +P does not divide the x coefficient +of this curve, the algorithm ends in step 9. Consider the translation of replacing a4 with + +22 +ANDREW YAO +a4+dπ3 +P and a6 with a6+(α1+α5πP)dπ3 +P for d ∈ LP,1. The previous parts of the algorithm +do not change, but the coefficient of x is increased by dπ3 +P. Therefore, π4 +P does not divide +the x coefficient with density QP −1 +Q9 +P . We have that δK(III∗, 2, 0; P) = QP −1 +Q9 +P +Assume π4 +P divides the coefficient of x of the curve. The total density for this case is +1 +Q9 +P . If π6 +P does not divide mna1 + ma3 + na4 + a6 + m2 + n3, Tate’s algorithm ends in +step 10. This occurs with density QP −1 +Q10 +P +from adding multiples of π6 +P to a6. We then have +that δK(II∗, 1, 0; P) = QP −1 +Q10 +P . +6.5. Subprocedure Density Calculations. We calculate the density of Kodaira types +r = I∗ +N for N ≥ 1 and Tamagawa numbers n = 2, 4. Note that previously, the curve was +reduced by removing a2 with a translation on x to obtain G(3) +P . However, here the density +is calculated in GP without the reduction. That is, the density is calculated for curves in +long Weierstrass form. +Let X be the set of elliptic curves E ∈ GP such that NP(E) = 0 and Tate’s algorithm +enters the step 7 subprocedure when used on E. For E ∈ X, let L(E) be the number of +iterations of the step 7 subprocedure that are completed when Tate’s algorithm is used +on E. For a nonnegative integer N, let XN be the set of E ∈ X such that L(E) ≥ N. +Suppose N is an even nonnegative integer. Assume that N = 0. In iteration N = 0, +there is a translation. Note that the double root of P(T) is the squareroot of w1. Because +of this, in step 7, we add γ0πP to n and lγ0πP to m for some γ0 ∈ LP,1 such that +(m + ln)a1 + la3 + a4 + n2 ≡ γ2 +0π2 +P +(mod π3 +P) +Next, assume N ≥ 2. Suppose iteration N of the step 7 subprocedure is reached and the +quadratic has a double root. Then, +vP((m + ln)a1 + la3 + a4 + n2) ≥ N + 6 +2 +. +Also, we add γNπ +N+2 +2 +P +to n and lγNπ +N+2 +2 +P +to m for some γN ∈ LP,1 such that +mna1 + ma3 + na4 + a6 + m2 + n3 ≡ (la1 + a2 + n + l2)γ2 +NπN+2 +P +(mod πN+4 +P +). +Note that vP(la1 + a2 + n + l2) = 1. +Suppose N is an odd nonnegative integer. Suppose iteration N of the step 7 subpro- +cedure is reached and the quadratic has a double root. Then, vP(na1 + a3) ≥ N+5 +2 . Also, +γNπ +N+3 +2 +P +is added to m for some γN ∈ LP,1 such that +mna1 + ma3 + na4 + a6 + m2 + n3 ≡ γ2 +NπN+3 +P +(mod πN+4 +P +) +Let N be a nonnegative integer. +Let YN be the set of curves y2 + a′ +1xy + a′ +3y = +x3 + a′ +2x2 + a′ +4x + a′ +6 with vP(a′ +1) ≥ 1, vP(a′ +2) = 1, vP(a′ +3) ≥ ⌊N+5 +2 ⌋, vP(a′ +4) ≥ ⌊N+6 +2 ⌋, and +vP(a′ +6) ≥ N + 4. +Suppose E ∈ XN and that the translations of Tate’s algorithm when it is used on E +are α1, α2, α3, α4, γ0, γ1, . . ., γN. Let TN(E) = (α1, α2, α3, α4, γ0, γ1, . . . , γN). Note that +because the characteristic of K is p = 2, TN(E) is well defined. Also, let θN(E) : XN → YN + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +23 +be E with x replaced by x + n and y replaced by y + lx + m, where +n = α1 + +⌊ N +2 ⌋ +� +i=0 +γ2iπi+1 +P , l = α3, m = α2 + α4πP + α3 +⌊ N +2 ⌋ +� +i=0 +γ2iπi+1 +P ++ +⌊ N−1 +2 ⌋ +� +i=0 +γ2i+1πi+2 +P . +Lemma 6.5. If U is an open subset of YN, µP(θ−1 +N (U)) = QN+5 +P +µP(U). +Proof. Let a = (α1, α2, α3, α4, γ0, γ1, . . . , γN)0≤i≤N be an element of LN+5 +P,1 . Suppose that +XN,a is the set of E ∈ XN such that TN(E) = a. Suppose that θN,a is θN restricted to +XN,a. Let U be an open subset of YN. Using a method similar to the proof of Lemma 4.3, +we have that +µP(θ−1 +N,a(U)) = µP(U). +Because there are QN+5 +P +choices of a, the result follows. +■ +Suppose N is a positive integer. With Lemma 6.5, we can compute the density for curves +that enter step 7 in the first iteration and have type I∗ +N. We have that µP(YN−1) = +QP −1 +Q2N+10 +P +, +and the Haar measure in G(3) +P +of curves that have type I∗ +N is then (QP −1)2 +QN+7 +P +. Particularly, +δK(I∗ +N, 2, 0; P) = δK(I∗ +N, 4, 0; P) = (QP −1)2 +2QN+7 +P +. +7. Local and Global Density Results +In Section 4, Section 5, and Section 6, we compute the local densities of Koidara types +and Tamagawa numbers for p ≥ 5, p = 3, and p = 2, respectively. The methods we use +involved first removing some terms from the equations of elliptic curves with translations, +and then using translations to compute the local densities. Let r be a Koidara type and n +be a positive integer. Note that δK(r, n; P) only depends on QP. Additionally, in [3], the +local densities of r and the Tamagawa number n for elliptic curves in short Weierstrass +form over Qr for primes r ≥ 5 have the same form as δK(r, n; P) for global function +fields K and P ∈ MK. In [1], the local densities of r and the Tamagawa number n for +elliptic curves in short Weierstrass form over completions of number fields at places that +lie above primes r ≥ 5 also have the same form as δK(r, n; P) for global function fields K +and P ∈ MK. +Next, we discuss some results about local and global densities, including a proof of +Theorem 1.2. Particularly, we compute the density of completing at most k ≥ 0 iterations +of Tate’s algorithm. +7.1. Proof of Theorem 1.2. Let U and V be the sets of elliptic curves E ∈ GP with +Kodaira type r and Tamagawa number n such that NP(E) = 0 and NP(E) = k, respec- +tively. Note that ϕ(U) and ϕ(V ) are the sets of curves E ∈ S0 with Kodaira type r and +Tamagawa number n such that NP(E) = 0 and NP(E) = k, respectively. +Suppose E ∈ GP and ϕ(E) ∈ ϕ(U). Then, E has Kodaira type r, Tamagawa number +n, and NP(E) = 0. This means that E ∈ U. From this, ϕ−1(ϕ(U)) ⊂ U. Moreover, +U ⊂ ϕ−1(ϕ(U)). It follows that ϕ−1(ϕ(U)) = U. Similarly, ϕ−1(ϕ(V )) = V . +We have that U and V are open sets. Moreover, ϕ(U) and ϕ(V ) are open sets. With +this, we have that µP(U) = µP(ϕ(U)) and µP(V ) = µP(ϕ(V )) for all characteristics p + +24 +ANDREW YAO +from Lemma 4.1, Lemma 5.1, and Lemma 6.1. Therefore, it suffices to prove that +µP(ϕ(V )) = +1 +Q10k +P +µP(ϕ(U)). +Suppose E ∈ ϕ(V ). We have that φk(E) has Kodaira type r, Tamagawa number n, +and NP(φk(E)) = 0. +Therefore, φk(E) ⊂ ϕ(U). +It follows that ϕ(V ) ⊂ φ−1 +k (ϕ(U)). +Next, suppose E ∈ Sk and φk(E) ∈ ϕ(U). Then, the Koidara type of E is r and the +Tamagawa number of E is n. Moreover, because NP(φk(E)) = 0, NP(E) = k. It follows +that E ∈ ϕ(V ). Therefore, φ−1 +k (ϕ(U)) ⊂ ϕ(V ). From this, φ−1 +k (ϕ(U)) = ϕ(V ). The result +then follows from Lemma 4.2, Lemma 5.4, and Lemma 6.4. +7.2. Density for Multiple Iterations. Let k be a nonnegative integer. For P ∈ MK, +let Uk +P denote the set of elliptic curves E in GP such that NP(E) ≥ k + 1. The following +proposition is important for the proof of Theorem 7.2. +Proposition 7.1. For P ∈ MK, µP(Uk +P) = +1 +Q10(k+1) +P +. +Proof. Suppose P ∈ Mk. From Lemma 4.2, Lemma 5.4, and Lemma 6.4 with k + 1 as k +and GP as U, we have that +µP(Uk +P) = +1 +Q10(k+1) +P +· µP(GP) = +1 +Q10(k+1) +P +. +This finishes the proof. +■ +Theorem 7.2. Let S be a finite nonempty subset of MK. Suppose U is the set of elliptic +curves in WS such that NP(E) ≤ k for all P ∈ SC. Then, +dS(U) = +1 +ζK(10(k + 1)) · +� +P ∈S +� +Q10(k+1) +P +Q10(k+1) +P +− 1 +� +. +Proof. For a positive integer M, let VM be the set of elliptic curves E ∈ WS such that +there exists P ∈ SC with degree at least M such that E ∈ Uk +P. From Proposition 3.3, we +have that limM→∞ dS(VM) = 0. Therefore, we can use Theorem 3.1 with Uk +P as UP for +P ∈ SC and T = {}. The result follows from Proposition 7.1. +■ +Example 7.3. We give an example of Theorem 7.2. Let K = Fq(t). Suppose P∞ is the +infinite place of Fq(t) and let S = {P∞}. Let k be a nonnegative integer and U be the set +of elliptic curves in WS such that NP(E) ≤ k for all P ∈ SC. From Theorem 5.9 of [5], +because the genus of K is 0, we have that ζK(10(k + 1)) = +q20k+19 +(q10k+9−1)(q10k+10−1). Because +P∞ has degree 1, from Theorem 7.2, dS(U) = 1 − +1 +q10k+9. + +DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +25 +References +[1] Yunseo Choi, Sean Li, Apoorva Panidapu, and Casia Siegel, Tamagawa products for elliptic curves +over number fields, arXiv, 2021. +[2] John Cremona and Mohammad Sadek, Local and global densities for Weierstrass models of elliptic +curves, arXiv, 2020. +[3] Michael Griffin, Ken Ono, and Wei-Lun Tsai, Tamagawa products of elliptic curves over Q, The +Quarterly Journal of Mathematics 72 (2021), no. 4, 1517–1543. +[4] Giacomo Micheli, A local to global principle for densities over function fields, arXiv, 2017. +[5] Michael Rosen, Number theory in function fields, 1st ed., Springer, 2002. +[6] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, 1st ed., Springer, 1994. +[7] John Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Func- +tions of One Variable IV, 1975, pp. 33–52. + diff --git a/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/load_file.txt b/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c565fcc8ef49dbc3ccd6e1fefc238a8e2838c5d4 --- /dev/null +++ b/GtFJT4oBgHgl3EQfEiwz/content/tmp_files/load_file.txt @@ -0,0 +1,1062 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf,len=1061 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='11437v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='NT] 26 Jan 2023 DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS ANDREW YAO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let K be a global function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using Haar measures, we compute the densities of the Kodiara types and Tamagawa numbers of elliptic curves over a completion of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, we prove results about the number of iterations of Tate’s algorithm that are completed when the algorithm is used on elliptic curves over a completion of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Introduction Let p be a prime and q = pn for a positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let K be a finite extension of Fq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define MK to be the set of places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose P ∈ MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let KP be the completion of K at P and RP be the valuation ring of KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E is an elliptic curve over K with equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 such that a1, a2, a3, a4, and a6 are elements of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' E has a long Weierstrass form, and if a1 = a2 = a3 = 0, E has a short Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We study densities for elliptic curves over K that have a long Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' As an elliptic curve over KP, E has a Kodaira type, which describes its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, E has a Tamagawa number cP = [E(KP) : E0(KP)] over KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' A method to determine the Kodaira type and Tamagawa number of an elliptic curve over KP is Tate’s algorithm ([6], [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The description of Tate’s algorithm in [6] is used in this paper to compute local densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Often, steps from this description of Tate’s algorithm are referred to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The papers [2] and [3] discuss densities of Kodaira types and Tamagawa products for elliptic curves over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In these papers, the densities at the nonarchimedean places of Q are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In [2] and [3], the densities are for elliptic curves in long and short Weierstrass form, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, [1] discusses densities of Kodaira types and Tamagawa products for elliptic curves over number fields in short Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that some of the methods for computing local densities with Tate’s algorithm used in Section 4, Section 5, and Section 6 of this paper are similar to methods used in [1], [2], and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Local densities over KP can be obtained using the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let N be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that KN P as an additive group is locally compact, and because of this, Haar’s theorem can be used on KN P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, suppose µP is the Haar measure on KN P such that µP(RN P ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let GP be the set of curves y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 over KP such that a1, a2, a3, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because the discriminant of an elliptic curve must be nonzero, not 1 2 ANDREW YAO all elements of GP are elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, note that GP can be considered to be R5 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The local densities for GP are obtained from the Haar measure on R5 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For an elliptic curve E ∈ GP, let NP(E) be the number of iterations of Tate’s algorithm that are completed when the algorithm is used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose T is the set of Kodaira types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let r be an element of T and n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define δK(r, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) to be the Haar measure of the set of elliptic curves E over KP with coefficients in RP such that E has Kodaira type r and the Tamagawa number of E is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For k ≥ 0, define δK(r, n, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) to be the Haar measure of the set of elliptic curves E over KP with coefficients in RP such that E has Kodaira type r, the Tamagawa number of E is n, and NP(E) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In this paper, we often consider the number of iterations that Tate’s algorithm completes when the algorithm is used on an elliptic curve over KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that in order to study this topic, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4 is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we give an important result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a Kodaira type r, positive integer n, and nonnegative integer k, δK(r, n, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = 1 Q10k P δK(r, n, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 by considering the cases p ≥ 5, p = 3, and p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the general method used to prove the theorem is to use translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The proof of this result is given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In Section 2, we introduce elliptic curves and Tate’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, in Section 3, for a nonempty finite subset S of MK and a positive integer N, we discuss how to obtain global densities for ON K,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, in Section 4, Section 5, and Section 6, we compute the local densities if the characteristic p of K is at least 5, equal to 2, and equal to 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Finally, in Section 7, we prove additional results about local and global densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose P is a place of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let the degree of P be [RP/πPRP : Fq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, let QP = |RP/πPRP|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, let πP be a uniformizer of P in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Denote vP to be the valuation vπP over KP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' note that vP is also a valuation over K because K ⊂ KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Additionally, for a nonnegative integer k, let LP,k be a set of representatives of the cosets of RP/πk PRP such that 0 ∈ LP,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose S is a finite nonempty subset of MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We let OK,S be the set of x ∈ K such that if P ∈ SC = MK\\S, vP(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, let WS be the set of curves y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 such that a1, a2, a3, a4, a6 ∈ OK,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For d ≥ 1, let Td be the number of places of P with degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The zeta function of K is ζK(s) = ∞ � d=1 � 1 − 1 qds �−Td .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose D is a divisor of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define L(D) as the set of x ∈ K such that x = 0 or x ̸= 0 and (x) + D ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This research was done in MIT SPUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The author would like to thank Hao Peng for providing useful guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, the author would like to thank Zhiyu Zhang for suggesting the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Additionally, the author would like to thank David Jerison and Ankur Moitra for giving advice about the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Elliptic Curves and Global Densities Suppose P is a place of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let E be an elliptic curve over KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' There exist a1, a2, a3, a4, a6 ∈ KP such that E has equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a1, a2, a3, a4, a6 ∈ KP satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Additionally, define b2(E) = a2 1 + 4a2, b4(E) = a1a3 + 2a4, b6(E) = a2 3 + 4a6, b8(E) = a2 1a6 + 4a2a6 − a1a3a4 + a2a2 3 − a2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, the discriminant of E is ∆(E) = −b2(E)2b8(E) − 8b4(E)3 − 27b6(E)2 + 9b2(E)b4(E)b6(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Elliptic curves E and F over KP are equivalent if there exists l, m, n, u ∈ KP such that u ̸= 0 and the equation for F can be obtained from the equation for E by first replacing x with u2x + n and y with u3y + lu2x + m and then dividing by u6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' An elliptic curve E over KP is minimal if the equation for E has coefficients in RP and if there does not exist an elliptic curve F over KP such that the equation for F has coefficients in RP, F is equivalent to E, and vP(∆(F)) < vP(∆(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The following proposition generalizes Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 of [7] to nonminimal equivalent el- liptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that this proposition is used later in the paper to compute local densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let E and F be elliptic curves over KP that have equations with co- efficients in RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that E and F are equivalent and satisfy vP(∆(E)) = vP(∆(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, there exists l, m, n, u ∈ RP such that vP(u) = 0 and the equation of F can be ob- tained from the equation of E by first replacing x with u2x+n and y with u3y +lu2x+m and then dividing by u6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 of [7] can be used to prove this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E is an elliptic curve over KP with equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 and assume that a1, a2, a3, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For l, m, n ∈ KP, let E′(l, m, n) be the elliptic curve that is E with x replaced by x + n and y replaced by y + lx + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' NP(E) ≥ k if and only if there exists l, m, n ∈ RP such that if E′(l, m, n) has equation E′(l, m, n) : y2 + a′ 1xy + a′ 3y = x3 + a′ 2x2 + a′ 4x + a′ 6, a′ i ∈ πki P RP for i ∈ {1, 2, 3, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose l, m, n exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let l, m, n satisfy the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Tate’s algorithm, we have that NP(E) = NP(E′(l, m, n)) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we prove that if NP(E) ≥ k, l, m, and n exist using induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The base case k = 0 is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let a be a nonnegative integer and assume the result is true for k = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove the result is true for k = a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume NP(E) ≥ a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because NP(E) ≥ a, 4 ANDREW YAO l, m, n ∈ RP exist such that if x is replaced with x + n and y is replaced with y + lx + m, the resulting curve E′(l, m, n) : y2+a′ 1xy+a′ 3y = x3+a′ 2x2+a′ 4x+a′ 6 has a′ i ≡ 0 (mod πia P ) for i ∈ {1, 2, 3, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose l, m, n ∈ RP satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that the curve that is obtained after Tate’s algorithm is used for a iterations on E′(l, m, n) is F : y2 + a′ 1 πa P xy + a′ 3 π3a P y = x3 + a′ 2 π2a P x2 + a′ 4 π4a P x + a′ 6 π6a P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that F is E with x replaced with π2a P x + n and y replaced with π3a P y + lπ2a P x + m divided by π6a P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because NP(E′(l, m, n)) = NP(E) ≥ a+ 1, F will complete at least one more iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' During this iteration, suppose x is replaced with x+n′ and y is replaced with y +l′x+m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that the resulting elliptic curve F ′ : y2 + a′′ 1xy + a′′ 3y = x3 + a′′ 2x2 + a′′ 4x + a′′ 6 has a′′ i ≡ 0 (mod πi P) for i ∈ {1, 2, 3, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, F ′ is E with x replaced with π2a P x + n + n′π2a P and y replaced with π3a P y + (l + l′πa P)π2a P x + m + m′π3a P + ln′π2a P divided by π6a P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The equation of E′(l + l′πa P, m + m′π3a P + ln′π2a P , n + n′π2a P ) is y2 + πa Pa′′ 1xy + π3a P a′′ 3y = x3 + π2a P a′′ 2x2 + π4a P a′′ 4x + π6a P a′′ 6, and πai P a′′ i ∈ π(a+1)i P RP for i ∈ {1, 2, 3, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Note that Tate’s algorithm cannot be used on a curve in GP with discriminant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, this is not considered in the calculations of local densities later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose r ∈ T, n is a positive integer, and k is a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The set U of elliptic curves E ∈ GP with Kodaira type r, Tamagawa number n, and M(E) = k is an open subset of GP, because if E ∈ U, if multiples of πM P are added to the coefficients of E for sufficiently positive large integers M, the resulting curve will be an element of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, the set of elliptic curves is an open subset of GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In the next proposition, we prove that the Haar measure of this set is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' note that it follows that the Haar measure of the set of curves in GP with discriminant 0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The Haar measure of the set of elliptic curves is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let M be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E : y2 +a1xy +a3y = x3 +a2x2 +a4x+a6, we see that the number of solutions for ai, i ∈ {1, 2, 3, 4, 6} modulo πM P to ∆(E) ≡ 0 (mod πM P ) is O(Q4M P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, the Haar measure of the set of elliptic curves with discriminant equal to 0 is at most O(Q4M P ) Q5M P = O( 1 QM P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The result follows from taking M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Global Densities Next, global densities are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Definitions and theorems from [4] are used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let S be a finite nonempty subset of MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, suppose N is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let Div(S) be the set of divisors � P ∈S nPP such that for P ∈ S, nP is a nonnegative integer, and there exists P ∈ S such that nP > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose U ⊂ ON K,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The upper density of U at S is dS(U) = lim sup D∈Div(S) |U ∩ L(D)N| |L(D)|N , and the lower density of U at S is dS(U) = lim inf D∈Div(S) |U ∩ L(D)N| |L(D)|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If dS(U) = dS(U), the density dS(U) of U at S exists, and equals dS(U) = dS(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1 ([4], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For P ∈ SC, let UP ⊂ KN P be a measurable set such that µP(∂UP ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a positive integer M, let VM be the set of x ∈ ON K,S such that x ∈ UP for some P ∈ SC with degree at least M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose limM→∞ dS(VM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let P : ON K,S → 2SC, P(a) = {P ∈ SC : a ∈ UP}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then: (1) � P ∈SC µP(UP) is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' (2) For T ⊂ 2SC, ν(T) := dS(P−1(T)) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, ν defines a measure on 2SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' (3) ν is concentrated at finite subsets of SC, and for a finite set T of places in SC, ν(T) = � P ∈T µP(UP) � P ∈SC\\T (1 − µP(UP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 ([4], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let f and g be polynomials in OK,S[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' , xd] that are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For M ≥ 1, let VM be the set of x ∈ ON K,S such that f(x) ≡ g(x) ≡ 0 (mod πP) for some P ∈ SC with degree at least M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, limM→∞ dS(VM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In this paper, we consider global densities for elliptic curves over K with coefficients in OK,S in long Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that WS can be considered to be O5 K,S, and particularly, the global density definitions from above for O5 K,S can be used on WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similar methods are used in [2] for elliptic curves over Q with coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that an elliptic curve must have a nonzero discriminant, meaning that not all curves in WS are elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, for D ∈ Div(S), the number of curves in WS with discriminant 0 that are elements of L(D)5, where WS is considered to be O5 K,S, is O(|L(D)|4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, if proportions over elliptic curves in WS is considered rather than the proportions over WS, the density is not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3 is about the global density of nonminimal elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the lemma is used to prove Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 6 ANDREW YAO Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a positive integer M, let VM be the set of elliptic curves E ∈ WS such that there exists P ∈ SC with degree at least M such that NP(E) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, limM→∞ dS(VM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove this with casework on the characteristic p of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that E is an elliptic curve in GP with equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 for a1, a2, a3, a4, a6 ∈ RP such that NP(E) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume p ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that E can be translated to the curve y2 = x3 + � −b2(E)2 48 + b4(E) 2 � x − b2(E)3 864 − b2(E)b4(E) 24 + b6(E) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because NP(E) ≥ 1, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, −b2(E)2 48 + b4(E) 2 ≡ 0 (mod πP) and −b2(E)3 864 − b2(E)b4(E) 24 + b6(E) 4 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 with f(x1, x2, x3, x4, x6) = −(x2 1 + 4x2)2 48 + x1x3 + 2x4 2 and g(x1, x2, x3, x4, x6) = −(x2 1 + 4x2)3 864 − (x2 1 + 4x2)(x1x3 + 2x4) 24 + x2 3 + 4x6 4 proves this proposition for p ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that E can be translated to the curve y2 = x3 + b2(E) 4 x2 + b4(E) 2 x + b6(E) 4 Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, b2(E) 4 ≡ 0 (mod πP) from the coefficient of x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Additionally, ∆(E) ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 with f(x1, x2, x3, x4, x6) = −(x2 1 + x2)2(x2 1x6 + x2x6 − x1x3x4 + x2x2 3 − x2 4) + (x1x3 + 2x4)3 and g(x1, x2, x3, x4, x6) = x2 1 + x2 proves this proposition for p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, a1 ≡ 0 (mod πP) from the coefficient of xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, ∆(E) ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2 with f(x1, x2, x3, x4, x6) = x4 1(x2 1x6 + x1x3x4 + x2x2 3 + x2 4) + x4 3 + x3 1x3 3 and g(x1, x2, x3, x4, x6) = x1 proves this proposition for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Local Densities for p ≥ 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that the characteristic of K is p ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let P be a place of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We compute the local densities over KP of Kodaira types r and Tamagawa numbers n for elliptic curves in GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let G(1) P be the set of curves y2 = x3 + a4x + a6 DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 7 over KP such that a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that G(1) P can be considered to be R2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define ϕ : GP → G(1) P as the function such that if E is a curve in GP, ϕ(E) is the curve in G(1) P with equation ϕ(E) : y2 = x3 + � −b2(E)2 48 + b4(E) 2 � x − b2(E)3 864 − b2(E)b4(E) 24 + b6(E) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If E is an elliptic curve, ϕ(E) is an elliptic curve equivalent to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(1) P , µP(ϕ−1(U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let V be the set of y2 = x3 + a′ 4x + a′ 6 with a′ 4 ∈ r4 + πn4 P RP and a′ 6 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It suffices to prove that µP(ϕ−1(V )) = µP(V ) = 1 Qn4+n6 because all open subsets of G(1) P can be written as a disjoint countable union of sets with the form of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6 ∈ GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ϕ(E) ∈ V if and only if −b2(E)2 48 + b4(E) 2 ∈ r4 + πn4 P RP and −b2(E)3 864 − b2(E)b4(E) 24 + b6(E) 4 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that ϕ(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let M = max(n4, n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' First, select a1, a2, and a3 modulo πM P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Each has QM P possible residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, a4 will have QM−n4 P residues modulo πM P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' select the residue for a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Finally, a6 has QM−n6 P residues modulo πM P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' select the residue for a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that if each of a1, a2, a3, a4, a6 are taken modulo πM P , the number of combinations of residues is Q5M−n4−n6 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, because ai is modulo πM P for i ∈ {1, 2, 3, 4, 6}, each combination of residues has a Haar measure of 1 Q5M P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Multiple Iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose Sk is the set of elliptic curves E ∈ G(1) P such that NP(E) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E is an elliptic curve in G(1) P with equation E : y2 = x3 + a4x + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume E ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, l, m, n ∈ RP exist such that � y + l πk P x + m π3k P �2 − � x + n π2k P �3 − a4 π4k P � x + n π2k P � − a6 π6k P ∈ RP[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The coefficient of xy is 2l πk P , giving that vP(l) ≥ k, and the coefficient of y is 2m π3k P , giving that vP(m) ≥ 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, the coefficient of x2 is 3n−l2 π2k P , giving that vP(n) ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, we have that vP(a4) ≥ 4k and vP(a6) ≥ 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define the function φk : Sk → S0, y2 = x3 + a4x + a6 �→ y2 = x3 + a4 π4k P x + a6 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that Sk ⊂ S0 ⊂ G(1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, µP(S0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we show how we can use φk to compute densities for Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(1) P , µP(φ−1 k (U)) = 1 Q10k P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 8 ANDREW YAO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose r4, r6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, suppose n4 and n6 are nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let V be the set of elliptic curves y2 = x3 + a′ 4x + a′ 6 with a′ 4 ∈ r4 + πn4 P RP and a′ 6 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because µP(S0) = 1, µP(V ) = 1 Qn4+n6 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' To prove the lemma, it suffices to prove that µP(φ−1 k (V )) = 1 Q10k P µP(V ) = 1 Qn4+n6+10k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E : y2 = x3 + a4x + a6 ∈ G(1) P is an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove that E ∈ Sk and φk(E) ∈ V if and only if a4 π4k P ∈ r4 + πn4 P RP and a6 π6k P ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If φk(E) ∈ V , then a4 π4k P ∈ r4 + πn4 P RP and a6 π6k P ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that a4 π4k P ∈ r4 + πn4 P RP and a6 π6k P ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Tate’s algorithm, we have that E ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, it is true that φk(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that E ∈ Sk and φk(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This is true if and only if a4 ∈ π4k P r4 + πn4+4k P R and a6 ∈ π6k P r6 + πn6+6k P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, because µP(S0) = 1, the density of curves y2 = x3 + a4x + a6 with discriminant 0 such that a4 ∈ π4k P r4 + πn4+4k P and a6 ∈ π6k P r6 + πn6+6k P is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because of this, µP(φ−1 k (V )) = 1 Qn4+n6+10k P , completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the density of a set of curves in G(1) P is the Haar measure of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In this subsection, we compute the density of the set of minimal elliptic curves with a given Kodaira type and Tamagawa number over G(1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This can be extended to nonminimal elliptic curves using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, in this subsection, we use that the set of curves in G(1) P that have a discriminant equal to 0 has a Haar measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose the discriminant is not divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We compute the density for this set by considering a4 and a6 modulo πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a4 ∈ r4 + πPRP and a6 ∈ r6 + πPRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We find the number of pairs (r4, r6) in L2 P,1 such that � r4 3 �3 + � r6 2 �2 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If r4 = 0, r6 has 1 choice, and if −r4 3 is a square modulo πP, r6 has 2 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Otherwise, r6 has 0 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that the number of pairs (r4, r6) is QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, where each pair (r4, r6) has a density of 1 Q2 P , the density of the discriminant not being divisible by πP is QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, Tate’s algorithm ends in step 1 and we get that δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume that the discriminant is divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Furthermore, assume that a4, a6 ̸≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because there are QP − 1 pairs (r4, r6) in L2 P,1 for this case, the total density is QP −1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let α be the element of LP,1 such that a4 ≡ −3α2 (mod πP) and a6 ≡ 2α3 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The singular point is (α, 0) and in step 2, x is replaced with x + n where n = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because α ̸≡ 0 (mod πP), Tate’s algorithm ends in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The quadratic considered in step 2 is T 2 − 3α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that for QP −1 2 values of α, this quadratic has roots in RP/πPRP and c = vP(∆(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Otherwise, c = 1 if vP(∆(E)) is odd and c = 2 if vP(∆(E)) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let N be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a4 ∈ r4 + πN P RP and a6 ∈ r6 + πN P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We find the number of pairs (r4, r6) in L2 P,1 such that � r4 3 �3 + �r6 2 �2 ≡ 0 (mod πN P ) and r4, r6 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because there are QN P −QN−1 P 2 nonzero residues that are squares modulo πM P , we have that DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 9 the number of pairs (r4, r6) is QN P − QN−1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, the density of vP(∆(E)) ≥ N for a4, a6 ̸≡ 0 (mod πP) is QP −1 QN+1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density of vP(∆(E)) = N is QP −1 QN+1 P − QP −1 QN+2 P = (QP −1)2 QN+2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We therefore have that δK(I1, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q3 P , δK(I2, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q4 P , and δK(IN, N, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK � IN, 2 �N 2 � − N + 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P � = (QP − 1)2 2QN+2 P for N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a4), vP(a6) ≥ 1, the singular point modulo πP from step 2 of Tate’s algorithm is (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a6) = 1, the algorithm ends in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this, we get δK(II, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that vP(a6) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a4) = 1, the algorithm ends in step 4, and we get that δK(III, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose vP(a4) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a6) = 2, the algorithm ends in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, we have that δK(IV, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(a6) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In step 6, the polynomial P(T) ∈ (RP/πPRP)[T] has coefficient of T 2 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From adding multiples of π2 P to a4, the choices for the coefficient of T are LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, from adding multiples of π3 P to a6, the choices for the constant term are LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, we have that each polynomial P(T) ∈ (RP/πPRP)[T] with coefficient of T 2 equal to 0 corresponds to a density of 1 Q7 P in G(1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume P(T) has distinct roots in RP/πPRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total number of P(T) for this case is Q2 P −QP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' therefore, the total density for this case is QP −1 Q6 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that Tate’s algorithm ends in step 6 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The number of P(T) with 0, 1, and 3 roots in RP/πPRP is Q2 P −1 3 , Q2 P −QP 2 , and Q2 P −3QP +2 6 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, δK(I∗ 0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −1 3Q7 P , δK(I∗ 0, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q6 P , and δK(I∗ 0, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −3QP +2 6Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume that P(T) has a double root and a simple root in RP/πPRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, Tate’s algorithm enters the subprocedure in step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the total number of P(T) is QP − 1 and the total density is therefore QP −1 Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, we compute that δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 2QN+7 P for all positive integers N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume P(T) has a triple root in RP/πPRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the total number of P(T) is 1 and the total density is therefore 1 Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because the coefficient of T 2 in P(T) is 0, the triple root is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a6) = 4, the algorithm ends in step 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, δK(IV ∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV ∗, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume that vP(a6) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a4) = 3, the algorithm ends in step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that δK(III∗, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 10 ANDREW YAO Suppose vP(a4) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(a6) = 5, the algorithm ends in step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, δK(II∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q10 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With density 1 Q10 P , we have that vP(a4) ≥ 4 and vP(a6) ≥ 6, meaning that the curve is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' That is, the curve will complete iteration 1 and continue iteration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the density of nonminimal curves calculated from the algorithm matches Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Subprocedure Density Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we study the densities for the sub- procedure in step 7 of Tate’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We compute the subprocedure densities by studying the translation of x in Tate’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In the step 7 subprocedure, because the coefficient of y is initially 0, there will be no translations of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let X be the set of elliptic curves E ∈ G(1) P such that NP(E) = 0 and Tate’s algorithm enters the step 7 subprocedure when used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E ∈ X, let L(E) be the number of iterations of the step 7 subprocedure that are completed when Tate’s algorithm is used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a nonnegative integer N, let XN be the set of E ∈ X such that L(E) ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is an even nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Iteration N of the step 7 subprocedure is completed if and only if n ∈ RP exists such that vP(n) = 1, vP(a4 + 3n2) ≥ N+6 2 , and vP(n3 + 3na4 + a6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n = n1 satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n = n2 also satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that n2 1 ≡ n2 2 (mod π N+6 2 P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This gives that n1 is equivalent to n2 or −n2 modulo π N+4 2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, because n3 1 + n1a4 ≡ n3 2 + n2a4 (mod πN+4 P ), we have that vP(n1 − n2) ≥ N+4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, if vP(n1 − n2) ≥ N+4 2 , n = n2 also satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose N is an odd nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Iteration N of the subprocedure is completed if and only if n ∈ RP exists such that vP(n) = 1, vP(a4 + 3n2 1) ≥ N+5 2 , and vP(n3 + na4 + a6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similarly, we have that if n = n1 satisfies the condition, n = n2 satisfies the condition if and only if vP(n1 − n2) ≥ N+3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n is an element of LP,⌊ N+4 2 ⌋ such that vP(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let Yn,N be the set of curves x3 + 3nx2 + a′ 4x + a′ 6 such that vP(a′ 4) ≥ � N+6 2 � and vP(a′ 6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that Yn,N can be considered to be an open subset of R2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E ∈ XN, let nN(E) be the unique value of n ∈ LP,⌊ N+4 2 ⌋ such that vP(n) = 1, vP(a4 + 3n2) ≥ � N+6 2 � , and vP(n3 + na4 + a6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let θN be the function such that if E : y2 = x3 + a4x + a6 is an element of XN, θN(E) : y2 = (x + nN(E))3 + a4(x + nN(E)) + a6 = x3 + 3nN(E)x2 + (a4 + 3nN(E)2)x + nN(E)a4 + a6 + nN(E)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of Yn,N, µP(θ−1 N (U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose r4, r6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, suppose n4 and n6 are nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that vP(r4), n4 ≥ ⌊N+4 2 ⌋ and vP(r6), n6 ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let V ⊂ Yn,N be the set of E′ : y2 = x3 + 3nx2 + a′ 4x + a′ 6 such that a′ 4 ∈ r4 + πn4 P RP and a′ 6 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It suffices to prove that µP(θ−1 N (V )) = µP(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E : y2 = x3 + a4x + a6 is an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove that that E ∈ XN and θN(E) ∈ V if and only if a4 + 3n2 ∈ r4 + πn4 P RP, na4 + a6 + n3 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 11 Assume that E ∈ XN and θN(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because θN(E) ∈ V , we have that nN(E) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, a4 + 3n2 ∈ r4 + πn4 P RP and na4 + a6 + n3 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume that a4 + 3n2 ∈ r4 + πn4 P and na4 + a6 + n3 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because vP(a4 + 3n2) ≥ � N+6 2 � and vP(na4 + a6 + n3) ≥ N + 4, E ∈ XN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that θN(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let M = max(n4, n6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Modulo πM P , there are QM−n4 P choices for the residue of a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' After choosing a4 modulo πM P , there are QM−n6 P choices for the residue of a6 modulo πM P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Each of these combinations of residues modulo πM P for a4 and a6 has a density of 1 Q2M P in G(1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The Haar measure of the Q2M−n4−n6 P combinations is 1 Qn4+n6 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because the set of curves in G(1) P with discriminant 0 has a Haar measure of 0, µP(θ−1 N (V )) = 1 Qn4+n6 P = µP(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Let N be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We compute the density of I∗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let n be an element of LP,⌊ N+3 2 ⌋ such that vP(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that the Haar measure of the set of E ∈ Yn,N−1 that do not complete iteration N is QP −1 Q⌊ N+5 2 ⌋+N+4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, because there are (QP − 1)Q⌊ N−1 2 ⌋ P values of n, the density of I∗ N is (QP −1)2 QN+7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From adding multiples of πN+4 P to a6, c = 2 and c = 4 have equal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP − 1)2 2QN+7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Local Densities for p = 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that the characteristic of K is p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let P be a place of K and G(2) P be the set of curves y2 = x3 + a2x2 + a4x + a6 over KP such that a2, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that G(2) P can be considered to be R3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define ϕ : GP → G(2) P as the function such that if E is a curve in GP, ϕ(E) is the curve in G(2) P with equation y2 = x3 + b2(E) 4 x2 + b4(E) 2 x + b6(E) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that if E is an elliptic curve, E and ϕ(E) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(2) P , µP(ϕ−1(U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This can be proved using a method similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Multiple Iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose Sk is the set of elliptic curves E ∈ G(2) P such that NP(E) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E ∈ Sk has equation E : y2 = x3 + a2x2 + a4x + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, l, m, n ∈ RP exist such that � y + l πk P x + m π3k P �2 = � x + n π2k P �3 + a2 π2k P � x + n π2k P �2 + a4 π4k P � x + n π2k P � + a6 π6k P 12 ANDREW YAO has coefficients in RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From the coefficient of xy, vP(l) ≥ k, and from the coefficient of y, vP(m) ≥ 3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, we have that y2 = � x + n π2k P �3 + a2 π2k P � x + n π2k P �2 + a4 π4k P � x + n π2k P � + a6 π6k P has coefficients in RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that vP(a2) ≥ 2k also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For an elliptic curve E ∈ G(2) P with equation E : y2 = x3 + a2x2 + a4x + a6, let Ak(E) be the set of n ∈ RP such that y2 = x3 + a2 π2k P x2 + 2na2 + a4 π4k P x + n2a2 + na4 + a6 + n3 π6k P has coefficients in RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The next proposition is useful for computing local densities for multiple iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let E be an elliptic curve in G(2) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' E ∈ Sk if and only if a unique element n ∈ LP,k exists such that n ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume a unique element n ∈ LP,k exists such that n ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, Ak(E) is nonempty, and using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, E ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume E ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, we have that Ak(E) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let the equation of E be E : y2 = x3 + a2x2 + a4x + a6 for a2, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From replacing x with x+n′ for n′ ∈ RP, we have that n+n′π2k P ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, n ∈ LP,k exists such that n ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we prove uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume n1, n2 ∈ Ak(E) ∩ LP,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let F : y2 = x3 + a2 π2k P x2 + a4 π4k P x + a6 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For 1 ≤ i ≤ 2, let Fi be F with x replaced by x + ni π2k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that F1, F2 ∈ G(2) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From the coefficients of x in F1 and F2, 2n1a2 + a4 ≡ 2n2a2 + a4 ≡ 0 (mod π4k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, from the constant terms of F1 and F2, n2 1a2 + n1a4 + n3 1 ≡ n2 2a2 + n2a4 + n3 2 (mod π6k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For the sake of contradiction, assume that vP(n1 − n2) < 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let a = vP(n1 − n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that vP(n3 1 − n3 2) = vP((n1 − n2)3) = 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that n2 1a2 + n1a4 − n2 2a2 − n2a4 = (n1 − n2)(n1a2 + n2a2 + a4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because a4 ≡ n1a2 ≡ n2a2 (mod π4k P ), n1a2 + n2a2 + a4 ≡ 3a4 ≡ 0 (mod π4k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, vP(n2 1a2 + n1a4 − n2 2a2 − n2a4) = vP((n1 − n2)(n1a2 + n2a2 + a4)) ≥ a + 4k > 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 13 Since vP(n3 1 − n3 2) = 3a, vP(n2 1a2 + n1a4 + n3 1 − n2 2a2 − n2a4 − n3 2) = 3a < 6k, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, vP(n1 − n2) ≥ 2k and n1 = n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2, for E ∈ Sk, let n(E) be the unique n ∈ LP,2k such that the n ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define φk : Sk → S0 to be the function such that if E ∈ Sk has equation E : y2 = x3 + a2x2 + a4x + a6, φk(E) ∈ S0 has equation φk(E) : y2 = x3 + a2 π2k P x2 + 2n(E)a2 + a4 π4k P x + n(E)2a2 + n(E)a4 + a6 + n(E)3 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that Sk ⊂ S0 ⊂ G(2) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, µP(S0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For n ∈ LP,2k, suppose Sk,n is the set of E ∈ Sk such that n(E) = n, and let φk,n be φk restricted to Sk,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n ∈ LP,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(2) P , µP(φ−1 k,n(U)) = 1 Q12k P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose r2, r4, r6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, suppose n2, n4, and n6 are nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let V be the set of y2 = x3 + a′ 2x2 + a′ 4x+ a′ 6 such that a′ 2 ∈ r2 + πn2 P RP, a′ 4 ∈ r4 + πn4 P RP, and a′ 6 ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E : y2 = x3 + a2x2 + a4x + a6 ∈ G(2) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' E ∈ Sk,n and φk,n(E) ∈ V if and only if a2 π2k P ∈ r2 + πn2 P RP, 2na2 + a4 π4k P ∈ r4 + πn4 P RP, n2a2 + na4 + a6 + n3 π6k P ∈ r6 + πn6 P RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that E ∈ Sk,n and φk,n(E) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let M = max(n2+2k, n4+4k, n6+6k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' There are QM−n2−2k P ways to pick a2 modulo πM P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, a4 will have QM−n4−4k P choices for the residue modulo πM P ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' pick a4 modulo πM P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, a6 has QM−n6−6k P choices for the residue modulo πM P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Select the residue for a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The number of combinations of residues is Q3M−n2−n4−n6−12k P and each combination of residues has a Haar measure of Q−3M P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, because µP(S0) = 1, the set of curves with discriminant 0 counted in these combinations of residues has a Haar measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, µP(φ−1 k,n(V )) = 1 Qn2+n4+n6+12k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, µP(φ−1 k,n(U)) = 1 Q12k P µP(U) for all open subsets U of G(2) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(2) P , µP(φ−1 k (U)) = 1 Q10k P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let U be an open subset of G(2) P We have that φ−1 k (U) = � n∈LP,2k φ−1 k,n(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, µP(φ−1 k (U)) = � n∈LP,2k µP(φ−1 k,n(U)) = � n∈LP,2k 1 Q12k P µP(U) = 1 Q10k P µP(U), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 14 ANDREW YAO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density Calculations for vP(a2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(a2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density for this case over G(2) P is QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The discriminant is −a3 2a6 + a2 2a2 4 − a3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From adding multiples of πP to a6, the set of curves with discriminant not divisible by πP has density (QP −1)2 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, we add (QP −1)2 Q2 P to δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume the discriminant is divisible by πP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The algorithm ends in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because vP(a2) = 0, the coefficient of a6 in the discriminant is not divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, we see that for N ≥ 0, the density over G(2) P of curves such that vP(a2) = 0 and vP(∆(E)) = N is (QP −1)2 QN+2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If a2 ≡ r2 (mod πP) for r2 ∈ LP,1 such that r2 ̸= 0, T 2 + a2 is irreducible over RP/πPRP for QP −1 2 values of r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using step 2 of Tate’s algorithm, we have that δK(I1, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q3 P , δK(I2, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q4 P , and δK(IN, N, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK � IN, 2 �N 2 � − N + 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P � = (QP − 1)2 2QN+2 P for N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density Calculations for vP(a2) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose vP(a2) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density for this is 1 QP and modulo πP, the discriminant is −a3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume the discriminant is not divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This occurs if and only if a4 is not divisible by πP, and the density of this case is QP −1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Adding this density to δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) gives that δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume the discriminant is divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for these cases will be 1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose α1 is an element of LP,1 such that a6 + α3 1 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' A singular point is (α1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that x is replaced with x + n where n = α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The resulting curve has equation y2 = (x + n)3 + a2(x + n)2 + a4(x + n) + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that n2a2 + na4 + a6 + n3 is not divisible by π2 P with density QP −1 Q3 P by adding multiples of πP to a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, δK(II, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume n2a2 + na4 + a6 + n3 is divisible by π2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density of vP(2na2 + a4) = 1 is QP −1 Q4 P from replacing a4 with a4 + πPd and a6 with a6 − α1πPd for d ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(2na2 + a4) = 1, the algorithm ends in step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that δK(III, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume 2na2 + a4 is divisible by π2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that vP(n2a2 + na4 + a6 + n3) = 2 with density QP −1 Q5 P from adding multiples of π2 P to a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If this is true, the algorithm ends in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, we have that δK(IV, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(n2a2 + na4 + a6 + n3) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In step 6, there is no translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a2 is replaced by a2 + d1πP, a4 is replaced with a4 − 2α1d1πP, and a6 is replaced with a6 + α2 1d1πP for d1 ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the previous parts of the algorithm will not be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, this changes the coefficient of x2 from a2 to a2 + d1πP , which changes the coefficient of T 2 of P(T) in step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, replace DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 15 a4 with a4 + d2π2 P and a6 with a6 − α1d2π2 P for d2 ∈ πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similarly, this does not change the previous parts of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, d2π2 P will be added to the coefficient of x, which adds d2 to the coefficient of T of P(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, replace a6 with a6 + d3π3 P for d3 ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This adds d3 to the constant term P(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, the choices for P(T) are the monic polynomials with degree 3 in (RP/πPRP)[T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' each choice for P(T) corresponds to a density of 1 Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, the number of P(T) with a double root and triple root are QP(QP − 1) and QP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume P(T) has distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that the algorithm ends in step 6, with δK(I∗ 0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −1 3Q7 P , δK(I∗ 0, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q6 P , and δK(I∗ 0, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −3QP +2 6Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume P(T) has a double root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, Tate’s algorithm ends in step 7 and the total density is QP −1 Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5, we compute that δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 2QN+7 P for all positive integers N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume P(T) has a triple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density for this case is 1 Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let α2 be the element of LP,1 such that n2a2 + na4 + a6 + n3 ≡ −π3 P α3 2 (mod π4 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, for the translation in step 8, we let n = α1+α2πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(n2a2+na4+a6+n3) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This occurs with density QP −1 Q8 P by adding multiples of π4 P to a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In this case, Tate’s algorithm ends in step 8, and δK(IV ∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV ∗, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume vP(n2a2 + na4 + a6 + n3) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Consider replacing a4 with a4 + dπ3 P and a6 with a6 − (α1 + α2πP)dπ3 P for d ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This does not change previous parts of the algorithm but adds dπ3 P to the coefficient of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, vP(2na2 + a4) = 3 with density QP −1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this, we have that Tate’s algorithm ends in step 9 and δK(III∗, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(2na2+a4) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density of this case is 1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From adding multiples of π6 P to a6, vP(n3 +a2n2 +a4n+a6) = 5 with density QP −1 Q10 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, if vP(n3 +a2n2 +a4n+ a6) = 5, the algorithm ends in step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This gives that δK(II∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q10 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Subprocedure Density Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let X be the set of elliptic curves E ∈ G(2) P such that NP(E) = 0 and Tate’s algorithm enters the step 7 subprocedure when used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E ∈ X, let L(E) be the number of iterations of the step 7 subprocedure that are completed when Tate’s algorithm is used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a nonnegative integer N, let XN be the set of E ∈ X such that L(E) ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is an even nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Iteration N of the step 7 subprocedure is completed if and only if n ∈ RP exists such that vP(a2) = 1, vP(2na2 + a4) ≥ N+6 2 , and vP(n3 + n2a2 + na4 + a6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume n = n1 satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n = n2 satisfies the condition also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because vP(a2) = 1, vP(n1 − n2) ≥ N+4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume vP(n1 − n2) ≥ N+4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We show that n = n2 also satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Clearly, vP(2n2a2 + a4) ≥ N+6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, we have that n2 2a2 + n2a4 = n2 1a2 + n1a4 + 1 2(n2 − n1)((2n1a2 + a4) + (2n2a2 + a4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 16 ANDREW YAO Therefore, vP(n3 2 +n2 2a2 +n2a4 +a6) ≥ N +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that n = n2 satisfies the condition if and only if vP(n1 − n2) ≥ N+4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose N is an odd nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Iteration N of the step 7 subprocedure is completed if and only if n ∈ RP exists such that vP(n2a2 + na4 + a6 + n3) ≥ N + 4 and vP(2na2 + a4) ≥ N+5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume n = n1 satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similarly to when N is even, we have that n = n2 also satisfies the condition if and only if vP(n1 − n2) ≥ N+3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let YN be the set of curves y2 = x3+a′ 2x2+a′ 4x+a′ 6 with vP(a′ 2) = 1, vP(a′ 4) ≥ � N+6 2 � , and vP(a′ 6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E ∈ XN, let nN(E) be the unique value of n in LP,⌊ N+4 2 ⌋ from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose θN(E), with θN : XN → YN, is the curve θN(E) : y2 = (x + nN(E))3 + a2(x + nN(E))2 + a4(x + nN(E)) + a6 = x3 + a2x2 + (2nN(E)a2 + a4)x + nN(E)2a2 + nN(E)a4 + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of YN, µP(θ−1 N (U)) = Q⌊ N+4 2 ⌋ P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose n ∈ LP,⌊ N+4 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let XN,n be the set of E ∈ XN with nN(E) = n and θN,n be θN restricted to XN,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose U is an open subset of YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using a method similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, we have that µP(θ−1 N,n(U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because there are Q⌊ N+4 2 ⌋ P values of n, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Suppose N is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5, we can compute the density of the curves E with NP(E) = 0 that have type I∗ N and Tamagawa number 2 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The Haar measure of the curves in YN−1 that end in iteration N is (QP −1)2 Q N+6+⌊ N+5 2 ⌋ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, we have that δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 2QN+7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Local Densities for p = 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that the characteristic of K is p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let P be a place of K and G(3) P be the set of curves y2 + a1xy + a3y = x3 + a4x + a6 over KP such that a1, a3, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that G(3) P can be considered to be R4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define ϕ : GP → G(3) P as the function such that if E is the curve in GP with equation E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, ϕ(E) is the curve in G(3) P with equation ϕ(E) : y2 + a1xy + � a3 − a1a2 3 � y = x3 + � a4 − a2 2 3 � x + 2a3 2 27 − a2a4 3 + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that if E is an elliptic curve, E and ϕ(E) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(3) P , µP(ϕ−1(U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This can be proved using a method similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Multiple Iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose Sk is the set of elliptic curves E ∈ G(3) P such that NP(E) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For an elliptic curve E ∈ G(3) P with equation E : y2 + a1xy + a3y = x3 + a4x + a6, let Ak(E) be the set of (l, m, n) ∈ R3 P such that if X = x + n π2k P and Y = y + l πk P x + m π3k P , � y + l πk P x + m π3k P �2 + a1 πk P � x + l2 + a1l π2k P � � y + l πk P x + m π3k P � + a3 π3k P � y + l πk P x + m π3k P � − � x + l2 + a1l π2k P �3 − a4 π4k P � x + l2 + a1l π2k P � − a6 π6k P ∈ RP[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let E be an elliptic curve in G(3) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' E ∈ Sk if and only if a unique pair (l, m) ∈ LP,k × LP,3k exists such that (l, m, l2 + a1l) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a unique pair (l, m) satisfying the conditions exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because Ak(E) is nonempty, E ∈ Sk from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume E ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, Ak(E) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let the equation of E be E : y2 + a1xy + a3y = x3 + a4x + a6 for a1, a3, a4, a6 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From replacing y with y + l′x for l′ ∈ RP, if (l, m, n) ∈ Ak(E), (l + l′πk P , m, n) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, there exist l ∈ LP,k and m, n ∈ RP such that (l, m, n) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, if (l, m, n) ∈ Ak(E), l2 + a1l + n ≡ 0 (mod π2k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, from replacing x with x+ l2+a1l+n π2k P , if (l, m, n) ∈ Ak(E), (l, m+l(l2 +a1l +n), l2 +a1l) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, there exist l ∈ LP,k and m ∈ RP such that (l, m, l2 + a1l) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, from replacing y with y + m′ for m′ ∈ RP, there exists l ∈ LP,k and m ∈ LP,3k such that (l, m, l2 + a1l) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we prove that (l, m) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that (l1, m1), (l2, m2) ∈ LP,k × LP,3k and (l1, m1, l2 1 + a1l1), (l2, m2, l2 2 + a1l2) ∈ Ak(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We prove that (l1, m1) = (l2, m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let F be the curve F : y2 + a1 πk P xy + a3 π3k P y = x3 + a4 πk P x + a6 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For 1 ≤ i ≤ 2, let Fi be F with x replaced by x + l2 i +a1li π2k P and y replaced by y + li πk P x + mi π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that Fi ∈ G(3) P because (li, mi, l2 i + aili) ∈ Ak(E) for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, a1 ≡ 0 (mod πk P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that F1 and F2 are equivalent and vP(∆(F1)) = vP(∆(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, let τ be a translation from the equation of F1 to the equation of F2 that replaces x with u2x + n′ and y with u3y + l′u2x + m′, where u, l′, m′, n′ ∈ RP and vP(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The coefficient of xy after τ is applied to the equation of F1 is a1 uπk P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, the coefficient of xy in F2 is a1 πk P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, u = 1 and a1 ≡ 0 (mod πk P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, the coefficient of y after τ is applied to the equation of F1 a1l2 1 + a2 1l1 + a3 + π2k P a1n′ π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, the coefficient of y in F2 is a1l2 2 + a2 1l2 + a3 π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 18 ANDREW YAO Therefore, l2 1 + a1l1 + π2k P n′ = l2 2 + a1l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because a1 ≡ 0 (mod πk P), we have that l1 ≡ l2 (mod πk P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, l1 = l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, n′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The coefficient of x2 after τ is applied to the equation of F1 is n′ + (l′)2 + a1l′ πk P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This equals the coefficient of x2 in F2, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because n′ = 0, we have that l′ = 0 or l′ = a1 πk P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From setting the coefficient of x after τ is applied to the equation of F1 equal to the coefficient of x in F2, a1 πk P � m1 π3k P + m′ � + a1(l2 1 + a1l1) + a3 π3k P l′ = a1 πk P m2 π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose l′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then m1 π3k P + m′ = m2 π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It follows that m1 ≡ m2 (mod π3k P ) and m1 = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose l′ = a1 πk P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that m1 π3k P + m′ + a1(l2 1 + a1l1) + a3 π3k P = m2 π3k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, using that the coefficient of y in F2 is an element of RP, a1(l2 1 + a1l1) + a3 ≡ a1(l2 2 + a1l2) + a3 ≡ 0 (mod π3k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, m1 ≡ m2 (mod π3k P ) and m1 = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From the coefficient of y in F2, we have that a3 ≡ 0 (mod π3k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, from the coefficients of x in F1 and F2, l4 1 + a3l1 ≡ l4 2 + a3l2 (mod π4k P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This gives that l1 = l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, from the constant terms of F1 and F2, m2 1 + a3m1 ≡ m2 2 + a3m2 (mod π6 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, we obtain that m1 = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2, for E ∈ Sk, let the unique pair (l, m) ∈ LP,k × LP,3k such that (l, m, l2 +a1l) ∈ Ak(E) be (l(E), m(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Define φk : Sk → S0 to be the function such that if E ∈ Sk has equation E : y2 + a1xy + a3y = x3 + a4x + a6, φk(E) has equation φk(E) : y2 + a1 πk P xy + a1(l(E)2 + a1l(E)) + a3 π3k P = x3+ l(E)2(l(E)2 + a1l(E)) + a1m(E) + a3l(E) + a4 π4k P x+ (a1m(E) + a4 + a2 1l(E)2 + l(E)4)(l(E)2 + a1l(E)) + a3m(E) + a6 + m(E)2 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 19 The equation for φk(E) is equivalent to � y + l(E) πk P x + m(E) π3k P �2 + a1 πk P � x + l(E)2 + a1l(E) π2k P � � y + l(E) πk P x + m(E) π3k P � + a3 π3k P � y + l(E) πk P x + m(E) π3k P � = � x + l(E)2 + a1l(E) π2k P �3 + a4 π4k P � x + l(E)2 + a1l(E) π2k P � + a6 π6k P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that S0 ⊂ G(3) P , and from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, µP(S0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For l ∈ LP,k and m ∈ LP,3k, let Sk,l,m be the set of E ∈ Sk such that l(E) = l and m(E) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that φk,l,m is φk restricted to Sk,l,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose l ∈ LP,k and m ∈ LP,3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(3) P , µP(φ−1 k,l,m(U)) = 1 Q14k P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This can be proved with a method that is similar to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of G(3) P , µP(φ−1 k (U)) = 1 Q10k P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let U be an open subset of G(3) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that φ−1 k (U) = � l∈LP,k,m∈LP,3k φ−1 k,l,m(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, µP(φ−1 k (U)) = � l∈LP,k � m∈LP,3k µP(φ−1 k,l,m(U)) = � l∈LP,k � m∈LP,3k 1 Q14k P µP(U) = 1 Q10k P µP(U), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density Calculations for vP(a1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that vP(a1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This case has density QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The discriminant is a4 1(a2 1a6 + a1a3a4 + a2 4) + a4 3 + a3 1a3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that by considering a6 modulo πP, the discriminant is not divisible by πP with density (QP −1)2 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the algorithm ends in step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, we add (QP −1)2 Q2 P to δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume the discriminant is divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let (α1, α2) be the singular point modulo πP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' it can be proven that α1, α2 ∈ RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, α1 ≡ −a3 a1 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In step 2, replace x by x + n and y by y + m with n = α1 and m = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, the coefficient of xy is a1, which is not divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The algorithm then ends in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that the discriminant is linear in a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, we have that vP(a1) = 0 and vP(∆(E)) = N with density (QP −1)2 QN+2 P for N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the polynomial considered in step 2 is T 2 + a1T + α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a1 ≡ r1 (mod πP) and a3 ≡ r3 (mod πP) for r1, r3 ∈ LP,1 such that r1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Given r1, T 2 + a1T + α1 is irreducible over RP/πPRP for QP 2 values of r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Afterwards, using step 2 of Tate’s algorithm, we get that in this case, 20 ANDREW YAO δK(I1, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q3 P , δK(I2, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 Q4 P , and δK(IN, N, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK � IN, 2 �N 2 � − N + 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P � = (QP − 1)2 2QN+2 P for N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density Calculations for vP(a1) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In this subsection, we assume that vP(a1) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density for this is 1 QP , and the discriminant modulo πP is a4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose vP(a3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The density for this case is QP −1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Here, the discriminant is not divisible by πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Tate’s algorithm then ends in step 1, and we add QP −1 Q2 P to δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Following this, we obtain that δK(I0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 QP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, assume vP(a3) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this is 1 Q2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The singular point modulo πP is (x, y) = (α1, α2) for α1, α2 ∈ LP,1 such that a4 ≡ α2 1 (mod πP) and a6 ≡ α2 2 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We replace x with x + n and y with y + m, where n = α1 and m = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The curve is (y + m)2 + a1(x + n)(y + m) + a3(y + m) = (x + n)3 + a4(x + n) + a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If π2 P does not divide mna1 +ma3 +na4+a6+m2 +n3, the algorithm ends in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' By adding multiples of πP to a6, this occurs with density QP −1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that δK(II, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume π2 P divides mna1 + ma3 + na4 + a6 + m2 + n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q3 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that b8 = n(na1 + a3)2 + (ma1 + a4 + n2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If b8 is not divisible by π3 P, the algorithm ends in step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' By adding multiples of πP to a4, we have that δK(III, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume b8 is divisible by π3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q4 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If vP(na1 + a3) = 1, the algorithm ends in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume a4 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, replace a3 with a3 + dπP and a4 with a4 + βdπP for β, d ∈ LP,1 such that β2 ≡ α1 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This will not affect previous parts of the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' particularly, this will not change b8 modulo π3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, na1 + a3 will be increased by dπP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, we have that vP(na1 + a3) = 1 with density QP −1 Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, δK(IV, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume vP(na1 + a3) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q5 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let α3 be the element of LP,1 such that n ≡ α2 3 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, let α4 be the element of LP,1 such that mna1 + ma3 + na4 + a6 + m2 + n3 ≡ α2 4π2 P (mod π3 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' After the transformation in step 6, the equation of the curve is (y + lx + m)2 + a1(x + n)(y + lx + m) + a3(y + lx + m) = (x + n)3 + a4(x + n) + a6, where n = α1, l = α3, and m = α2 + α4πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that in step 6, the polynomial P(T) ∈ (RP/πPRP)[T] is P(T) = T 3 + w2T 2 + w1T + w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a4 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because 0 ∈ LP,1, we have that n = l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This means that w2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, we can replace a4 with a4 + d1π2 P for d1 ∈ LP,1, and the previous DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 21 parts of the algorithm will not be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, the choices for w1 modulo πP are the elements of LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Following this, from replacing a6 with a6 + d2π3 P for d2 ∈ LP,1, the choices for w0 modulo πP are the elements of LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that the number of P(T) with a double root and no roots in RP/πPRP are QP − 1 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, we have that the number of P(T) with 3 distinct roots in RP/πPRP and 0 roots, 1 root, and 3 roots in RP/πPRP are Q2 P −1 3 , Q2 P −QP 2 , and Q2 P −3QP +2 6 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a4 ̸≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Consider the translation of replacing a1 with a1 + d1πP, a3 with a3 + α1d1πP, a4 with a4 + (α2 + α4πP)d1πP, and a6 with a6 + α1(α2 + α4πP)d1πP for d1 ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' After this, the parts of the algorithm before step 6 do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In step 6, w0 and w1 do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, w2 increases by α3d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because α3 ̸= 0, the choices for w2 are the elements of LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, replace a6 with a6 + d2π3 P for d2 ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, the choices for w0 are also the elements of LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The number of P(T) with a double root and no roots in RP/πPRP are the same as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, the number of P(T) with 3 distinct roots in RP/πPRP and 0 roots, 1 root, and 3 roots in RP/πPRP are the same as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose P(T) has distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the total density is QP −1 Q6 P and Tate’s algorithm ends in step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We see that δK(I∗ 0, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −1 3Q7 P , δK(I∗ 0, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q6 P , and δK(I∗ 0, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = Q2 P −3QP +2 6Q7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that P(T) has a double root and a simple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the total density is QP −1 Q7 P and Tate’s algorithm ends in step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5, we compute that δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 2QN+7 P for all positive integers N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose P(T) has a triple root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For this case, the density is 1 Q7 P , and the root of P(T) is √w1 modulo πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If a4 ≡ 0 (mod πP), the triple root is 0 modulo πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let α5 be an element of LP,1 such that (m + ln)a1 + la3 + a4 + n2 ≡ α2 5π2 P (mod π3 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, the translation in step 8 sets n to be n = α1 + α5πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose a4 ≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Replace a3 with a3 + dπ2 P and a6 with a6 + (α2 + α4πP)dπ2 P for some d ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, note that the previous parts of the algorithm, including P(T), are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, the coefficient of y increases by dπ2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that for one value of d, the coefficient of y is divisible by π3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose a4 ̸≡ 0 (mod πP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Replace a1 with a1 + dπ2 P and a4 with a4 + (α2 + α4πP)dπ2 P for some d ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The previous parts of the algorithm, including P(T), are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, the coefficient of y increases by (α1 + α5πP)dπ2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similarly, we have that for one value of d, the coefficient of y is divisible by π3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, we get that the coefficient of y is not divisible by π3 P and the algorithm ends in step 8 with density QP −1 Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that δK(IV ∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(IV ∗, 3, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 2Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume the coefficient of y is divisible by π3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density of this case is 1 Q8 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let α6 be the element of LP,1 such that mna1 + ma3 + na4 + a6 + m2 + n3 ≡ α2 6π4 P (mod π5 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, m is set to m = α2 + α4πP + α6π2 P in step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If π4 P does not divide the x coefficient of this curve, the algorithm ends in step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Consider the translation of replacing a4 with 22 ANDREW YAO a4+dπ3 P and a6 with a6+(α1+α5πP)dπ3 P for d ∈ LP,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The previous parts of the algorithm do not change, but the coefficient of x is increased by dπ3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, π4 P does not divide the x coefficient with density QP −1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that δK(III∗, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q9 P Assume π4 P divides the coefficient of x of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The total density for this case is 1 Q9 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If π6 P does not divide mna1 + ma3 + na4 + a6 + m2 + n3, Tate’s algorithm ends in step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This occurs with density QP −1 Q10 P from adding multiples of π6 P to a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We then have that δK(II∗, 1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = QP −1 Q10 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Subprocedure Density Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We calculate the density of Kodaira types r = I∗ N for N ≥ 1 and Tamagawa numbers n = 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that previously, the curve was reduced by removing a2 with a translation on x to obtain G(3) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' However, here the density is calculated in GP without the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' That is, the density is calculated for curves in long Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let X be the set of elliptic curves E ∈ GP such that NP(E) = 0 and Tate’s algorithm enters the step 7 subprocedure when used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For E ∈ X, let L(E) be the number of iterations of the step 7 subprocedure that are completed when Tate’s algorithm is used on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a nonnegative integer N, let XN be the set of E ∈ X such that L(E) ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is an even nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Assume that N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In iteration N = 0, there is a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that the double root of P(T) is the squareroot of w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because of this, in step 7, we add γ0πP to n and lγ0πP to m for some γ0 ∈ LP,1 such that (m + ln)a1 + la3 + a4 + n2 ≡ γ2 0π2 P (mod π3 P) Next, assume N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose iteration N of the step 7 subprocedure is reached and the quadratic has a double root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, vP((m + ln)a1 + la3 + a4 + n2) ≥ N + 6 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, we add γNπ N+2 2 P to n and lγNπ N+2 2 P to m for some γN ∈ LP,1 such that mna1 + ma3 + na4 + a6 + m2 + n3 ≡ (la1 + a2 + n + l2)γ2 NπN+2 P (mod πN+4 P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that vP(la1 + a2 + n + l2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose N is an odd nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose iteration N of the step 7 subpro- cedure is reached and the quadratic has a double root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, vP(na1 + a3) ≥ N+5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, γNπ N+3 2 P is added to m for some γN ∈ LP,1 such that mna1 + ma3 + na4 + a6 + m2 + n3 ≡ γ2 NπN+3 P (mod πN+4 P ) Let N be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let YN be the set of curves y2 + a′ 1xy + a′ 3y = x3 + a′ 2x2 + a′ 4x + a′ 6 with vP(a′ 1) ≥ 1, vP(a′ 2) = 1, vP(a′ 3) ≥ ⌊N+5 2 ⌋, vP(a′ 4) ≥ ⌊N+6 2 ⌋, and vP(a′ 6) ≥ N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E ∈ XN and that the translations of Tate’s algorithm when it is used on E are α1, α2, α3, α4, γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=', γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let TN(E) = (α1, α2, α3, α4, γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' , γN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that because the characteristic of K is p = 2, TN(E) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Also, let θN(E) : XN → YN DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 23 be E with x replaced by x + n and y replaced by y + lx + m, where n = α1 + ⌊ N 2 ⌋ � i=0 γ2iπi+1 P , l = α3, m = α2 + α4πP + α3 ⌊ N 2 ⌋ � i=0 γ2iπi+1 P + ⌊ N−1 2 ⌋ � i=0 γ2i+1πi+2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' If U is an open subset of YN, µP(θ−1 N (U)) = QN+5 P µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let a = (α1, α2, α3, α4, γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' , γN)0≤i≤N be an element of LN+5 P,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that XN,a is the set of E ∈ XN such that TN(E) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose that θN,a is θN restricted to XN,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let U be an open subset of YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Using a method similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, we have that µP(θ−1 N,a(U)) = µP(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because there are QN+5 P choices of a, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Suppose N is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='5, we can compute the density for curves that enter step 7 in the first iteration and have type I∗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that µP(YN−1) = QP −1 Q2N+10 P , and the Haar measure in G(3) P of curves that have type I∗ N is then (QP −1)2 QN+7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, δK(I∗ N, 2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = δK(I∗ N, 4, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) = (QP −1)2 2QN+7 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Local and Global Density Results In Section 4, Section 5, and Section 6, we compute the local densities of Koidara types and Tamagawa numbers for p ≥ 5, p = 3, and p = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The methods we use involved first removing some terms from the equations of elliptic curves with translations, and then using translations to compute the local densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let r be a Koidara type and n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that δK(r, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) only depends on QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Additionally, in [3], the local densities of r and the Tamagawa number n for elliptic curves in short Weierstrass form over Qr for primes r ≥ 5 have the same form as δK(r, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) for global function fields K and P ∈ MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' In [1], the local densities of r and the Tamagawa number n for elliptic curves in short Weierstrass form over completions of number fields at places that lie above primes r ≥ 5 also have the same form as δK(r, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' P) for global function fields K and P ∈ MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, we discuss some results about local and global densities, including a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Particularly, we compute the density of completing at most k ≥ 0 iterations of Tate’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let U and V be the sets of elliptic curves E ∈ GP with Kodaira type r and Tamagawa number n such that NP(E) = 0 and NP(E) = k, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Note that ϕ(U) and ϕ(V ) are the sets of curves E ∈ S0 with Kodaira type r and Tamagawa number n such that NP(E) = 0 and NP(E) = k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E ∈ GP and ϕ(E) ∈ ϕ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, E has Kodaira type r, Tamagawa number n, and NP(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This means that E ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, ϕ−1(ϕ(U)) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, U ⊂ ϕ−1(ϕ(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It follows that ϕ−1(ϕ(U)) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Similarly, ϕ−1(ϕ(V )) = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that U and V are open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, ϕ(U) and ϕ(V ) are open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' With this, we have that µP(U) = µP(ϕ(U)) and µP(V ) = µP(ϕ(V )) for all characteristics p 24 ANDREW YAO from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1, and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, it suffices to prove that µP(ϕ(V )) = 1 Q10k P µP(ϕ(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose E ∈ ϕ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We have that φk(E) has Kodaira type r, Tamagawa number n, and NP(φk(E)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, φk(E) ⊂ ϕ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It follows that ϕ(V ) ⊂ φ−1 k (ϕ(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Next, suppose E ∈ Sk and φk(E) ∈ ϕ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, the Koidara type of E is r and the Tamagawa number of E is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Moreover, because NP(φk(E)) = 0, NP(E) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' It follows that E ∈ ϕ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, φ−1 k (ϕ(U)) ⊂ ϕ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From this, φ−1 k (ϕ(U)) = ϕ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The result then follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Density for Multiple Iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For P ∈ MK, let Uk P denote the set of elliptic curves E in GP such that NP(E) ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The following proposition is important for the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For P ∈ MK, µP(Uk P) = 1 Q10(k+1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose P ∈ Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4, and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='4 with k + 1 as k and GP as U, we have that µP(Uk P) = 1 Q10(k+1) P µP(GP) = 1 Q10(k+1) P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let S be a finite nonempty subset of MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose U is the set of elliptic curves in WS such that NP(E) ≤ k for all P ∈ SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Then, dS(U) = 1 ζK(10(k + 1)) · � P ∈S � Q10(k+1) P Q10(k+1) P − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' For a positive integer M, let VM be the set of elliptic curves E ∈ WS such that there exists P ∈ SC with degree at least M such that E ∈ Uk P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3, we have that limM→∞ dS(VM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Therefore, we can use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1 with Uk P as UP for P ∈ SC and T = {}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' The result follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' ■ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' We give an example of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let K = Fq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Suppose P∞ is the infinite place of Fq(t) and let S = {P∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Let k be a nonnegative integer and U be the set of elliptic curves in WS such that NP(E) ≤ k for all P ∈ SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='9 of [5], because the genus of K is 0, we have that ζK(10(k + 1)) = q20k+19 (q10k+9−1)(q10k+10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Because P∞ has degree 1, from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content='2, dS(U) = 1 − 1 q10k+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' DENSITIES FOR ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 25 References [1] Yunseo Choi, Sean Li, Apoorva Panidapu, and Casia Siegel, Tamagawa products for elliptic curves over number fields, arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [2] John Cremona and Mohammad Sadek, Local and global densities for Weierstrass models of elliptic curves, arXiv, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [3] Michael Griffin, Ken Ono, and Wei-Lun Tsai, Tamagawa products of elliptic curves over Q, The Quarterly Journal of Mathematics 72 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 4, 1517–1543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [4] Giacomo Micheli, A local to global principle for densities over function fields, arXiv, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [5] Michael Rosen, Number theory in function fields, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=', Springer, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [6] Joseph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' Silverman, Advanced topics in the arithmetic of elliptic curves, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=', Springer, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' [7] John Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Func- tions of One Variable IV, 1975, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} +page_content=' 33–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFJT4oBgHgl3EQfEiwz/content/2301.11437v1.pdf'} diff --git a/J9AyT4oBgHgl3EQfTffD/content/tmp_files/2301.00108v1.pdf.txt b/J9AyT4oBgHgl3EQfTffD/content/tmp_files/2301.00108v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1aef949a5633ac8aed40a41a40d417b8e1bb0dc --- /dev/null +++ b/J9AyT4oBgHgl3EQfTffD/content/tmp_files/2301.00108v1.pdf.txt @@ -0,0 +1,2194 @@ +Targeted 𝑘-node Collapse Problem: Towards Understanding the +Robustness of Local 𝑘-core Structure +Yuqian Lv +Zhejiang University of Technology +lvyuqian_email@163.com +Bo Zhou +Zhejiang University of Technology +wxjs201@163.com +Jinhuan Wang +Zhejiang University of Technology +jhwang@zjut.edu.cn +Qi Xuan +Zhejiang University of Technology +xuanqi@zjut.edu.cn +ABSTRACT +The concept of𝑘-core, which indicates the largest induced subgraph +where each node has 𝑘 or more neighbors, plays a significant role +in measuring the cohesiveness and the engagement of a network, +and it is exploited in diverse applications, e.g., network analysis, +anomaly detection, community detection, etc. Recent works have +demonstrated the vulnerability of 𝑘-core under malicious pertur- +bations which focuses on removing the minimal number of edges +to make a whole 𝑘-core structure collapse. However, to the best of +our knowledge, there is no existing research concentrating on how +many edges should be removed at least to make an arbitrary node +in 𝑘-core collapse. Therefore, in this paper, we make the first at- +tempt to study the Targeted 𝑘-node Collapse Problem (TNCP) with +four novel contributions. Firstly, we offer the general definition of +TNCP problem with the proof of its NP-hardness. Secondly, in order +to address the TNCP problem, we propose a heuristic algorithm +named TNC and its improved version named ATNC for implemen- +tations on large-scale networks. After that, the experiments on 16 +real-world networks across various domains verify the superiority +of our proposed algorithms over 4 baseline methods along with de- +tailed comparisons and analyses. Finally, the significance of TNCP +problem for precisely evaluating the resilience of 𝑘-core structures +in networks is validated. +PVLDB Reference Format: +Yuqian Lv, Bo Zhou, Jinhuan Wang, and Qi Xuan. Targeted 𝑘-node +Collapse Problem: Towards Understanding the Robustness of Local 𝑘-core +Structure. PVLDB, 16(1): XXX-XXX, 2022. +doi:XX.XX/XXX.XX +PVLDB Artifact Availability: +The source code, data, and/or other artifacts have been made available at +https://github.com/Yocenly/TNCP. +1 +INTRODUCTION +Networks or graphs play significant roles in describing various +complex systems from numerous domains, e.g., social networks[11, +27, 31, 43], citation networks[16, 26], biological networks[11, 22, 47] +This work is licensed under the Creative Commons BY-NC-ND 4.0 International +License. Visit https://creativecommons.org/licenses/by-nc-nd/4.0/ to view a copy of +this license. For any use beyond those covered by this license, obtain permission by +emailing info@vldb.org. Copyright is held by the owner/author(s). Publication rights +licensed to the VLDB Endowment. +Proceedings of the VLDB Endowment, Vol. 16, No. 1 ISSN 2150-8097. +doi:XX.XX/XXX.XX +1 +3 +2 +6 +8 +4 +5 +7 +9 +b +a +c +d +1 +3 +4 +6 +7 +10 +9 +2 +5 +8 +11 +12 +13 +14 +3-core +2-core +15 +16 +18 +17 +1-core +2-node +3-node +1-node +Figure 1: An example of 𝑘-core distribution in a graph. Each +subgraph in the dotted box with a certain color represents +the 𝑘-core and the color of each node represents its core +value. +and power networks[2, 36]. Therefore, understanding the topolog- +ical information of graphs is what matters in the study of graph +theory. Due to the advantages of simplicity and efficiency [20], the +concept of 𝑘-core, which denotes the maximal induced subgraph +where each node within it occupies at least 𝑘 neighbors [9], has +stood out as an important metric for describing the global struc- +tural engagement of networks from massive evaluation metrics. +As shown in Figure 1, an example graph with 18 nodes and 28 +edges is given where 3 cores exist, i.e., 1-core, 2-core and 3-core +which are surrounded by dotted boxes with different colors. As +more and more researchers devoted themselves to the study of +𝑘-core, 𝑘-core has been used in a broad variety of important ap- +plications [32]. For example, in ecological networks, Morone et al. +[35] exploited the 𝑘-core as a predictor to estimate the structural +collapse in mutualistic ecosystems, and Burleson-Lesser et al. [7] +presented a new approach for characterizing the stability and ro- +bustness of networks with all-positive interactions by studying the +distribution of the 𝑘-core of the underlying network. Besides, in +social networks, Wang et al. [44] considered the pruning process of +𝑘-core to measure the vulnerability and resilience of social engage- +ment and further studied its equilibrium statistical mechanics. And +in biological networks, Luo et al. [30] studied the core structure of +protein-protein interactions networks with an interesting discovery +that core structures help to reveal the existence of multiple levels of +protein expression dynamics, and Isaac et al. [17] discovered that +arXiv:2301.00108v1 [cs.SI] 31 Dec 2022 + +residues belonging to inner cores are more conserved than those at +the periphery of the network with the evidence that these groups +are functionally and structurally critical. +With the rapidly increasing number of applications based on +𝑘-core structures, the robustness (or also called resilience) of 𝑘-core +have gradually attracted the attention of researchers. For instance, +Zhou et al.[51] studied the robustness of 𝑘-shell, a subset of 𝑘-core, +and demonstrated that 𝑘-shell is vulnerable under the disturbance +of edge rewiring. Their optimal-based experimental results showed +that the 𝑘-core distributions of graphs can be drastically changed +even a small proportion of edges are rewired. Zhou et al.[52] also +studied the minimal budgets of removed edges for the collapse of +the innermost 𝑘-core. They provided a proof of its NP-hardness +and offered effective heuristic algorithms to cover this problem. +Furthermore, Chen et al.[8] focused on the 𝑘-core minimization +problem and suggested three sub-problems, i.e., KNM, KEM and +KCM. They further proposed several heuristic algorithms through +edge removal to cover these sub-problems respectively. Medya et +al. [34] also concentrated on the 𝑘-core minimization problem and +proposed a novel algorithm inspired by shapley value, a cooperative +game-theoretic concept. Their algorithm could leverage the strong +interdependencies in the effects of edges removal in the search +space. Besides, Zhang et al.[49] studied the collapsed𝑘-core problem +which aims to find a set of nodes whose detachment will lead to +the minimal size of the resulting collapsed 𝑘-core. +However, as we can see, throughout the previous works, all of +them considered the 𝑘-core as a whole to evaluate its robustness, +while none of them focused on the robustness of an individual node +within 𝑘-core. Thus it brings us a question that how many edges +should we disconnect at least to make an arbitrary node contained +in 𝑘-core collapse? As far as we know, there is no existing work +dedicating to the study of this problem. In this paper, we make +the first attempt to study this problem and name it as Targeted +𝑘-node Collapse Problem (TNCP). Our main contributions can be +summarized as below. +• We offer a general definition of TNCP problem with a proof +of its NP-hardness. We demonstrate that the naive exhaustive +method will lead to the exponential explosion of the time com- +plexity. Therefore, a series of theorems are provided to narrow +down the search space of candidates. +• Combined with the theorems, we propose a heuristic algorithm +named TNC to cover the TNCP problem. However, we find that +TNC algorithm is not suitable for large-scale networks. Thus, +an improved algorithm named ATNC with less time complexity +is proposed based on TNC algorithm. +• We verify the superiority of our proposed algorithms over 4 +baseline methods through experiments on 16 real-world net- +works collected from different public platforms along with +detailed comparisons and analyses. +• We demonstrate that the research of TNCP problem is helpful +for precisely evaluating the resilience of the 𝑘-core structures +in networks. +The remaining sections of this paper are structured as follows. +In Section 2, a brief review on the previous works about 𝑘-core is +illustrated. In Section 3, the statement of TNCP problem and basic +definitions, which will be used in the rest of this paper, are intro- +duced along with the theorems for candidate reduction. In Section +4, we introduce our proposed methods TNC and ATNC with their +time complexity analyses. In Section 5, we give the introductions +about the datasets being used, the baseline methods for compar- +isons and the metrics for evaluations. In Section 6, experimental +results on all mentioned datasets are shown along with detailed +comparisons and analyses between our proposed algorithms and +the baseline methods. In Section 7, the significance of TNCP prob- +lem for precisely evaluating the resilience of 𝑘-core in networks is +validated. Finally, our work is concluded in Section 8. +2 +RELATED WORKS +The researches on the 𝑘-core structure of networks have been +enduring, and those most related to our work are introduced as +below, including core decomposition, core robustness/resilience, +and core percolation. +Core Decomposition. Hajnal et al. [14] gave the first 𝑘-core +related concept and defined the degeneracy of a graph as the maxi- +mum core number of a node. Then, Seidman [40], as well as Matula +and Beck [33], defined the 𝑘-core subgraph as the maximal con- +nected subgraph where each node has at least 𝑘 neighbors. Khaouid +et al. [18] explored whether𝑘-core decomposition of large networks +can be computed using a consumer-grade PC. Sariyüce et al. [39] +proposed the first incremental 𝑘-core decomposition algorithms +for streaming graph data. Hébert-Dufresne et al. [15] proposed +onion decomposition which is derived from 𝑘-core decomposition. +Eidsaa and Almaas [10] presented 𝑠-core analysis, a generalization +of 𝑘-core analysis, for weighted networks. +Core Robustness/Resilience. In addition to works mentioned +in the last section, Adiga and Vullikanti [1] examined the robustness +of the top core sets in perturbed/sampled graphs. Zdeborová et +al. [48] used 𝑘-core as a heuristic tool in the process of graph +decycling and dismantling. Laishram et al. [23] proposed metrics +for measuring the core resilience of a network under the situations +of node/edge removals. +Core Percolation. Azimi-Tafreshi et al. [3] generalized the the- +ory of 𝑘-core percolation on complex networks to k-core percola- +tion on multiplex networks, where k = (𝑘𝑎,𝑘𝑏, ...). Whi et al. [46] +revealed the hierarchical structure of functional connectivity on +resting-state fMRI (rsfMRI) through the method of 𝑘-core perco- +lation. Wang et al. [45] proposed a generalized 𝑘-core percolation +model to investigate the robustness of the higher-order dependent +networks. Zheng et al. [50] studied the robustness of multiplex net- +works with interdependent and interconnected links under 𝑘-core +percolation. Guo et al. [13] applied 𝑘-core percolation analysis on +brain structural network, suggesting that the brain networks are +mostly reliable against random or 𝑘-core-based percolation with +their structure design. +3 +PROBLEM STATEMENT +In this section, the descriptions of commonly used definitions and +fundamental concepts will be discussed in the following contents +along with the statement of TNCP problem and the proofs of our +proposed theorems. +2 + +3.1 +Preliminaries +In this paper, a network or a graph (these two concepts will be +used indiscriminately) is indicated as 𝐺 = (𝑉, 𝐸), where 𝑉 and +𝐸 ⊆ (𝑉 × 𝑉 ) represent the sets of nodes and edges respectively, +which are extracted from real-world entities and the relationships +between any pair of entities. As a prerequisite, we only focus on +those unweighted and undirected graphs without self-loops or +isolated nodes. Here, we present some fundamental definitions and +related concepts which are relevant to the subsequent discussions. +In Table 1, we compile a list of principal symbols and notations for +convenient query. +Table 1: Summary of notations. +Notation +Definition +𝐺𝑘 +the 𝑘-core subgraph of 𝐺 +𝑑(𝑖,𝐺𝑘) +the degree of node 𝑖 in 𝐺𝑘 +𝐶(𝑖,𝐺) +the core value of node 𝑖 +𝑆𝑁(𝑖,𝑘,𝐺) +the supportive neighbors of the node 𝑖 in 𝐺𝑘 +𝑆𝑁(𝑖,𝐺) +the simplification of 𝑆𝑁(𝑖,𝐶(𝑖,𝐺),𝐺) +N(𝑖,𝐺𝑘) +the one-hop neighbors of node 𝑖 in 𝐺𝑘 +𝐶𝑆(𝑖,𝐺) +core strength of node 𝑖 +𝑁𝑅(𝑖,𝐺) +node robustness of node 𝑖 +𝑃(𝑖,𝐺) +the corona pedigree of node 𝑖 +𝐸𝑃 +(𝑖,𝐺) +those edges connected with nodes in 𝑃(𝑖,𝐺) +Definition 1. 𝑘-core. For a given graph 𝐺, its 𝑘-core, denoted +as 𝐺𝑘 = (𝑉𝑘, 𝐸𝑘) where 𝑉𝑘 ⊆ 𝑉 and 𝐸𝑘 ⊆ 𝐸, means the maximal +induced subgraph whose nodes occupy at least 𝑘 neighbors within 𝐺𝑘, +i.e., ∀𝑖 ∈ 𝑉𝑘,𝑑(𝑖,𝐺𝑘) ≥ 𝑘, where 𝑑(𝑖,𝐺𝑘) is the degree of 𝑖 in 𝐺𝑘. +Definition 2. Core Value of Node. With the concept of 𝑘-core, +we can also describe the core value of a given node 𝑖 within 𝐺 by +𝐶(𝑖,𝐺), which represents the maximum core value of the 𝑘-core where +node 𝑖 exists, i.e., 𝐶(𝑖,𝐺) satisfies that 𝑖 ∈ 𝐺𝐶(𝑖,𝐺) but 𝑖 ∉ 𝐺𝐶(𝑖,𝐺)+1. +The nodes whose core values are equal to 𝑘 are named as 𝑘-nodes. +In accordance with Definition 1, the existence of a given node 𝑖 +within 𝐺𝑘 relies on its neighbor nodes who overlap with 𝐺𝑘. We +can also realize that those neighbors with core values less than 𝑘 +are not included in 𝐺𝑘. Be a result, those neighbors helping support +the existence of 𝑖 in 𝐺𝑘 are referred to as Supportive Neighbors of 𝑖 +which is recorded as 𝑆𝑁(𝑖,𝑘,𝐺) = {𝑗|𝑗 ∈ N(𝑖,𝐺),𝐶(𝑗,𝐺) ≥ 𝑘}, where +N(𝑖,𝐺) represents the one-hop neighbors of 𝑖 within 𝐺. In this way, +the following theorem could be deduced. +Theorem 1. Core Support Condition. Node 𝑖 can remain in 𝐺𝑘 +if and only if it satisfies |𝑆𝑁(𝑖,𝑘,𝐺) | ≥ 𝑘; otherwise, it will be squeezed +out of 𝐺𝑘. +Proof. According to the definition of supportive neighbors of +node 𝑖, 𝑆𝑁(𝑖,𝑘,𝐺) actually denotes the intersection of N(𝑖,𝐺) and +𝑉𝑘. Based on Definition 1, it is clear that only the satisfaction of +𝑑(𝑖,𝐺𝑘) ≥ 𝑘 can remain the existence of 𝑖 in 𝐺𝑘. In this way, if node +𝑖 ∈ 𝐺𝑘, there is |𝑆𝑁(𝑖,𝑘,𝐺) | = |N(𝑖,𝐺) ∩𝑉𝑘 | = |N(𝑖,𝐺𝑘) | = 𝑑(𝑖,𝐺𝑘) ≥ 𝑘, +which shows that node 𝑖 could be contained in 𝐺𝑘 if and only if +Delete +edge (5, 7) +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +Delete +edge (4, 8) +Delete +edge (1, 3) +2-node +3-node +Figure 2: Given a graph with 9 nodes and 17 edges, we can +find that all nodes stay in 3-core. Take node 5 as our target +node, (i) the removal of edge (5, 7) does not make any ef- +fect to the 𝑘-core distribution; (ii) the removal of edge (4, 8) +makes node 4 being squeezed out of 3-core while makes no +effect to node 5; (iii) the removal of edge (1, 3) makes the tar- +get node 5 being squeezed out of 3-core. +at least 𝑘 neighbors whose core values are not less than 𝑘 are +connected with it. +□ +Example 1. As illustrated in Figure 1, for node 13 who lives in 𝐺2, +it has 𝑆𝑁(13,2,𝐺) = 2 which allows it to satisfy Theorem 1, while it +has 𝑆𝑁(13,3,𝐺) = 1 < 3 so that it cannot exist in 𝐺3. +Theorem 1 provides us with a sufficient and necessary condition +to determine whether a certain node exists in 𝐺𝑘. Derived from this, +Laishram et al.[23] exploited a naive and easily-computed metric +called Core Strength to measure the most conservative number +of disconnected neighbors of node 𝑖 for squeezing 𝑖 out of 𝐺𝐶(𝑖,𝐺) , +which is formulated as +𝐶𝑆(𝑖,𝐺) = |𝑆𝑁(𝑖,𝐶(𝑖,𝐺),𝐺) | − 𝐶(𝑖,𝐺) + 1. +(1) +This metric describes that if any 𝐶𝑆(𝑖,𝐺) of supportive neighbors +are disconnected with the target node 𝑖, it will absolutely be in +violation of Theorem 1 and be squeezed out of 𝐺𝐶(𝑖,𝐺) . For instance, +as shown in Figure 2, we set node 5 ∈ 𝐺3 as the target node. That +is easy to find that the target node has 5 supportive neighbors +𝑆𝑁(5,3,𝐺) = {1, 2, 4, 6, 7} and core strength 𝐶𝑆(5,𝐺) = 3. We arbitrar- +ily select 3 supportive neighbors to disconnect, e.g. {4, 6, 7}, then +the number of its supportive neighbors will be reduced to 2 which +is against what Theorem 1 restricts. Please notice that in the rest +of this paper, if 𝑘 = 𝐶(𝑖,𝐺), we will use 𝑆𝑁(𝑖,𝐺) instead of 𝑆𝑁(𝑖,𝑘,𝐺) +for the sake of simplicity. +3 + +3.2 +Problem Definition +As mentioned in the aforementioned contents, the core strength +metric describes the most conservative number of edges we should +disconnect for target-node collapse. Because of so-called cascade +phenomenon or domino phenomenon of 𝑘-core collapse [12], how- +ever, this metric cannot estimate the exact number of edges that +must be deleted which may be less than that quantified by core +strength. As an illustration, let us turn our sights back to Figure +2, the deletion of edge (1, 3) will practically make node 5 with +𝐶𝑆(5,𝐺) = 3 collapse from 𝐺3 to 𝐺2. From here, we can derive the +problem named Targeted 𝑘-node Collapse Problem (TNCP) aiming +to quantify the minimal number of edges to remove for downgrad- +ing the core value of a target node. +Proposition 1. For a given 𝐺 and a target node 𝑖 ∈ 𝑉 with +𝐶(𝑖,𝐺) = 𝑘, TNCP problem aims to find a set 𝑒 ⊆ 𝐸 containing the +least number of edges such that𝐶(𝑖,𝐺′) < 𝐶(𝑖,𝐺), where𝐺′ = (𝑉, 𝐸\𝑒), +and can be formulated as: +𝑒∗ = arg min +𝑒 +|𝑒| , +𝑠.𝑡.𝐶(𝑖,𝐺′) < 𝐶(𝑖,𝐺). +(2) +The minimal size of 𝑒 is named as Node Robustness which dis- +plays the fewest number of removed edges for the collapse of +the target node under elaborate perturbations and is recorded as +𝑁𝑅(𝑖,𝐺) = 𝑒∗. Furthermore, those nodes whose core strengths are +larger than their node robustness are referred to as Bubble Nodes +which are recorded as 𝐵𝑁 = {𝑖|𝑖 ∈ 𝑉,𝐶𝑆(𝑖,𝐺) > 𝑁𝑅(𝑖,𝐺)}. +Theorem 2. The TNCP problem is NP-hard for 𝐶(𝑖,𝐺) ≥ 2. +Proof. First, when 𝐶(𝑖,𝐺) = 1, according to Definition 1, it is +easy to realize that some node will always remain in 𝐺1 as long as +at least one neighbor is connected with it. In this way, if we want +a node to collapse from 𝐺1 to 𝐺0, we have to disconnect all of its +adjacent neighbors and make it isolated from 𝐺, where the cost of +operations is in polynomial time. +Then, when 𝐶(𝑖,𝐺) ≥ 2, considering the cascade phenomenon of +𝑘-core collapse, a slight disturbance is able to lead a huge variation +to the target node on weakening the number of its supportive +neighbors. Therefore, in such a situation, the Set Cover Problem +(SCP) which has been proved to be NP-hard [21] can be reduced +to TNCP problem. Given a universe collection 𝑆𝑁(𝑖,𝐺) and a set of +candidates 𝐸 which contains all edges within 𝐺 under the condition +of target node𝑖. In order to cover the TNCP problem, we have to find +out a minimal-size set of edges 𝑒 ⊆ 𝐸 such that |𝑆𝑁(𝑖,𝐺) \ Φ(𝑒)| < +𝐶(𝑖,𝐺), where Φ(𝑒) represents those collapsed nodes whose core +values will be changed after the removal of 𝑒 from 𝐺. +Additionally, paying attention to the complexity of TNCP prob- +lem, without any prior information, we have to traverse all pos- +sible combinations of the already existing edges, whose mathe- +matical expression can be formulated as 𝑓 = �𝛿 +𝑚=1 +�|𝐸 | +𝑚 +�, where +𝛿 = 𝐶𝑆(𝑖,𝐺). Based on the induction formulas of �𝑛 +𝑚 +� = �𝑛−1 +𝑚 +� + �𝑛−1 +𝑚−1 +� +and �𝑀 +𝑚=0 +�𝑀 +𝑚 +� = 2𝑀, the above equation could be written as +𝑓 = O(|𝐸|𝛿−1) +�𝛿 +0 +� ++ O(|𝐸|𝛿−2) +�𝛿 +1 +� ++ · · · + +�𝛿 +𝛿 +� += O(|𝐸|𝛿−1) + O(|𝐸|𝛿−2) · 21 + · · · + 2𝛿 += 2𝛿 + +𝛿 +∑︁ +𝑚=1 +O(|𝐸|𝑚−1) · 2𝛿−𝑚 +(3) +With the complexity in the amount of the exponential increase, +it is evident that traversing all combinations takes non-polynomial +time. Combining the aforementioned approaches, the TNCP prob- +lem cannot be addressed in polynomial time when 𝐶(𝑖,𝐺) ≥ 2. +□ +Example 2. As seen in Figure 1 covering 18 nodes and 28 edges, +node 6 is chosen to be the target node for𝑘-node collapse. As mentioned +before, there is just one edge, like (1, 2), should be removed in order +to achieve the collapse of node 6. However, without the omniscient +knowledge, it is difficult to locate which edge or edges are necessar- +ily deleted. From the descriptions above, it is naturally realized that +𝑁𝑅(6,𝐺) ≤ 𝐶𝑆(6,𝐺), thus we need to visit all �2 +𝑚=1 +�28 +𝑚 +� combinations +to identify the key edge or edges useful for 𝑘-node collapse under the +worst situation. Fortunately, in this scenario, the computational com- +plexity is not high because of the previous information of 𝑁𝑅(6,𝐺) = 1 +with the removal of edge (4, 6). +However, the robustness of the target node will always be equal +to 1, like 𝑁𝑅(7,𝐺) = 2 under the removal of (1, 2) and (7, 8) as well +as 𝑁𝑅(8,𝐺) = 2 under the removal of (4, 8) and (7, 8) in Figure 2. +In real-world networks, the robustness of some nodes may reach +tens or even hundreds, which can probably lead to an exponential +increase in time consumption. Additionally, real-world networks +often contain thousands or even millions of edges, making it chal- +lenging to find a feasible solution within a reasonable amount of +time. Therefore, it is important to design an effective heuristic +algorithm to solve the TNCP problem. +3.3 +Candidate Reduction +As mentioned above, the naive exhaustive method for solving the +TNCP problem is highly complex, making it difficult to implement +in practice. In order to obtain a feasible solution within a reasonable +amount of time, we need to reduce the number of candidate edges. +In this section, we will introduce and prove some theorems that +can be used to achieve this reduction in candidates. +Theorem 3. ∀(𝑖, 𝑗) ∈ 𝐸, when 𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺), it satisfies that +𝑖 ∈ 𝑆𝑁(𝑗,𝐺) ∧ 𝑗 ∉ 𝑆𝑁(𝑖,𝐺), and when 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺), it satisfies that +𝑖 ∈ 𝑆𝑁(𝑗,𝐺) ∧ 𝑗 ∈ 𝑆𝑁(𝑖,𝐺). +Proof. Based on the definition of supportive neighbors, it is +evident that only those neighbors with core values greater than or +equal to 𝐶(𝑖,𝐺) can be contained within the supportive neighbors +of node 𝑖, i.e., {𝑗|𝑗 ∈ 𝐺,𝐶(𝑗,𝐺) < 𝐶(𝑖,𝐺)} ∩ 𝑆𝑁(𝑖,𝐺) = ∅. For the +same reason, considering 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺), there exists that {𝑖, 𝑗} ∈ +𝑆𝑁(𝑖,𝐺) ∩ 𝑆𝑁(𝑗,𝐺). +□ +In other words, nodes with low core values could never establish +relationships that would be supportive to nodes with high core val- +ues, while nodes with high core values establish one-way relation- +ships that would be supportive of their connected nodes with low +4 + +core values. Additionally, connected nodes with the same core value +become supportive neighbors to each other. This suggests that the +removal of edges bridging node pairs with different core values may +only affect the side holding a low core value, while the removal of +edges bridging node pairs with the same core values may affect both +sides. Combining the description of Theorem 3, those relationships +bridging nodes with higher core values and lower core values are +named as one-way supportive relationships, and those relationships +bridging nodes with the same core value are named as bidirectional +supportive relationships. In this way, the neighbors who control +the bidirectional supportive relationships with an arbitrary node 𝑖 +are recorded as � +𝑆𝑁 (𝑖,𝐺) = {𝑗|𝑗 ∈ 𝑁(𝑖,𝐺),𝐶(𝑗,𝐺) = 𝐶(𝑖,𝐺)}. +Theorem 4. If an edge (𝑖, 𝑗) ∈ 𝐸 is removed, for all nodes in 𝐺, +only those with core values equal to min(𝐶(𝑖,𝐺),𝐶(𝑗,𝐺)) may have +their core values changed. +Proof. It might be assumed that 𝐶(𝑖,𝐺) ≥ 𝐶(𝑗,𝐺) = 𝑘𝑚𝑖𝑛 and +be marked that 𝐺′ = 𝐺 \ {(𝑖, 𝑗)}. In accordance with Theorem +1, the removal of edge (𝑖, 𝑗) will surely make node 𝑗 collapse if +and only if 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛,𝐺) = 𝑘𝑚𝑖𝑛. After the elimination of (𝑖, 𝑗), +there exists that 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛,𝐺′) ≤ 𝑘𝑚𝑖𝑛 − 1 which is absolutely in +violation with Theorem 1 and makes node 𝑗 excluded from 𝐺𝑘𝑚𝑖𝑛. +In addition, Sariyüce et al. [39] and Li et al. [25] have proved that +the core value of some node can decrease at most 1 when one of +its supportive neighbors is lost. Benefiting from this, node 𝑗 will +still remain in 𝐺′ +𝑘𝑚𝑖𝑛−1 and satisfy that 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛−1,𝐺′) ≥ 𝑘𝑚𝑖𝑛 − 1. +According to Theorem 3, the collapse of node 𝑗 from 𝐺𝑘𝑚𝑖𝑛 to +𝐺′ +𝑘𝑚𝑖𝑛−1 probably leads to the collapse of those nodes contained in +� +𝑆𝑁 (𝑗,𝐺𝑘𝑚𝑖𝑛 ). Following like this, based on the cascade phenomenon, +it is easy to find that only nodes whose core values equal to𝑘𝑚𝑖𝑛 will +probably collapse from 𝐺𝑘𝑚𝑖𝑛 to 𝐺′ +𝑘𝑚𝑖𝑛−1 in the case of eliminating +edge (𝑖, 𝑗). Besides, for those nodes with core values larger than +𝑘𝑚𝑖𝑛, according to Theorem 3,𝑘𝑚𝑖𝑛-nodes make no contributions to +supporting their presence in 𝐺𝑘𝑚𝑖𝑛+1 so that no effect will work on +them after edge (𝑖, 𝑗) is removed. Meanwhile, due to the existence +of those collapsed nodes in 𝐺′ +𝑘𝑚𝑖𝑛−1, on the basis of Theorem 3, +they still establish supportive relationships with those nodes with +core values less than 𝑘𝑚𝑖𝑛 whose number of supportive neighbors +remains the same so that no change happens to their core values +after edge (𝑖, 𝑗) is removed. +□ +Benefiting from Theorem 4, only the removal of edges contained +in 𝐸𝑘\𝑘+1 = 𝐸𝑘 \𝐸𝑘+1 = {(𝑢, 𝑣)|(𝑢, 𝑣) ∈ 𝐸,𝑚𝑖𝑛(𝐶(𝑢,𝐺),𝐶(𝑣,𝐺)) = 𝑘} +will have the probability to make the target node 𝑖 with 𝐶(𝑖,𝐺) = +𝑘 collapse, which allows us to reduce the candidates from 𝐸 to +𝐸𝑘\𝑘+1. As illustrated in Figure 2 where only a 3-core exists, in +order to make node 5 with 𝐶(5,𝐺) = 3 collapse, we should take +𝐸3\4 = 𝐸3 = 𝐸 into consideration. However, we may notice that +the removal of edge (1, 3) leads to the collapse of node 5 while +none of nodes contained in this graph collapse after the removal +of edge (5, 7), which shows a substantial difference. Therefore, the +following theorem is presented to further narrow down the search +space of candidate edges. +Theorem 5. A given edge (𝑖, 𝑗) ∈ 𝐸 whose elimination could make +nodes within 𝐺 collapse requires both of the following two conditions +to be satisfied: (i)𝑚𝑖𝑛{𝐶𝑆(𝑖,𝐺),𝐶𝑆(𝑗,𝐺)} = 1; (ii) (𝐶𝑆(𝑖,𝐺) −𝐶𝑆(𝑗,𝐺)) · +(𝐶(𝑖,𝐺) − 𝐶(𝑗,𝐺)) ≥ 0. +Proof. Firstly, the condition (i) will not be satisfied if and only +if neither 𝐶𝑆(𝑖,𝐺) nor 𝐶𝑆(𝑗,𝐺) is equal to 1, i.e., 𝐶𝑆(𝑖,𝐺) ≥ 2 and +𝐶𝑆(𝑗,𝐺) ≥ 2. In such a case, node𝑖 and node 𝑗 satisfy that |𝑆𝑁(𝑖,𝐺) | ≥ +𝐶(𝑖,𝐺) +1 and |𝑆𝑁(𝑗,𝐺) | ≥ 𝐶(𝑗,𝐺) +1. The removal of edge (𝑖, 𝑗) will +absolutely not make node 𝑖 or node 𝑗 to violate Theorem 1. +Next, assume that the first condition has been satisfied, it might +be supposed that 𝐶𝑆(𝑖,𝐺) ≥ 𝐶𝑆(𝑗,𝐺) = 1 since edge (𝑖, 𝑗) is equiva- +lent to edge (𝑗,𝑖) in𝐺. For the core values of node𝑖 and node 𝑗, there +are three cases to consider, i.e., 𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺), 𝐶(𝑖,𝐺) < 𝐶(𝑗,𝐺) +and 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺). According to Theorem 3, the removal of edge +(𝑖, 𝑗) will surely make node 𝑗 collapse because of the violation of +Theorem 1 in the cases of𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺) and𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺) while +no node will collapse in the case of 𝐶𝑖,𝐺 < 𝐶(𝑗,𝐺). +□ +Combining the findings derived by Theorem 4 and Theorem 5, +for the targeted collapse mission of a given node 𝑖 with 𝐶(𝑖,𝐺) = 𝑘, +those edges existing in 𝐸𝑘\𝑘+1 and connecting to 𝑉𝐶 +(𝑘,𝐺) = {𝑢|𝑢 ∈ +𝑉,𝐶(𝑢,𝐺) = 𝑘 ∧𝐶𝑆(𝑢,𝐺) = 1} are what we should focus on and take +into candidates. Actually, nodes contained in 𝑉𝐶 +(𝑘,𝐺) are so-called +corona nodes of 𝐺𝑘 [4, 5, 52], which denotes that these nodes have +exactly 𝑘 one-hop neighbors in 𝐺𝑘. However, the subgraph con- +structed by corona nodes may not be connected and will probably +be divided into several disconnected components. As shown in Fig- +ure 1 where six corona nodes {1, 3, 4, 5, 9, 10} exist, the component +constructed by nodes {1, 3, 4} is disconnected with that constructed +by nodes {9, 10}, and so does that constructed by node {5}. There- +fore, for simplicity of representation, we provide the following +definition to represent the corona component in which a particular +corona node 𝑖 exists. +Definition 3. Corona Pedigree. For a corona node 𝑖 ∈ 𝐺 with +𝐶(𝑖,𝐺) = 𝑘, the corona pedigree of 𝑖, denoted as 𝑃(𝑖,𝐺), represents the +largest-connected subgraph containing 𝑖 as its component and satisfies +that ∀𝑗 ∈ 𝑃(𝑖,𝐺),𝐶(𝑗,𝐺) = 𝑘 ∧ 𝐶𝑆(𝑗,𝐺) = 1. +Example 3. As shown in Figure 1, there exist three corona pedigrees +in 𝐺3, e.g., 𝑃(4,𝐺) contains nodes {1, 3, 4} and edges {(1, 3), (1, 4)}, +𝑃(5,𝐺) contains node {5}, 𝑃(10,𝐺) contains nodes {9, 10} and edge +{(9, 10)}. +Note that 𝑃(𝑗,𝐺) is equivalent to 𝑃(𝑖,𝐺) if it satisfies that 𝑗 ∈ 𝑃(𝑖,𝐺). +Then, those edges adjacent to 𝑃(𝑖,𝐺) are represented as 𝐸𝑃 +(𝑖,𝐺) = +{(𝑢, 𝑣)|(𝑢, 𝑣) ∈ 𝐸,𝑢 ∈ 𝑃(𝑖,𝐺)∨𝑣 ∈ 𝑃(𝑖,𝐺)} and the following theorem +could be deduced. +Theorem 6. The removal of an arbitrary edge within 𝐸𝑃 +(𝑖,𝐺) will +absolutely make all nodes within 𝑃(𝑖,𝐺) collapse. +Proof. According to Definition 3, each node within 𝑃(𝑖,𝐺) pos- +sesses its core strength of 1 which means the disconnection of any +supportive neighbor will make this node collapse. Besides, each +edge within 𝐸𝑃 +(𝑖,𝐺) actually bridges some corona node within 𝑃(𝑖,𝐺) +with one of its supportive neighbors. In this way, if one of edges in +𝐸𝑃 +(𝑖,𝐺) is removed, the corona node (or corona nodes) adjacent to it +will surely collapse. Because of the cascade phenomenon, the other +nodes contained in 𝑃(𝑖,𝐺) will collapse follow. +□ +5 + +Example 4. As shown in Figure 2, taking 𝑃(1,𝐺) where nodes {1, 3} +exist as example, We get 𝐸𝑃 +(1,𝐺) = {(1, 2), (1, 3), (1, 5), (2, 3), (3, 7)}. +Node 3 will be absolutely squeezed out of 3-core after the removal of +an arbitrary edge contained in 𝐸𝑃 +(1,𝐺), like (2, 3), and then node 1 will +also collapse from 𝐺3 because of the cascade phenomenon. +4 +METHODOLOGIES +In this section, in order to address the TNCP problem, we propose +an effective heuristic algorithm called Targeted 𝑘-Node Collapse +(TNC) as the first solution. Additionally, based on TNC algorithm, +we design an optimized strategy called Adjacent Targeted 𝑘-Node +Collapse (ATNC) to further reduce computational complexity, mak- +ing it suitable for large-scale networks. +4.1 +TNC Algorithm +To solve the TNCP problem, we propose the TNC algorithm, which +itreatively removes one edge that can lead to the greatest impact +on the target node 𝑖 until the target node collapses. The impact +on the target node is determined by maximizing (i) the number of +collapsed nodes within 𝑆𝑁(𝑖,𝐺), and (ii) the number of nodes whose +core strengths change within 𝑆𝑁(𝑖,𝐺). As discussed earlier, those +edges existing in 𝐸𝑘\𝑘+1 and connecting to 𝑉𝐶 +(𝑘,𝐺) play significant +roles in the collapse of target node 𝑖 with 𝐶(𝑖,𝐺) = 𝑘. Then, ac- +cording to Theorem 6, for a corona node 𝑢 ∈ 𝑉𝐶 +(𝑘,𝐺) and its corona +pedigree 𝑃(𝑢,𝐺) ∈ 𝐺𝑘, it is easy to realize that the disconnection of +the relationship between node𝑢 and one of its supportive neighbors +will actually make all nodes within 𝑃(𝑢,𝐺) collapse from 𝐺𝑘 and +then make all edges within 𝐸𝑃 +(𝑢,𝐺) be excluded from 𝐸𝑘\𝑘+1. In this +manner, in order to avoid unnecessary duplicate operations, we +only need to select the corona pedigree 𝑃(𝑣,𝐺) whose detachment +leads to the greatest impact on the target node and removes one of +edges existing in 𝐸𝑃 +(𝑣,𝐺) in each iteration until the target node col- +lapses. Figure 3 illustrates the overall framework of TNC algorithm +along with the detailed operations shown in Algorithm 1. Words +for further descriptions are given as following. +As shown in Algorithm 1, Line 4, the corona nodes, 𝐶𝑜𝑟𝑜𝑛𝑎𝑠, +are firstly extracted from 𝐺′ +𝑘 as candidates where 𝐺′ is initialized +as 𝐺 in Line 1. After that, in Lines 6-7, by exploiting an assistant +algorithm called CalculateImpact which will be introduced in the +following paragraphs, the impact which will be made on the target +node 𝑖 is measured by 𝐹 [𝑢] and 𝐼 [𝑢] if corona node 𝑢 ∈ 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 +collapses, and𝐶𝑜𝑟𝑜𝑛𝑎𝑠 is updated according to Theorem 6. Next, we +select the top corona node 𝑣 sorted according to 𝐹 [·] (first priority) +and 𝐼 [·] (second priority) in Line 8. Then, one of edges contained +in 𝐸𝑃 +(𝑣,𝐺′) is added to 𝑒 with the update of 𝐺′ in Lines 13-14. The +above process will continue until there is the violation of Theorem +1 to make the target node 𝑖 collapse. Note that if the collapse of 𝑣 +makes no supportive neighbors of node 𝑖 collapse, we will remove +the edge bridging the target node and its supportive neighbor with +the minimal core strength in 𝐺′ as instead, in Lines 9-11. +CalculateImpact Algorithm. After the collapse of node𝑢 ∈ 𝑉 , +for all nodes in 𝐺, those nodes whose core strength decreases are +named as Influenced Nodes, those whose core value decreases are +named as Followed Nodes, and those whose core strengths and +core values remain the same are named as Uninfluenced Nodes. To +effectively measure the impact that the collapse of a corona node 𝑛 +can make on the target node 𝑖, we offer CalculateImpact algorithm +which is based on Depth-First Search (DFS) and whose details are +shown in Algorithm 2. +As shown in Algorithm 2, Line 1, S is defined to store the nodes +waiting to be visited, F and I are defined to store the followed +nodes and the influenced nodes, respectively. Besides, in Line 2, a +dictionary T with default value of 0 is defined to record the decrease +in the number of supportive neighbors of each node in 𝐺 after the +input node 𝑛 collapses. In this way, for a visited node 𝑢 popped +from S, if T [𝑢] > 0, it will be marked as an influenced node and +be added into I in Line 7; furthermore, if 𝐶𝑆(𝑢,𝐺) ≤ T [𝑢], it will +also be marked as a followed node and be added into F in Line 9. +Besides, if node 𝑢 has been marked as a followed node, on the basis +of Theorem 4, those nodes contained in � +𝑆𝑁 (𝑢,𝐺) and satisfying +𝐶𝑆(·,𝐺) > T [·] will be pushed into S in Line 10. Please note that +those nodes marked as followed nodes will be excluded from S in +Line 11. The above process will be repeated iteratively until S is +empty. +For example, contents shown in the dotted box of Figure 3 exhibit +the detailed process of Algorithm 2 where node 1 with 𝐶𝑆(1,𝐺) = 1 +is taken as the input node. First, in the initial-state graph, F and +I are initialized as empty sets and S = {1}. Next, in the second +graph, node 1 is popped from S with the update of T [1] = 1 and +be added into I. It is apparent that node 1 is also added into F +because of the satisfaction of 𝐶𝑆(1,𝐺) ≤ T [1], and its neighbors +{2, 5, 3} are pushed into S. After that, in the third graph, node 2 +with 𝐶𝑆(2,𝐺) = 2 is popped, and we get T [2] = 1 with the addition +of node 2 into I. Then, the next iteration will be triggered directly +because of 𝐶𝑆(2,𝐺) > T [2]. Continuing in this flow, we finally +achieve that I = {1, 2, 5, 3, 4, 7, 6} and F = {1, 3, 5, 2, 4}, and further +get that |N(5,𝐺) ∩ F | = 4 and |N(5,𝐺) ∩ I| = 5. +Time Complexity. As shown in Algorithm 1, first, in order to +extract the corona nodes 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 of 𝐺𝑘 from 𝐺, it takes the time +in the order of O(|𝑉 |) in Line 3. Then, from Line 5 to Line 7, one +corona node within each corona pedigree in 𝐺𝑘 is assigned weights +through 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝐼𝑚𝑝𝑎𝑐𝑡 algorithm which takes the time in the +order of O(𝐶𝑆𝑘\𝑘+1·|𝐶𝑜𝑟𝑜𝑛𝑎𝑠|) where𝐶𝑆𝑘\𝑘+1 = +� +𝑣∈𝑉𝑘 \𝑉𝑘+1 𝐶𝑆(𝑣,𝐺) +|𝑉𝑘\𝑉𝑘+1 | +. +After that, considering the worst condition, 𝐶𝑆(𝑖,𝐺) iterations are +executed with the total time complexity in the order of O(𝐶𝑆(𝑖,𝐺) · +(|𝑉 | + 𝐶𝑆𝑘\𝑘+1 · |𝐶𝑜𝑟𝑜𝑛𝑎𝑠|)). +4.2 +ATNC Algorithm +In the previous part, we give the introduction of TNC algorithm +which iteratively removes one edge that connected to the corona +pedigree whose detachment could cause the greatest impact on the +target node for addressing the TNCP problem. However, in each +iteration, TNC algorithm needs to traverse all nodes within 𝐺′ to +extract the corona nodes of 𝐺′ +𝑘 and then visit each corona pedigree +through CalculateImpact algorithm to filter out the most impacted +one. Clearly, the process is highly time-consuming for large-scale +networks which pushes the expectation of a heuristic algorithm +with less time complexity. In this part, we offer Adjacent Targeted +𝑘-Node Collapse (ATNC) improved from TNC which actually takes +the strategy of adjacent search to exploit the local information of +6 + +Delete one edge +from 𝑬(𝟏,𝑮) +𝑷 +Re-extract corona nodes +(No) +Extract +corona nodes +1 +3 +5 +6 +9 +8 +2 +4 +7 +1 +3 +5 +6 +9 +8 +2 +4 +7 +Take node 1 as input for example +If target node +has collapsed +(Yes) +Original Graph +Adversarial Graph +Target Node +1 +3 +9 +8 +4 +Node +F[·] +I[·] +4 +5 +4 +5 +2 +3 +2 +3 +1 +2 +1 +3 +8 +9 +4 +1 +3 +5 +6 +9 +8 +2 +4 +7 +Get F[·], I[·] by +CalculateImpact +Initial State +={}, ={}, +={1} +𝒖=1 +={1}, ={1}, +={2,5,3} +𝒖=2 +={1,2}, ={1}, +={5,3} +𝒖=5 +={1,2,5}, +={1}, +={3} +Terminated State +={1,2,5,3,4,7,6}, +={1,3,5,2,4}, +={} +𝒖=3 +={1,2,5,3}, +={1,3}, +={5,2} +𝒖=5 +={1,2,5,3}, +={1,3,5}, +={2,4,6,2} +𝒖=2 +={1,2,5,3}, +={1,3,5,2}, +={4,4,6} +𝒖=4 +={1,2,5,3,4}, +={1,3,5,2,4}, +={7,6} +𝒖=7 +={1,2,5,3,4,7}, +={1,3,5,2,4}, +={6} +Influenced Node +Followed Node +UnInfluenced Node +Sort corona nodes by F[·] and I[·] +Figure 3: The framework of TNC algorithm. Given a target node 5 with 𝐶𝑆(5,𝐺) = 3, we first extract the corona nodes from 𝐺3, +and then evaluate the impact of each corona node on the target node through CalculateImpact algorithm. After that, the most +impacted node 1 is filtered out and one edge existing in 𝐸𝑃 +(1,𝐺) is deleted. If the target node has collapsed, the adversarial graph +will be output and the removed edges will be returned; otherwise, we re-extract the corona nodes and repeat the above process. +The contents shown in the dotted box display the detailed operations of CalculateImpact algorithm. The detailed descriptions +will be presented in Section 4.1. +Algorithm 1: TNC +input :the given graph 𝐺, the target node 𝑖; +output:the removed edges 𝑒. +1 𝑒 ← empty set; 𝐺′ ← 𝐺; 𝑘 ← 𝐶(𝑖,𝐺); +2 𝐹, 𝐼 ← dictionaries with default value of 0; +3 while |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 do +4 +𝐶𝑜𝑟𝑜𝑛𝑎𝑠 ← {𝑢|𝑢 ∈ 𝐺′ +𝑘, N(𝑢,𝐺′ +𝑘) = 𝑘}; +5 +foreach 𝑢 ∈ 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 do +6 +𝐹 [𝑢], 𝐼 [𝑢] ← CalculateImpact(𝐺′,𝑖,𝑢); +7 +𝐶𝑜𝑟𝑜𝑛𝑎𝑠 ← 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 \ 𝑃(𝑢,𝐺′); +8 +𝑣 ← The top corona node sorted according to 𝐹 [·] (first +priority) and 𝐼 [·] (second priority); +9 +if 𝐹 [𝑣] = 0 then +10 +𝑚 ← The supportive neighbor of 𝑖 in 𝐺′ with the +lowest core strength; +11 +𝑒 ← 𝑒 ∪ {(𝑖,𝑚)}; +12 +else +13 +𝑒 ← 𝑒 ∪ {∀(𝑚,𝑛) ∈ 𝐸𝑃 +(𝑣,𝐺′)}; +14 +𝐺′ ← 𝐺 \ 𝑒; +15 return 𝑒 +the target node. The details of ATNC are shown in Algorithm 3 +along with its descriptions as following. +As shown in Algorithm 3, Line 3, instead of extracting all corona +nodes within 𝐺′ +𝑘 by TNC algorithm, ATNC only exploits those +corona nodes adjacent to the target node which are named as corona +neighbors𝐶𝑜𝑟𝑁𝑏𝑟𝑠. Next, in Lines 5-8, through the same operations +Algorithm 2: CalculateImpact +input :the given graph 𝐺, the target node 𝑖, the input node +𝑛; +output:the number of followed nodes in N(𝑖,𝐺), the +number of influenced nodes in N(𝑖,𝐺). +1 S ← empty stack; F ← empty set; I ← empty set; +2 T ← a dictionary with default value of 0; +3 S.𝑝𝑢𝑠ℎ(𝑛); +4 while S is not empty do +5 +𝑢 ← S.𝑝𝑜𝑝(); +6 +T [𝑢] ← T [𝑢] + 1; +7 +I ← I ∪ {𝑢}; +8 +if 𝐶𝑆(𝑢,𝐺) ≤ T [𝑢] then +9 +F ← F ∪ {𝑢}; +10 +V ← {𝑣|𝑣 ∈ � +𝑆𝑁 (𝑢,𝐺),𝐶𝑆(𝑣,𝐺) > T [𝑣]}; +11 +S.𝑝𝑢𝑠ℎ(V); +12 +S ← S \ F ; +13 return |N(𝑖,𝐺) ∩ F |, |N(𝑖,𝐺) ∩ I| +as those of TNC, the top corona node 𝑣 is filtered out. After that, we +add edge (𝑖, 𝑣) into 𝑒 with the update of 𝐺′ and the re-extraction +of 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 in Lines 9-10. The above process will continue until +𝐶𝑜𝑟𝑁𝑏𝑟𝑠 is empty or there is the violation of Theorem 1 for the +target node 𝑖. Note that if the above loop quits with 𝑆𝑁(𝑖,𝑘,𝐺′) > 𝑘 +which means that |𝐶𝑜𝑟𝑁𝑏𝑟𝑠| = 0 and the target node still remains +in 𝐺𝑘, then we will randomly sample 𝐶𝑆(𝑖,𝐺′) supportive neighbors +7 + +from 𝑆𝑁(𝑖,𝑘,𝐺′) and make the target node 𝑖 disconnected with them +in Lines 12-14. +Time Complexity. Similar to the time complexity of TNC, since +only the corona nodes existing in the one-hop neighbors of the +target node will be selected as candidates, the time for collecting the +candidates is in the order of O(|N(𝑖,𝐺) |) in Algorithm 3, Line 3 at +first. Then, from Line 5 to Line 7, each corona pedigree contained in +𝐺𝑘 is traversed with the quantification of their impact to the target +node which takes the time in the order of O(𝐶𝑆 (𝑘\𝑘+1) · |𝐶𝑜𝑟𝑁𝑏𝑟𝑠|). +After that, considering the worst condition, 𝐶𝑆(𝑖,𝐺) iterations are +executed with the total time complexity in the order of O(𝐶𝑆(𝑖,𝐺) · +(|𝑉 | + 𝐶𝑆𝑘\𝑘+1 · |𝐶𝑜𝑟𝑁𝑏𝑟𝑠|)). +Algorithm 3: ATNC +Input: the given graph 𝐺, the target node 𝑖; +Output: the removed edges 𝑒. +1 𝑒 ← empty set; 𝐺′ ← 𝐺; 𝑘 ← 𝐶(𝑖,𝐺); +2 𝐹, 𝐼 ← dictionaries with default value of 0; +3 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 ← {𝑗|𝑗 ∈ � +𝑆𝑁 (𝑖,𝐺′),𝐶𝑆(𝑗,𝐺′) = 1}; +4 while |𝐶𝑜𝑟𝑁𝑏𝑟𝑠| > 0 and |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 do +5 +foreach 𝑢 ∈ 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 do +6 +𝐹 [𝑢], 𝐼 [𝑢] ← CalculateImpact(𝐺′,𝑖,𝑢); +7 +𝐶𝑜𝑟𝑁𝑏𝑟𝑠 ← 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 \ 𝑃(𝑢,𝐺′); +8 +𝑣 ← The top corona node sorted according to 𝐹 [·] (first +priority) and 𝐼 [·] (second priority); +9 +𝑒 ← 𝑒 ∪ {(𝑖, 𝑣)}; +10 +𝐺′ ← 𝐺 \ 𝑒; +11 +Re-extract 𝐶𝑜𝑟𝑁𝑏𝑟𝑠; +12 if |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 then +13 +𝑒′ ← Sample({(𝑖, 𝑗)|𝑗 ∈ 𝑆𝑁(𝑖,𝐺′)},𝐶𝑆(𝑖,𝐺′)); +14 +𝑒 ← 𝑒 ∪ 𝑒′; +15 return 𝑒 +5 +EXPERIMENTS +In this section, our experiments will be conducted on 16 real-world +network datasets collected from various domains to demonstrate +the performance of TNC and ATNC. We also include 4 baseline +methods for comparisons. All of our experiments are deployed on +a server with Intel(R) Xeon(R) Gold 5218R CPU @ 2.10GHz and +377GB RAM, which installs Linux Ubuntu 20.04.4. +5.1 +Datasets +The basic properties of 16 real-world networks from various do- +mains, e.g., Social Network (SN), Collaboration Network (CN), In- +frastructure Network (IN) and Web Network (WN), are presented +in Table 2. Different labels are exploited to distinguish the different +public platforms where networks are collected. For example, those +marked with stars are collected from https://networkrepository. +com/ [38] and those marked with circles are collected from http: +//snap.stanford.edu/ [24]. Please note that all networks used in the +following experiments are converted to undirected and unweighted +graphs, with no self-loops or isolated nodes. Due to the space limita- +tion, more detailed information of these networks could be achieved +on the mentioned websites. +Table 2: Basic properties of mentioned networks containing +the number of nodes |𝑉 |, the number of edges |𝐸|, the maxi- +mal value of 𝑘-core 𝑘𝑚𝑎𝑥 and the average degree 𝑑𝑎𝑣𝑔. +Network +|𝑉 | +|𝐸| +𝑘𝑚𝑎𝑥 +𝑑𝑎𝑣𝑔 +SN +TVShow★ +3892 +17239 +56 +8.8587 +LastFM◦ +7624 +27806 +20 +7.2943 +Facebook◦ +22470 +170823 +56 +15.2045 +DeezerEU◦ +28281 +92752 +12 +6.5593 +Gowalla◦ +196591 +950327 +51 +9.6681 +CN +HepPh★ +12006 +118489 +238 +19.7383 +AstroPh★ +18771 +198050 +56 +21.1017 +CondMat★ +21363 +91286 +25 +8.5462 +Citeseer★ +227320 +814134 +86 +7.1629 +IN +USAir★ +332 +2126 +26 +12.8072 +USPower★ +4941 +6594 +5 +2.6691 +RoadNet★ +1965206 +2766607 +3 +2.8156 +WN +EDU★ +3031 +6474 +29 +4.2719 +Indo★ +11358 +47606 +49 +8.3828 +Arabic★ +163598 +1747269 +101 +21.3605 +Google◦ +875713 +4322051 +44 +9.8709 +5.2 +Baselines +Given that we are the first work to study the TNCP problem, there +is no ready-made method that can be used as a comparison ex- +periment. For this reason, we design two random-based baseline +methods and adjust two existing algorithms which are originally +proposed to solve the 𝑘-core minimization problem. Their details +are shown as follows. +• Random Edge Deletion (RED) arbitrarily selects an edge +within 𝐸 to remove and then updates the core values of nodes +within 𝑉 . These two steps will be performed iteratively until +the target node collapses successfully. +• Random Neighbor Disconnection (RND) arbitrarily removes +an edge connected to the target node and then updates the core +values of nodes within 𝑉 . These two steps will be performed +iteratively until the target node collapses successfully. +• KNM was proposed by [8] as a solution to the 𝑘-core mini- +mization problem. It works by iteratively removing the edge +whose detachment will lead to the maximal number of nodes +who collapse from 𝐺𝑘. This process continues until the pertur- +bation budget is reached or 𝐺𝑘 = ∅. In this paper, we adapt the +termination condition of KNM algorithm to the collapse of the +target node. +• SV was proposed by [34] for covering the 𝑘-core minimization +problem which exploits the shapley value, a cooperative game- +theoretic concept. It assigns weights to the candidate edges +and then chooses the top 𝑏 edges to remove. In this paper, +considering the consumption of time, we set 𝐸𝑘\𝑘+1 as the +candidate edges instead of 𝐸𝑘 which is originally used by [34], +and we set the hyperparameter 𝜖2 = 0.1. Then we remove +8 + +candidate edges one by one according to their weights until +the target node collapses without the budget limitation of 𝑏. +In order to evaluate the transferability not only among various +networks but also among various individual nodes, we will apply +all baseline methods as well as our proposed algorithms on each +node within every network to achieve its node robustness. Then +we will evaluate the effectiveness of these algorithms by several +global metrics which will be introduced in Section 5.3. Additionally, +it is necessary to be noted that both of RED and RND will be +performed 10 times independently on each node in order to reduce +the randomness and the mean value is recorded as the robustness +of each node. +5.3 +Metrics +We propose the following metrics, Number of Bubble Nodes (NBN), +Sum of Reduced Cost (SRC), Weighted Average Reduction (WAR), and +Reduction Proportion (RP) to evaluate the effectiveness of various +methods. +• NBN: Through a particular algorithm, we are interested in how +many bubble nodes can be explored from 𝐺. Thus, the total +number of explored bubble nodes is recorded as NBN which is +formulated as below: +NBN = |𝐵𝑁 |. +(4) +The higher NBN is, the more transferable the algorithm is +among various nodes in a graph. +• SRC: For a bubble node𝑖, the decrease between its core strength +and node robustness is named as Reduced Cost which is quan- +tified as 𝑅𝐶(𝑖,𝐺) = 𝐶𝑆(𝑖,𝐺) − 𝑁𝑅(𝑖,𝐺). Therefore, the sum of +reduced cost of all explored bubble nodes in 𝐺 could be formu- +lated as below: +SRC = +∑︁ +𝑖 ∈𝐵𝑁 +𝑅𝐶(𝑖,𝐺). +(5) +• WAR: In order to illustrate the average cost reduction of ex- +plored bubble nodes in a network through some algorithm, we +propose WAR which is formulated as below: +WAR = +� +𝑟 ∈U +𝑝−1 +𝑟 +· 𝑟 +� +𝑟 ∈U +𝑝−1 +𝑟 +. +(6) +where U contains the unique elements of {𝑅𝐶(𝑖,𝐺) |𝑖 ∈ 𝐵𝑁 } +and 𝑝𝑟 = |{𝑖 |𝑖 ∈𝐵𝑁,𝑅𝐶(𝑖,𝐺)=𝑟 }| +|𝐵𝑁 | +denotes the probability of those +nodes whose reduced cost equal to 𝑟 appearing in 𝐵𝑁. And the +reason why we do not use arithmetic average will be explained +in Section 6.1 with specific examples. +• RP: We are also interested in the reduction proportion of node +robustness relative to core strength on all nodes in 𝐺 and pro- +pose RP for measuring, which is formulated as below: +RP = +SRC +� +𝑖 ∈𝑉 𝐶𝑆(𝑖,𝐺) +× 100%. +(7) +In addition, RP can be used to describe the redundancy of core +strength with respect to node robustness. The higher the RP, +the more redundant the core strength. +Table 3: The experimental results of RED and RND. Those +datasets that could not be covered within 105 seconds are +marked as /. Attention that for RED and RND, the robust- +ness of each node is assigned as the average of the results +achieved by 10 independent experiments. +Network +RED +RND +NBN SRC WAR RP(%) NBN +SRC +WAR RP(%) +TVShow +8 +1.8 +0.2 +0.02 +375 +204.9 +1.6894 +2.75 +LastFM +23 +20.1 1.2346 +0.15 +263 +350.4 +5.2813 +2.7 +Facebook +9 +4.7 +0.4083 +0.01 +1346 +1022.5 +4.3603 +2.0 +DeezerEU +1 +0.4 +0.4 +0.001 +648 +324.8 +2.3962 +0.62 +Gowalla +/ +/ +/ +/ +5215 +3015.2 +4.1356 +1.02 +HepPh +159 308.9 5.4106 +1.51 +1181 +2773.4 13.3372 13.59 +AstroPh +94 +90.2 2.1622 +0.24 +2234 +4337.7 10.7611 11.48 +CondMat +13 +2.4 +0.2441 0.007 +2439 +2209.4 +4.3676 +6.05 +Citeseer +/ +/ +/ +/ +23285 20848.6 12.4333 6.37 +USAir +2 +0.3 +0.15 +0.05 +10 +3.9 +0.4714 +0.66 +USPower +1 +0.4 +0.4 +0.005 +97 +50.2 +0.9693 +0.62 +RoadNet +/ +/ +/ +/ +/ +/ +/ +/ +EDU +3 +0.4 +0.1667 0.005 +14 +11.9 +0.9455 +0.16 +Indo +20 +3.7 +0.7049 +0.02 +754 +703.4 +3.8977 +4.68 +Arabic +9 +1.8 +0.3032 0.0009 4510 +5086.6 +6.9982 +2.42 +Google +/ +/ +/ +/ +53222 50569.3 18.2091 +3.1 +6 +RESULTS AND ANALYSES +The experimental results are exhibited in Table 3 and Table 4, which +contrastively shows the performance of TNC and ATNC, compared +with 4 baseline methods on 16 real-world networks mentioned +before. Meanwhile, the detailed comparisons and analyses are pre- +sented as follows. +6.1 +Performance Evaluation +Comparisons Among Baselines. Let us concentrate on Table 3 +in which the experimental results of RED and RND are exhibited. +Notice that the robustness of each node in networks is achieved +by the average of 10 independent experimental results. From the +table, it is easy to find that the NBN of RED is far fewer than that +of RND on all networks which represents that RED is unable to +explore bubble nodes and fails to cover the TNCP problem. On +the contrary, RND performs much better than RED on all used +metrics which demonstrates that the strategy of adjacent search +for candidate reduction is helpful for covering the TNCP problem. +Additionally, we can realize that RED is not able to complete the +search missions on the 4 networks whose number of nodes is more +than 105, i.e. Gowalla, Citeseer, RoadNet and Google, while RND +only fails on RoadNet, a network with millions of nodes. It also +proves that the strategy of adjacent search can effectively reduce +the time complexity of the algorithm. +After that, let us turn our sights to the experimental results +achieved by KNM and SV which are illustrated in Table 4. Neither +KNM nor SV displays powerful transferability among different +nodes in a network compared to RND. For instance, RND detects +2439 bubble nodes on CondMat network, whereas this number is +259 and 184 induced by KNM and SV, respectively, which reveals +9 + +Table 4: The experiment results of KNM, SV, TNC and ATNC. Those datasets that could not be completed by the method within +105 seconds are marked as /. The best results are bolded and the second-best results are underlined. +Network +KNM +SV +TNC +ATNC +NBN +SRC +WAR +RP(%) NBN +SRC +WAR +RP(%) +NBN +SRC +WAR +RP(%) +NBN +SRC +WAR +RP(%) +TVShow +188 +502 +8.8894 +6.73 +142 +403 +8.0841 +5.40 +543 +1104 +10.4244 14.80 +514 +1055 +10.4196 +14.14 +LastFM +201 +1025 17.0836 +7.90 +164 +895 +16.4166 +6.90 +458 +1652 +17.7163 12.74 +404 +1485 +17.6044 +11.45 +Facebook +828 +5335 35.8553 +10.42 +618 +3040 28.4149 +5.90 +2709 11691 40.5852 22.83 +2468 +10033 +37.4845 +19.59 +DeezerEU +182 +606 +11.3760 +1.15 +94 +303 +9.139 +0.58 +1162 +2185 +15.4195 +4.15 +1062 +1967 +15.1067 +3.74 +Gowalla +/ +/ +/ +/ +828 +9068 58.8298 +3.06 +/ +/ +/ +/ +7719 +22864 +62.2077 +7.72 +HepPh +562 +3664 31.4897 +17.96 +481 +3244 31.4435 +15.90 +1381 +5142 +31.9080 25.19 +1332 +5026 +31.7788 +24.63 +AstroPh +1276 7702 30.2089 +20.38 +816 +5191 29.0587 +13.73 +2984 12637 32.9459 33.43 +2727 +11382 +31.6881 +30.11 +CondMat +259 +797 +12.2096 +2.18 +184 +574 +10.3739 +1.57 +3008 +6760 +13.1719 18.50 +2801 +6231 +13.0097 +17.05 +Citeseer +/ +/ +/ +/ +514 +1884 +18.377 +0.58 +/ +/ +/ +/ +25244 +50203 +29.3033 15.34 +USAir +30 +111 +3.8897 +18.91 +21 +59 +2.6262 +10.05 +36 +120 +3.9747 +20.44 +36 +120 +3.9747 +20.44 +USPower +21 +27 +2.7141 +0.33 +16 +20 +2.5852 +0.25 +109 +130 +2.9294 +1.60 +109 +127 +2.9212 +1.56 +RoadNet +/ +/ +/ +/ +19 +21 +2.8945 +0.0006 +/ +/ +/ +/ +4105 +4175 +2.8658 +0.11 +EDU +11 +16 +2.5805 +0.21 +12 +16 +2.5516 +0.21 +59 +67 +2.8268 +0.89 +59 +67 +2.8268 +0.89 +Indo +73 +157 +9.4433 +1.04 +69 +140 +9.1799 +0.93 +779 +1212 +12.4205 +8.06 +772 +1177 +12.6278 +7.83 +Arabic +354 +550 +7.3287 +0.26 +214 +486 +8.8685 +0.23 +5088 +8232 +14.4142 +3.92 +5001 +8049 +14.423 +3.83 +Google +/ +/ +/ +/ +868 +4309 35.6256 +0.26 +/ +/ +/ +/ +77928 222404 75.7784 13.63 +TVShow +LastFM Facebook DeezerEU +HepPh +AstroPh CondMat +USAir +USPower +EDU +Indo +Arabic +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Time Consumption (seconds) +TNC +KNM +SV +ATNC +Figure 4: Comparisons of the running time of TNC, KNM, SV and ATNC. The subfigure exhibits the running time of SV and +ATNC on those large-scale networks separately. We can see that ATNC is significantly more efficient than the other methods +on the time consumption. +a difference of almost 10 times. Similarly, RND is able to filter out +4510 bubble nodes on Arabic network, while KNM and SV could +only find 354 and 214 nodes. However, the other metrics, i.e., SRC, +WAR and RP, are much higher for KNM and SV compared to those +for RND. For example, on Facebook network, the SRC of KNM is 5 +times larger than that of RND and on AstroPh network, the WAR +of KNM is 3 times larger than that of RND. These results tell us that +the heuristic methods enable the target node to collapse at a lower +budget compared to the random-based methods, although they can +only work on part of bubble nodes. Analysis from the principle +of these two algorithms, neither of them exploits the information +associated with the target node to guide the removal of edges which +leads to the unsatisfied performance on solving the TNCP problem. +Benefits of Our Proposed Methods. Next, turning to the re- +sults generated by TNC and ATNC shown in Table 4, TNC and +ATNC achieve the best and second-best performance on majority +of the datasets with significant benefits over KNM and SV. For ex- +ample, on CondMat network, only 259 and 184 bubble nodes could +be detected through KNM and SV, respectively, while there are +3008 and 2801 bubble nodes found by TNC and ATNC, respectively, +resulting in a difference of more than 10-fold between the two sides. +This definitely demonstrates that in the comparison to KNM and +SV, TNC and ATNC have stronger transferability across different +nodes and different networks. Besides, our proposed algorithms +also perform better than KNM and SV considering SRC, WAR and +RP metrics. However, we notice that the WAR of SV is a little larger +10 + +10 +10 +Google +Gowalla +Citeseer +RoadNet0 +10 +20 +Number of Deleted Edges +10 +20 +30 +Number of Support Neighbors + k=11, CS=27, NR=17 +(a) LastFM, Node 6101 +0 +10 +20 +Number of Deleted Edges +10 +20 +30 +Number of Support Neighbors + k=11, CS=23, NR=14 +(b) LastFM, Node 3103 +0 +5 +10 +15 +20 +Number of Deleted Edges +10 +15 +20 +25 +30 +Number of Support Neighbors + k=10, CS=21, NR=17 +(c) DeezerEU, Node 17963 +0 +10 +20 +30 +40 +Number of Deleted Edges +10 +20 +30 +40 +Number of Support Neighbors + k=8, CS=40, NR=12 +(d) DeezerEU, Node 24062 +0 +5 +10 +15 +Number of Deleted Edges +5 +10 +15 +20 +25 +Number of Support Neighbors + k=9, CS=18, NR=6 +(e) CondMat, Node 1233 +0 +5 +10 +15 +Number of Deleted Edges +10 +15 +20 +25 +Number of Support Neighbors + k=8, CS=18, NR=8 +(f) CondMat, Node 13621 +0 +1 +2 +3 +4 +Number of Deleted Edges +4 +6 +8 +10 +12 +Number of Support Neighbors + k=9, CS=4, NR=1 +(g) Indo, Node 2721 +0 +5 +10 +15 +Number of Deleted Edges +5 +10 +15 +20 +Number of Support Neighbors + k=5, CS=18, NR=10 +(h) Indo, Node 4712 +TNC +KNM +SV +ATNC +Figure 5: Case study on individual nodes from 4 mentioned networks operated by TNC, KNM, SV and ATNC. Each method is +marked with a unique label and the red dotted line in each subfigure indicates the critical value of the number of supportive +neighbors for current target node. +than that of ATNC on RoadNet. After the observation of the bubble +nodes found by SV and ATNC, there exists the situation that among +the 19 bubble nodes detected by SV, 1 node has reduced cost of 3 +and 18 nodes has reduced cost of 1; while for the 4105 bubble nodes +detected by ATNC, there are 8 nodes with reduced cost of 3, 54 +nodes with 2 and even 4043 nodes with 1. In the calculation of WAR +for ATNC, the bubble nodes with reduced cost of 1, which make +up nearly 98% of the total, surely have a significant diluting impact +on the final result. Actually, the existence of bubble nodes with +low reduced cost is common in the other networks. For instance, +on Facebook network, about 65% of the bubble nodes detected by +ATNC have their reduced cost less than 4 while there are 30 nodes +with reduced cost larger than 30, and on Indo network, 90% of the +bubble nodes detected by ATNC have their reduced cost less than +3 with 4 nodes whose reduced cost larger than 10. This is why we +use weighted averaging instead of arithmetic averaging to quantify +the average reduced cost of each bubble node in the network. +Comparison between TNC and ATNC. Reviewing what is +discussed in Section 4, it is easy to be realized that the candidates +waiting to be filtered of ATNC is a subset of those of TNC. Unsur- +prisingly, considering the comparison between TNC and ATNC in +Table 4, the performance of TNC is better than that of ATNC on +the majority of networks. We also notice that on Indo and Arabic, +the WAR of ATNC is slightly higher than that of TNC while the +other metrics of ATNC are less than those of TNC. Taking Indo as +example for analysis, we find that there are 3 nodes with reduced +cost of 8 among the 779 bubble detected by TNC while none of +these nodes with reduced cost of 8 explored by ATNC. This situ- +ation leads to an unfair weighting process of TNC compared to +ATNC in the calculation of WAR and causes the slight difference in +the final results. For the similar reason, the slight variations in the +number of bubble nodes with high reduced cost lead to the differ- +ence in the final result of WAR. However, the performance of TNC +is completely superior to that of ATNC on the whole. Besides, it is +easy to find that TNC is not suitable for those large-scale networks, +e.g., Gowalla, Citeseer, RoadNet and Google, due to the huge size +of the candidates. On the contrary, ATNC is able to complete these +tasks and receives appreciable results. The detailed comparisons of +efficiency will be discussed in the following contents. +Redundancy of Core Strength Metric. As introduced before, +the RP metric measures the redundancy of core strength with re- +spect to node robustness. As mentioned in Section 3.1, we have +shown that the core strength metric does not accurately quantify +the number of necessarily removed edges for making the target +𝑘-node collapse. From the results of ATNC in Table 4, there are +more than half of the networks whose RP is larger than 10% and +even part of them owning RP larger than 20%. For example, the +RP of Facebook is nearby 20% and the RP of AstroPh is more than +30%. These results undoubtedly demonstrate that the core strength +metric is not suitable for measuring the least number of edges to +remove for leading the collapse to a target node. +Efficiency of Different Methods. The visualization for the +time consumption of implementing KNM, SV, TNC and ATNC +across all the mentioned networks is illustrated in Figure 4. Overall, +we can find that TNC and KNM have similar performance since they +both traverse all corona nodes for edge removal in each iteration. +Then, we can find that SV performs better than KNM and TNC +on most of the networks except for HepPh network and USAir +11 + +0 +PRC +KNM +? +SV +...APRC.network. For HepPh network, its maximal core value 𝑘𝑚𝑎𝑥 = 238 +is much higher than that of the other networks which is up to +101. For USAir network, its size if much smaller than the others +and causes the operations of SV are much more time-consuming +than those of KNM and TNC. Besides, ATNC occupies the best +efficiency with significant time-consumption reduction compared +to the other methods. For example, on DeezerEU network, the time +consumption of SV method is about 10 times larger than that of +ARPC and the time consumption of TNC is even more than 100 +times larger than that of ATNC. And for large-scale networks, e.g., +Gowalla, Citeseer, RoadNet and Google, neither TNC nor KNM +can calculate the robustness for each node in those networks in +the limitation of 105 seconds, e.g., TNC even fails to complete the +calculation of 0.1% of total nodes on Google network within 105 +seconds, while ATNC is able to cover the task in an appreciable +amount of time. +In a word, our proposed methods TNC and ATNC have signifi- +cant advantages over the other baseline methods. And considering +the much lower time complexity of ATNC compared to TNC, ATNC +is more suitable to be deployed on large-scale networks for solving +TNCP problem, although the effect of TNC is slightly better than +that of ATNC. +6.2 +Case Study +In the previous section, we provide the performance of different +methods from a macroscopic perspective. Here, in this part, we +offer a microscopic point of view as a case study. We visualize the +variation in the number of supportive neighbors of the target node +when the implementation is processing. As illustrated in Figure 5, +8 individual target nodes collected from 4 of the mentioned net- +works are visualized. In each subfigure, the horizontal coordinate +indicates the number of removed edges during the process, the +vertical coordinate indicates the number of remaining supportive +neighbors of the target node after the removal. Different imple- +mented methods are marked with different labels. Meanwhile, the +red dotted line in each subfigure represents the critical number +of supportive neighbors for the target node which is equal to its +core value. The collapse of the target node happens when the curve +drops below the red dotted line since the violation of Theorem 1. +From the examples, it is clear that fewer removed edges is needed +through TNC and ATNC compared to those of KNM and SV, and +TNC is able to remove fewer edges than ATNC in some cases. +7 +APPLICATION +Currently, 𝑘-core has been widely used in numerous downstream +tasks, e.g., anomaly detection [41, 42], community detection [37], +detection of influential spreaders [6, 19, 28, 29], etc. Laishram et +al.[23] demonstrated that the performance of those downstream +tasks is highly relative to the resilience of the 𝑘-core structure in a +network. They proposed a heuristic metric named CIS whose cal- +culation is based on the core strength metric. They indicated that +the resilience of 𝑘-core is positively correlated with CIS. However, +as mentioned before, we have demonstrated that the core strength +metric is highly redundant for measuring the robustness of indi- +vidual 𝑘-nodes in real-world networks. Thus, the CIS calculated +from core strength, named as CS-based CIS, probably overestimates +TVShow +LastFM +Facebook +DeezerEU +Gowalla +HepPh +AstroPh +CondMat +Citeseer +USAir +USPower +RoadNet EDU +Indo +Arabic +Google +0 +2 +4 +6 +8 +CIS +CS-based +NR-based +Figure 6: Comparisons of CS-based CIS and NR-based CIS +on all mentioned networks. It is clear that, on most net- +works, NR-based CIS is able to measure the resilience of 𝑘- +core more precisely than CS-based CIS. +the resilience of 𝑘-core structures in a network. For the above rea- +sons, we replace core strength with the node robustness achieved +by ATNC algorithm in the calculation of CIS, which is named as +NR-based CIS. The results of CS-based CIS and NR-based CIS on +real-world networks are shown in Figure 6. It is clear that on most +networks, NR-based CIS is much smaller than CS-based CIS and is +able to precisely measure the resilience of the 𝑘-core in a network. +Besides, combining the information illustrated in Table 4, we can +find that the difference between NR-based CIS and CS-based CIS is +proportional to the RP metric, e.g., on Facebook network, ATNC +provides RP=19.59% and there is a two-fold difference between CS- +based CIS and NR-based CIS; while the difference on EDU network +who receives RP=0.89% by ATNC is negligible. From this, it is clear +that the node robustness metric has better performance, compared +to the core strength metric, in precisely describing the resilience of +𝑘-core structures in networks. +8 +CONCLUSION +In this paper, we engage in the first work on studying the robustness +of individual nodes within 𝑘-core. We propose the TNCP problem, +which aims to remove the minimal number of edges for making the +target node collapse, and we also provide a proof of its NP-hardness. +In order to solve TNCP problem, we propose two heuristic algo- +rithms including TNC algorithm which exploits corona nodes to +improve search efficiency, and ATNC algorithm which introduces +adjacent-search strategy to further lower down computational com- +plexity on large-scale networks. Extensive experimental results +on various real-world networks, together with thorough analyses, +demonstrate the superiority of our proposed methods over the +baseline methods. Meanwhile, we offer the detailed processes of +different algorithms being implemented on various target nodes for +case study. Finally, we demonstrate that studying TNCP problem is +helpful for precisely estimating the resilience of 𝑘-core in networks. +ACKNOWLEDGMENTS +This work was supported in part by the Key R&D Program of +Zhejiang under Grant 2022C01018, by the National Natural Science +Foundation of China under Grants 61973273 and U21B2001, by the +National Key R&D Program of China under Grant 2020YFB1006104, +and by The Major Key Project of PCL under Grants PCL2022A03, +PCL2021A02, and PCL2021A09. +12 + +REFERENCES +[1] Abhijin Adiga and Anil Kumar S Vullikanti. 2013. How robust is the core of +a network?. In Joint European Conference on Machine Learning and Knowledge +Discovery in Databases. Springer, Berlin, Heidelberg, 541–556. +[2] DO Akinyele and RK Rayudu. 2014. Review of energy storage technologies for +sustainable power networks. 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IEEE Transactions on Computational +Social Systems (2022), 1–15. +13 + diff --git a/J9AyT4oBgHgl3EQfTffD/content/tmp_files/load_file.txt b/J9AyT4oBgHgl3EQfTffD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..629467292ffd19a2fb0def54cc13fc465a770600 --- /dev/null +++ b/J9AyT4oBgHgl3EQfTffD/content/tmp_files/load_file.txt @@ -0,0 +1,1127 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf,len=1126 +page_content='Targeted 𝑘-node Collapse Problem: Towards Understanding the Robustness of Local 𝑘-core Structure Yuqian Lv Zhejiang University of Technology lvyuqian_email@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='com Bo Zhou Zhejiang University of Technology wxjs201@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='com Jinhuan Wang Zhejiang University of Technology jhwang@zjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='cn Qi Xuan Zhejiang University of Technology xuanqi@zjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='cn ABSTRACT The concept of𝑘-core, which indicates the largest induced subgraph where each node has 𝑘 or more neighbors, plays a significant role in measuring the cohesiveness and the engagement of a network, and it is exploited in diverse applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', network analysis, anomaly detection, community detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Recent works have demonstrated the vulnerability of 𝑘-core under malicious pertur- bations which focuses on removing the minimal number of edges to make a whole 𝑘-core structure collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, to the best of our knowledge, there is no existing research concentrating on how many edges should be removed at least to make an arbitrary node in 𝑘-core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, in this paper, we make the first at- tempt to study the Targeted 𝑘-node Collapse Problem (TNCP) with four novel contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Firstly, we offer the general definition of TNCP problem with the proof of its NP-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Secondly, in order to address the TNCP problem, we propose a heuristic algorithm named TNC and its improved version named ATNC for implemen- tations on large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, the experiments on 16 real-world networks across various domains verify the superiority of our proposed algorithms over 4 baseline methods along with de- tailed comparisons and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Finally, the significance of TNCP problem for precisely evaluating the resilience of 𝑘-core structures in networks is validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' PVLDB Reference Format: Yuqian Lv, Bo Zhou, Jinhuan Wang, and Qi Xuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Targeted 𝑘-node Collapse Problem: Towards Understanding the Robustness of Local 𝑘-core Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' PVLDB, 16(1): XXX-XXX, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' doi:XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='XX/XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='XX PVLDB Artifact Availability: The source code, data, and/or other artifacts have been made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='com/Yocenly/TNCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 1 INTRODUCTION Networks or graphs play significant roles in describing various complex systems from numerous domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', social networks[11, 27, 31, 43], citation networks[16, 26], biological networks[11, 22, 47] This work is licensed under the Creative Commons BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Visit https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='org/licenses/by-nc-nd/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='0/ to view a copy of this license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For any use beyond those covered by this license, obtain permission by emailing info@vldb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Copyright is held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Publication rights licensed to the VLDB Endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proceedings of the VLDB Endowment, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 16, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 1 ISSN 2150-8097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' doi:XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='XX/XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='XX 1 3 2 6 8 4 5 7 9 b a c d 1 3 4 6 7 10 9 2 5 8 11 12 13 14 3-core 2-core 15 16 18 17 1-core 2-node 3-node 1-node Figure 1: An example of 𝑘-core distribution in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Each subgraph in the dotted box with a certain color represents the 𝑘-core and the color of each node represents its core value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' and power networks[2, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, understanding the topolog- ical information of graphs is what matters in the study of graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Due to the advantages of simplicity and efficiency [20], the concept of 𝑘-core, which denotes the maximal induced subgraph where each node within it occupies at least 𝑘 neighbors [9], has stood out as an important metric for describing the global struc- tural engagement of networks from massive evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Figure 1, an example graph with 18 nodes and 28 edges is given where 3 cores exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', 1-core, 2-core and 3-core which are surrounded by dotted boxes with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As more and more researchers devoted themselves to the study of 𝑘-core, 𝑘-core has been used in a broad variety of important ap- plications [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, in ecological networks, Morone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [35] exploited the 𝑘-core as a predictor to estimate the structural collapse in mutualistic ecosystems, and Burleson-Lesser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [7] presented a new approach for characterizing the stability and ro- bustness of networks with all-positive interactions by studying the distribution of the 𝑘-core of the underlying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, in social networks, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [44] considered the pruning process of 𝑘-core to measure the vulnerability and resilience of social engage- ment and further studied its equilibrium statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' And in biological networks, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [30] studied the core structure of protein-protein interactions networks with an interesting discovery that core structures help to reveal the existence of multiple levels of protein expression dynamics, and Isaac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [17] discovered that arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='00108v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='SI] 31 Dec 2022 residues belonging to inner cores are more conserved than those at the periphery of the network with the evidence that these groups are functionally and structurally critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' With the rapidly increasing number of applications based on 𝑘-core structures, the robustness (or also called resilience) of 𝑘-core have gradually attracted the attention of researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For instance, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [51] studied the robustness of 𝑘-shell, a subset of 𝑘-core, and demonstrated that 𝑘-shell is vulnerable under the disturbance of edge rewiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Their optimal-based experimental results showed that the 𝑘-core distributions of graphs can be drastically changed even a small proportion of edges are rewired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [52] also studied the minimal budgets of removed edges for the collapse of the innermost 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' They provided a proof of its NP-hardness and offered effective heuristic algorithms to cover this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Furthermore, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [8] focused on the 𝑘-core minimization problem and suggested three sub-problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', KNM, KEM and KCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' They further proposed several heuristic algorithms through edge removal to cover these sub-problems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Medya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [34] also concentrated on the 𝑘-core minimization problem and proposed a novel algorithm inspired by shapley value, a cooperative game-theoretic concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Their algorithm could leverage the strong interdependencies in the effects of edges removal in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [49] studied the collapsed𝑘-core problem which aims to find a set of nodes whose detachment will lead to the minimal size of the resulting collapsed 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, as we can see, throughout the previous works, all of them considered the 𝑘-core as a whole to evaluate its robustness, while none of them focused on the robustness of an individual node within 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Thus it brings us a question that how many edges should we disconnect at least to make an arbitrary node contained in 𝑘-core collapse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As far as we know, there is no existing work dedicating to the study of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this paper, we make the first attempt to study this problem and name it as Targeted 𝑘-node Collapse Problem (TNCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Our main contributions can be summarized as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We offer a general definition of TNCP problem with a proof of its NP-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We demonstrate that the naive exhaustive method will lead to the exponential explosion of the time com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, a series of theorems are provided to narrow down the search space of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Combined with the theorems, we propose a heuristic algorithm named TNC to cover the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, we find that TNC algorithm is not suitable for large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Thus, an improved algorithm named ATNC with less time complexity is proposed based on TNC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We verify the superiority of our proposed algorithms over 4 baseline methods through experiments on 16 real-world net- works collected from different public platforms along with detailed comparisons and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We demonstrate that the research of TNCP problem is helpful for precisely evaluating the resilience of the 𝑘-core structures in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The remaining sections of this paper are structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 2, a brief review on the previous works about 𝑘-core is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 3, the statement of TNCP problem and basic definitions, which will be used in the rest of this paper, are intro- duced along with the theorems for candidate reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 4, we introduce our proposed methods TNC and ATNC with their time complexity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 5, we give the introductions about the datasets being used, the baseline methods for compar- isons and the metrics for evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 6, experimental results on all mentioned datasets are shown along with detailed comparisons and analyses between our proposed algorithms and the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Section 7, the significance of TNCP prob- lem for precisely evaluating the resilience of 𝑘-core in networks is validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Finally, our work is concluded in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 2 RELATED WORKS The researches on the 𝑘-core structure of networks have been enduring, and those most related to our work are introduced as below, including core decomposition, core robustness/resilience, and core percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Core Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Hajnal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [14] gave the first 𝑘-core related concept and defined the degeneracy of a graph as the maxi- mum core number of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, Seidman [40], as well as Matula and Beck [33], defined the 𝑘-core subgraph as the maximal con- nected subgraph where each node has at least 𝑘 neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Khaouid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [18] explored whether𝑘-core decomposition of large networks can be computed using a consumer-grade PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Sariyüce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [39] proposed the first incremental 𝑘-core decomposition algorithms for streaming graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Hébert-Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [15] proposed onion decomposition which is derived from 𝑘-core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Eidsaa and Almaas [10] presented 𝑠-core analysis, a generalization of 𝑘-core analysis, for weighted networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Core Robustness/Resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In addition to works mentioned in the last section, Adiga and Vullikanti [1] examined the robustness of the top core sets in perturbed/sampled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Zdeborová et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [48] used 𝑘-core as a heuristic tool in the process of graph decycling and dismantling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Laishram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [23] proposed metrics for measuring the core resilience of a network under the situations of node/edge removals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Core Percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Azimi-Tafreshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [3] generalized the the- ory of 𝑘-core percolation on complex networks to k-core percola- tion on multiplex networks, where k = (𝑘𝑎,𝑘𝑏, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Whi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [46] revealed the hierarchical structure of functional connectivity on resting-state fMRI (rsfMRI) through the method of 𝑘-core perco- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [45] proposed a generalized 𝑘-core percolation model to investigate the robustness of the higher-order dependent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [50] studied the robustness of multiplex net- works with interdependent and interconnected links under 𝑘-core percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [13] applied 𝑘-core percolation analysis on brain structural network, suggesting that the brain networks are mostly reliable against random or 𝑘-core-based percolation with their structure design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3 PROBLEM STATEMENT In this section, the descriptions of commonly used definitions and fundamental concepts will be discussed in the following contents along with the statement of TNCP problem and the proofs of our proposed theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 Preliminaries In this paper, a network or a graph (these two concepts will be used indiscriminately) is indicated as 𝐺 = (𝑉, 𝐸), where 𝑉 and 𝐸 ⊆ (𝑉 × 𝑉 ) represent the sets of nodes and edges respectively, which are extracted from real-world entities and the relationships between any pair of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As a prerequisite, we only focus on those unweighted and undirected graphs without self-loops or isolated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Here, we present some fundamental definitions and related concepts which are relevant to the subsequent discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In Table 1, we compile a list of principal symbols and notations for convenient query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Table 1: Summary of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Notation Definition 𝐺𝑘 the 𝑘-core subgraph of 𝐺 𝑑(𝑖,𝐺𝑘) the degree of node 𝑖 in 𝐺𝑘 𝐶(𝑖,𝐺) the core value of node 𝑖 𝑆𝑁(𝑖,𝑘,𝐺) the supportive neighbors of the node 𝑖 in 𝐺𝑘 𝑆𝑁(𝑖,𝐺) the simplification of 𝑆𝑁(𝑖,𝐶(𝑖,𝐺),𝐺) N(𝑖,𝐺𝑘) the one-hop neighbors of node 𝑖 in 𝐺𝑘 𝐶𝑆(𝑖,𝐺) core strength of node 𝑖 𝑁𝑅(𝑖,𝐺) node robustness of node 𝑖 𝑃(𝑖,𝐺) the corona pedigree of node 𝑖 𝐸𝑃 (𝑖,𝐺) those edges connected with nodes in 𝑃(𝑖,𝐺) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For a given graph 𝐺, its 𝑘-core, denoted as 𝐺𝑘 = (𝑉𝑘, 𝐸𝑘) where 𝑉𝑘 ⊆ 𝑉 and 𝐸𝑘 ⊆ 𝐸, means the maximal induced subgraph whose nodes occupy at least 𝑘 neighbors within 𝐺𝑘, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', ∀𝑖 ∈ 𝑉𝑘,𝑑(𝑖,𝐺𝑘) ≥ 𝑘, where 𝑑(𝑖,𝐺𝑘) is the degree of 𝑖 in 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Core Value of Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' With the concept of 𝑘-core, we can also describe the core value of a given node 𝑖 within 𝐺 by 𝐶(𝑖,𝐺), which represents the maximum core value of the 𝑘-core where node 𝑖 exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', 𝐶(𝑖,𝐺) satisfies that 𝑖 ∈ 𝐺𝐶(𝑖,𝐺) but 𝑖 ∉ 𝐺𝐶(𝑖,𝐺)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The nodes whose core values are equal to 𝑘 are named as 𝑘-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In accordance with Definition 1, the existence of a given node 𝑖 within 𝐺𝑘 relies on its neighbor nodes who overlap with 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We can also realize that those neighbors with core values less than 𝑘 are not included in 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Be a result, those neighbors helping support the existence of 𝑖 in 𝐺𝑘 are referred to as Supportive Neighbors of 𝑖 which is recorded as 𝑆𝑁(𝑖,𝑘,𝐺) = {𝑗|𝑗 ∈ N(𝑖,𝐺),𝐶(𝑗,𝐺) ≥ 𝑘}, where N(𝑖,𝐺) represents the one-hop neighbors of 𝑖 within 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way, the following theorem could be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Core Support Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 𝑖 can remain in 𝐺𝑘 if and only if it satisfies |𝑆𝑁(𝑖,𝑘,𝐺) | ≥ 𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' otherwise, it will be squeezed out of 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' According to the definition of supportive neighbors of node 𝑖, 𝑆𝑁(𝑖,𝑘,𝐺) actually denotes the intersection of N(𝑖,𝐺) and 𝑉𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Based on Definition 1, it is clear that only the satisfaction of 𝑑(𝑖,𝐺𝑘) ≥ 𝑘 can remain the existence of 𝑖 in 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' if node 𝑖 ∈ 𝐺𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' there is |𝑆𝑁(𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝐺) | = |N(𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝐺) ∩𝑉𝑘 | = |N(𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝐺𝑘) | = 𝑑(𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝐺𝑘) ≥ 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' which shows that node 𝑖 could be contained in 𝐺𝑘 if and only if Delete edge (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 7) 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 Delete edge (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 8) Delete edge (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3) 2-node 3-node Figure 2: Given a graph with 9 nodes and 17 edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' we can find that all nodes stay in 3-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Take node 5 as our target node, (i) the removal of edge (5, 7) does not make any ef- fect to the 𝑘-core distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (ii) the removal of edge (4, 8) makes node 4 being squeezed out of 3-core while makes no effect to node 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (iii) the removal of edge (1, 3) makes the tar- get node 5 being squeezed out of 3-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' at least 𝑘 neighbors whose core values are not less than 𝑘 are connected with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As illustrated in Figure 1, for node 13 who lives in 𝐺2, it has 𝑆𝑁(13,2,𝐺) = 2 which allows it to satisfy Theorem 1, while it has 𝑆𝑁(13,3,𝐺) = 1 < 3 so that it cannot exist in 𝐺3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 1 provides us with a sufficient and necessary condition to determine whether a certain node exists in 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Derived from this, Laishram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [23] exploited a naive and easily-computed metric called Core Strength to measure the most conservative number of disconnected neighbors of node 𝑖 for squeezing 𝑖 out of 𝐺𝐶(𝑖,𝐺) , which is formulated as 𝐶𝑆(𝑖,𝐺) = |𝑆𝑁(𝑖,𝐶(𝑖,𝐺),𝐺) | − 𝐶(𝑖,𝐺) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (1) This metric describes that if any 𝐶𝑆(𝑖,𝐺) of supportive neighbors are disconnected with the target node 𝑖, it will absolutely be in violation of Theorem 1 and be squeezed out of 𝐺𝐶(𝑖,𝐺) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For instance, as shown in Figure 2, we set node 5 ∈ 𝐺3 as the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' That is easy to find that the target node has 5 supportive neighbors 𝑆𝑁(5,3,𝐺) = {1, 2, 4, 6, 7} and core strength 𝐶𝑆(5,𝐺) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We arbitrar- ily select 3 supportive neighbors to disconnect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' {4, 6, 7}, then the number of its supportive neighbors will be reduced to 2 which is against what Theorem 1 restricts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Please notice that in the rest of this paper, if 𝑘 = 𝐶(𝑖,𝐺), we will use 𝑆𝑁(𝑖,𝐺) instead of 𝑆𝑁(𝑖,𝑘,𝐺) for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2 Problem Definition As mentioned in the aforementioned contents, the core strength metric describes the most conservative number of edges we should disconnect for target-node collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Because of so-called cascade phenomenon or domino phenomenon of 𝑘-core collapse [12], how- ever, this metric cannot estimate the exact number of edges that must be deleted which may be less than that quantified by core strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As an illustration, let us turn our sights back to Figure 2, the deletion of edge (1, 3) will practically make node 5 with 𝐶𝑆(5,𝐺) = 3 collapse from 𝐺3 to 𝐺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From here, we can derive the problem named Targeted 𝑘-node Collapse Problem (TNCP) aiming to quantify the minimal number of edges to remove for downgrad- ing the core value of a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For a given 𝐺 and a target node 𝑖 ∈ 𝑉 with 𝐶(𝑖,𝐺) = 𝑘, TNCP problem aims to find a set 𝑒 ⊆ 𝐸 containing the least number of edges such that𝐶(𝑖,𝐺′) < 𝐶(𝑖,𝐺), where𝐺′ = (𝑉, 𝐸\\𝑒), and can be formulated as: 𝑒∗ = arg min 𝑒 |𝑒| , 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝐶(𝑖,𝐺′) < 𝐶(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (2) The minimal size of 𝑒 is named as Node Robustness which dis- plays the fewest number of removed edges for the collapse of the target node under elaborate perturbations and is recorded as 𝑁𝑅(𝑖,𝐺) = 𝑒∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Furthermore, those nodes whose core strengths are larger than their node robustness are referred to as Bubble Nodes which are recorded as 𝐵𝑁 = {𝑖|𝑖 ∈ 𝑉,𝐶𝑆(𝑖,𝐺) > 𝑁𝑅(𝑖,𝐺)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The TNCP problem is NP-hard for 𝐶(𝑖,𝐺) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' First, when 𝐶(𝑖,𝐺) = 1, according to Definition 1, it is easy to realize that some node will always remain in 𝐺1 as long as at least one neighbor is connected with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way, if we want a node to collapse from 𝐺1 to 𝐺0, we have to disconnect all of its adjacent neighbors and make it isolated from 𝐺, where the cost of operations is in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, when 𝐶(𝑖,𝐺) ≥ 2, considering the cascade phenomenon of 𝑘-core collapse, a slight disturbance is able to lead a huge variation to the target node on weakening the number of its supportive neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, in such a situation, the Set Cover Problem (SCP) which has been proved to be NP-hard [21] can be reduced to TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Given a universe collection 𝑆𝑁(𝑖,𝐺) and a set of candidates 𝐸 which contains all edges within 𝐺 under the condition of target node𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In order to cover the TNCP problem, we have to find out a minimal-size set of edges 𝑒 ⊆ 𝐸 such that |𝑆𝑁(𝑖,𝐺) \\ Φ(𝑒)| < 𝐶(𝑖,𝐺), where Φ(𝑒) represents those collapsed nodes whose core values will be changed after the removal of 𝑒 from 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, paying attention to the complexity of TNCP prob- lem, without any prior information, we have to traverse all pos- sible combinations of the already existing edges, whose mathe- matical expression can be formulated as 𝑓 = �𝛿 𝑚=1 �|𝐸 | 𝑚 �, where 𝛿 = 𝐶𝑆(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Based on the induction formulas of �𝑛 𝑚 � = �𝑛−1 𝑚 � + �𝑛−1 𝑚−1 � and �𝑀 𝑚=0 �𝑀 𝑚 � = 2𝑀, the above equation could be written as 𝑓 = O(|𝐸|𝛿−1) �𝛿 0 � + O(|𝐸|𝛿−2) �𝛿 1 � + · · · + �𝛿 𝛿 � = O(|𝐸|𝛿−1) + O(|𝐸|𝛿−2) · 21 + · · · + 2𝛿 = 2𝛿 + 𝛿 ∑︁ 𝑚=1 O(|𝐸|𝑚−1) · 2𝛿−𝑚 (3) With the complexity in the amount of the exponential increase, it is evident that traversing all combinations takes non-polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Combining the aforementioned approaches, the TNCP prob- lem cannot be addressed in polynomial time when 𝐶(𝑖,𝐺) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As seen in Figure 1 covering 18 nodes and 28 edges, node 6 is chosen to be the target node for𝑘-node collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As mentioned before, there is just one edge, like (1, 2), should be removed in order to achieve the collapse of node 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, without the omniscient knowledge, it is difficult to locate which edge or edges are necessar- ily deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From the descriptions above, it is naturally realized that 𝑁𝑅(6,𝐺) ≤ 𝐶𝑆(6,𝐺), thus we need to visit all �2 𝑚=1 �28 𝑚 � combinations to identify the key edge or edges useful for 𝑘-node collapse under the worst situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Fortunately, in this scenario, the computational com- plexity is not high because of the previous information of 𝑁𝑅(6,𝐺) = 1 with the removal of edge (4, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, the robustness of the target node will always be equal to 1, like 𝑁𝑅(7,𝐺) = 2 under the removal of (1, 2) and (7, 8) as well as 𝑁𝑅(8,𝐺) = 2 under the removal of (4, 8) and (7, 8) in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In real-world networks, the robustness of some nodes may reach tens or even hundreds, which can probably lead to an exponential increase in time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, real-world networks often contain thousands or even millions of edges, making it chal- lenging to find a feasible solution within a reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, it is important to design an effective heuristic algorithm to solve the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3 Candidate Reduction As mentioned above, the naive exhaustive method for solving the TNCP problem is highly complex, making it difficult to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In order to obtain a feasible solution within a reasonable amount of time, we need to reduce the number of candidate edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this section, we will introduce and prove some theorems that can be used to achieve this reduction in candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' ∀(𝑖, 𝑗) ∈ 𝐸, when 𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺), it satisfies that 𝑖 ∈ 𝑆𝑁(𝑗,𝐺) ∧ 𝑗 ∉ 𝑆𝑁(𝑖,𝐺), and when 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺), it satisfies that 𝑖 ∈ 𝑆𝑁(𝑗,𝐺) ∧ 𝑗 ∈ 𝑆𝑁(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Based on the definition of supportive neighbors, it is evident that only those neighbors with core values greater than or equal to 𝐶(𝑖,𝐺) can be contained within the supportive neighbors of node 𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', {𝑗|𝑗 ∈ 𝐺,𝐶(𝑗,𝐺) < 𝐶(𝑖,𝐺)} ∩ 𝑆𝑁(𝑖,𝐺) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For the same reason, considering 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺), there exists that {𝑖, 𝑗} ∈ 𝑆𝑁(𝑖,𝐺) ∩ 𝑆𝑁(𝑗,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ In other words, nodes with low core values could never establish relationships that would be supportive to nodes with high core val- ues, while nodes with high core values establish one-way relation- ships that would be supportive of their connected nodes with low 4 core values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, connected nodes with the same core value become supportive neighbors to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' This suggests that the removal of edges bridging node pairs with different core values may only affect the side holding a low core value, while the removal of edges bridging node pairs with the same core values may affect both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Combining the description of Theorem 3, those relationships bridging nodes with higher core values and lower core values are named as one-way supportive relationships, and those relationships bridging nodes with the same core value are named as bidirectional supportive relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way, the neighbors who control the bidirectional supportive relationships with an arbitrary node 𝑖 are recorded as � 𝑆𝑁 (𝑖,𝐺) = {𝑗|𝑗 ∈ 𝑁(𝑖,𝐺),𝐶(𝑗,𝐺) = 𝐶(𝑖,𝐺)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' If an edge (𝑖, 𝑗) ∈ 𝐸 is removed, for all nodes in 𝐺, only those with core values equal to min(𝐶(𝑖,𝐺),𝐶(𝑗,𝐺)) may have their core values changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It might be assumed that 𝐶(𝑖,𝐺) ≥ 𝐶(𝑗,𝐺) = 𝑘𝑚𝑖𝑛 and be marked that 𝐺′ = 𝐺 \\ {(𝑖, 𝑗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In accordance with Theorem 1, the removal of edge (𝑖, 𝑗) will surely make node 𝑗 collapse if and only if 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛,𝐺) = 𝑘𝑚𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After the elimination of (𝑖, 𝑗), there exists that 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛,𝐺′) ≤ 𝑘𝑚𝑖𝑛 − 1 which is absolutely in violation with Theorem 1 and makes node 𝑗 excluded from 𝐺𝑘𝑚𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In addition, Sariyüce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [39] and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [25] have proved that the core value of some node can decrease at most 1 when one of its supportive neighbors is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Benefiting from this, node 𝑗 will still remain in 𝐺′ 𝑘𝑚𝑖𝑛−1 and satisfy that 𝑆𝑁(𝑗,𝑘𝑚𝑖𝑛−1,𝐺′) ≥ 𝑘𝑚𝑖𝑛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' According to Theorem 3, the collapse of node 𝑗 from 𝐺𝑘𝑚𝑖𝑛 to 𝐺′ 𝑘𝑚𝑖𝑛−1 probably leads to the collapse of those nodes contained in � 𝑆𝑁 (𝑗,𝐺𝑘𝑚𝑖𝑛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Following like this, based on the cascade phenomenon, it is easy to find that only nodes whose core values equal to𝑘𝑚𝑖𝑛 will probably collapse from 𝐺𝑘𝑚𝑖𝑛 to 𝐺′ 𝑘𝑚𝑖𝑛−1 in the case of eliminating edge (𝑖, 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, for those nodes with core values larger than 𝑘𝑚𝑖𝑛, according to Theorem 3,𝑘𝑚𝑖𝑛-nodes make no contributions to supporting their presence in 𝐺𝑘𝑚𝑖𝑛+1 so that no effect will work on them after edge (𝑖, 𝑗) is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Meanwhile, due to the existence of those collapsed nodes in 𝐺′ 𝑘𝑚𝑖𝑛−1, on the basis of Theorem 3, they still establish supportive relationships with those nodes with core values less than 𝑘𝑚𝑖𝑛 whose number of supportive neighbors remains the same so that no change happens to their core values after edge (𝑖, 𝑗) is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ Benefiting from Theorem 4, only the removal of edges contained in 𝐸𝑘\\𝑘+1 = 𝐸𝑘 \\𝐸𝑘+1 = {(𝑢, 𝑣)|(𝑢, 𝑣) ∈ 𝐸,𝑚𝑖𝑛(𝐶(𝑢,𝐺),𝐶(𝑣,𝐺)) = 𝑘} will have the probability to make the target node 𝑖 with 𝐶(𝑖,𝐺) = 𝑘 collapse, which allows us to reduce the candidates from 𝐸 to 𝐸𝑘\\𝑘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As illustrated in Figure 2 where only a 3-core exists, in order to make node 5 with 𝐶(5,𝐺) = 3 collapse, we should take 𝐸3\\4 = 𝐸3 = 𝐸 into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, we may notice that the removal of edge (1, 3) leads to the collapse of node 5 while none of nodes contained in this graph collapse after the removal of edge (5, 7), which shows a substantial difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, the following theorem is presented to further narrow down the search space of candidate edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' A given edge (𝑖, 𝑗) ∈ 𝐸 whose elimination could make nodes within 𝐺 collapse requires both of the following two conditions to be satisfied: (i)𝑚𝑖𝑛{𝐶𝑆(𝑖,𝐺),𝐶𝑆(𝑗,𝐺)} = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (ii) (𝐶𝑆(𝑖,𝐺) −𝐶𝑆(𝑗,𝐺)) · (𝐶(𝑖,𝐺) − 𝐶(𝑗,𝐺)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Firstly, the condition (i) will not be satisfied if and only if neither 𝐶𝑆(𝑖,𝐺) nor 𝐶𝑆(𝑗,𝐺) is equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', 𝐶𝑆(𝑖,𝐺) ≥ 2 and 𝐶𝑆(𝑗,𝐺) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In such a case, node𝑖 and node 𝑗 satisfy that |𝑆𝑁(𝑖,𝐺) | ≥ 𝐶(𝑖,𝐺) +1 and |𝑆𝑁(𝑗,𝐺) | ≥ 𝐶(𝑗,𝐺) +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The removal of edge (𝑖, 𝑗) will absolutely not make node 𝑖 or node 𝑗 to violate Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Next, assume that the first condition has been satisfied, it might be supposed that 𝐶𝑆(𝑖,𝐺) ≥ 𝐶𝑆(𝑗,𝐺) = 1 since edge (𝑖, 𝑗) is equiva- lent to edge (𝑗,𝑖) in𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For the core values of node𝑖 and node 𝑗, there are three cases to consider, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', 𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺), 𝐶(𝑖,𝐺) < 𝐶(𝑗,𝐺) and 𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' According to Theorem 3, the removal of edge (𝑖, 𝑗) will surely make node 𝑗 collapse because of the violation of Theorem 1 in the cases of𝐶(𝑖,𝐺) > 𝐶(𝑗,𝐺) and𝐶(𝑖,𝐺) = 𝐶(𝑗,𝐺) while no node will collapse in the case of 𝐶𝑖,𝐺 < 𝐶(𝑗,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ Combining the findings derived by Theorem 4 and Theorem 5, for the targeted collapse mission of a given node 𝑖 with 𝐶(𝑖,𝐺) = 𝑘, those edges existing in 𝐸𝑘\\𝑘+1 and connecting to 𝑉𝐶 (𝑘,𝐺) = {𝑢|𝑢 ∈ 𝑉,𝐶(𝑢,𝐺) = 𝑘 ∧𝐶𝑆(𝑢,𝐺) = 1} are what we should focus on and take into candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Actually, nodes contained in 𝑉𝐶 (𝑘,𝐺) are so-called corona nodes of 𝐺𝑘 [4, 5, 52], which denotes that these nodes have exactly 𝑘 one-hop neighbors in 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, the subgraph con- structed by corona nodes may not be connected and will probably be divided into several disconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Fig- ure 1 where six corona nodes {1, 3, 4, 5, 9, 10} exist, the component constructed by nodes {1, 3, 4} is disconnected with that constructed by nodes {9, 10}, and so does that constructed by node {5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' There- fore, for simplicity of representation, we provide the following definition to represent the corona component in which a particular corona node 𝑖 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Corona Pedigree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For a corona node 𝑖 ∈ 𝐺 with 𝐶(𝑖,𝐺) = 𝑘, the corona pedigree of 𝑖, denoted as 𝑃(𝑖,𝐺), represents the largest-connected subgraph containing 𝑖 as its component and satisfies that ∀𝑗 ∈ 𝑃(𝑖,𝐺),𝐶(𝑗,𝐺) = 𝑘 ∧ 𝐶𝑆(𝑗,𝐺) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Figure 1, there exist three corona pedigrees in 𝐺3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', 𝑃(4,𝐺) contains nodes {1, 3, 4} and edges {(1, 3), (1, 4)}, 𝑃(5,𝐺) contains node {5}, 𝑃(10,𝐺) contains nodes {9, 10} and edge {(9, 10)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Note that 𝑃(𝑗,𝐺) is equivalent to 𝑃(𝑖,𝐺) if it satisfies that 𝑗 ∈ 𝑃(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, those edges adjacent to 𝑃(𝑖,𝐺) are represented as 𝐸𝑃 (𝑖,𝐺) = {(𝑢, 𝑣)|(𝑢, 𝑣) ∈ 𝐸,𝑢 ∈ 𝑃(𝑖,𝐺)∨𝑣 ∈ 𝑃(𝑖,𝐺)} and the following theorem could be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The removal of an arbitrary edge within 𝐸𝑃 (𝑖,𝐺) will absolutely make all nodes within 𝑃(𝑖,𝐺) collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' According to Definition 3, each node within 𝑃(𝑖,𝐺) pos- sesses its core strength of 1 which means the disconnection of any supportive neighbor will make this node collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, each edge within 𝐸𝑃 (𝑖,𝐺) actually bridges some corona node within 𝑃(𝑖,𝐺) with one of its supportive neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way, if one of edges in 𝐸𝑃 (𝑖,𝐺) is removed, the corona node (or corona nodes) adjacent to it will surely collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Because of the cascade phenomenon, the other nodes contained in 𝑃(𝑖,𝐺) will collapse follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' □ 5 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Figure 2, taking 𝑃(1,𝐺) where nodes {1, 3} exist as example, We get 𝐸𝑃 (1,𝐺) = {(1, 2), (1, 3), (1, 5), (2, 3), (3, 7)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 3 will be absolutely squeezed out of 3-core after the removal of an arbitrary edge contained in 𝐸𝑃 (1,𝐺), like (2, 3), and then node 1 will also collapse from 𝐺3 because of the cascade phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 4 METHODOLOGIES In this section, in order to address the TNCP problem, we propose an effective heuristic algorithm called Targeted 𝑘-Node Collapse (TNC) as the first solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, based on TNC algorithm, we design an optimized strategy called Adjacent Targeted 𝑘-Node Collapse (ATNC) to further reduce computational complexity, mak- ing it suitable for large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 TNC Algorithm To solve the TNCP problem, we propose the TNC algorithm, which itreatively removes one edge that can lead to the greatest impact on the target node 𝑖 until the target node collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The impact on the target node is determined by maximizing (i) the number of collapsed nodes within 𝑆𝑁(𝑖,𝐺), and (ii) the number of nodes whose core strengths change within 𝑆𝑁(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As discussed earlier, those edges existing in 𝐸𝑘\\𝑘+1 and connecting to 𝑉𝐶 (𝑘,𝐺) play significant roles in the collapse of target node 𝑖 with 𝐶(𝑖,𝐺) = 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, ac- cording to Theorem 6, for a corona node 𝑢 ∈ 𝑉𝐶 (𝑘,𝐺) and its corona pedigree 𝑃(𝑢,𝐺) ∈ 𝐺𝑘, it is easy to realize that the disconnection of the relationship between node𝑢 and one of its supportive neighbors will actually make all nodes within 𝑃(𝑢,𝐺) collapse from 𝐺𝑘 and then make all edges within 𝐸𝑃 (𝑢,𝐺) be excluded from 𝐸𝑘\\𝑘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this manner, in order to avoid unnecessary duplicate operations, we only need to select the corona pedigree 𝑃(𝑣,𝐺) whose detachment leads to the greatest impact on the target node and removes one of edges existing in 𝐸𝑃 (𝑣,𝐺) in each iteration until the target node col- lapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Figure 3 illustrates the overall framework of TNC algorithm along with the detailed operations shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Words for further descriptions are given as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Algorithm 1, Line 4, the corona nodes, 𝐶𝑜𝑟𝑜𝑛𝑎𝑠, are firstly extracted from 𝐺′ 𝑘 as candidates where 𝐺′ is initialized as 𝐺 in Line 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, in Lines 6-7, by exploiting an assistant algorithm called CalculateImpact which will be introduced in the following paragraphs, the impact which will be made on the target node 𝑖 is measured by 𝐹 [𝑢] and 𝐼 [𝑢] if corona node 𝑢 ∈ 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 collapses, and𝐶𝑜𝑟𝑜𝑛𝑎𝑠 is updated according to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Next, we select the top corona node 𝑣 sorted according to 𝐹 [·] (first priority) and 𝐼 [·] (second priority) in Line 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, one of edges contained in 𝐸𝑃 (𝑣,𝐺′) is added to 𝑒 with the update of 𝐺′ in Lines 13-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The above process will continue until there is the violation of Theorem 1 to make the target node 𝑖 collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Note that if the collapse of 𝑣 makes no supportive neighbors of node 𝑖 collapse, we will remove the edge bridging the target node and its supportive neighbor with the minimal core strength in 𝐺′ as instead, in Lines 9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CalculateImpact Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After the collapse of node𝑢 ∈ 𝑉 , for all nodes in 𝐺, those nodes whose core strength decreases are named as Influenced Nodes, those whose core value decreases are named as Followed Nodes, and those whose core strengths and core values remain the same are named as Uninfluenced Nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' To effectively measure the impact that the collapse of a corona node 𝑛 can make on the target node 𝑖, we offer CalculateImpact algorithm which is based on Depth-First Search (DFS) and whose details are shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Algorithm 2, Line 1, S is defined to store the nodes waiting to be visited, F and I are defined to store the followed nodes and the influenced nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, in Line 2, a dictionary T with default value of 0 is defined to record the decrease in the number of supportive neighbors of each node in 𝐺 after the input node 𝑛 collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this way, for a visited node 𝑢 popped from S, if T [𝑢] > 0, it will be marked as an influenced node and be added into I in Line 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' furthermore, if 𝐶𝑆(𝑢,𝐺) ≤ T [𝑢], it will also be marked as a followed node and be added into F in Line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, if node 𝑢 has been marked as a followed node, on the basis of Theorem 4, those nodes contained in � 𝑆𝑁 (𝑢,𝐺) and satisfying 𝐶𝑆(·,𝐺) > T [·] will be pushed into S in Line 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Please note that those nodes marked as followed nodes will be excluded from S in Line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The above process will be repeated iteratively until S is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, contents shown in the dotted box of Figure 3 exhibit the detailed process of Algorithm 2 where node 1 with 𝐶𝑆(1,𝐺) = 1 is taken as the input node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' First, in the initial-state graph, F and I are initialized as empty sets and S = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Next, in the second graph, node 1 is popped from S with the update of T [1] = 1 and be added into I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It is apparent that node 1 is also added into F because of the satisfaction of 𝐶𝑆(1,𝐺) ≤ T [1], and its neighbors {2, 5, 3} are pushed into S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, in the third graph, node 2 with 𝐶𝑆(2,𝐺) = 2 is popped, and we get T [2] = 1 with the addition of node 2 into I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, the next iteration will be triggered directly because of 𝐶𝑆(2,𝐺) > T [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Continuing in this flow, we finally achieve that I = {1, 2, 5, 3, 4, 7, 6} and F = {1, 3, 5, 2, 4}, and further get that |N(5,𝐺) ∩ F | = 4 and |N(5,𝐺) ∩ I| = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Time Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Algorithm 1, first, in order to extract the corona nodes 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 of 𝐺𝑘 from 𝐺, it takes the time in the order of O(|𝑉 |) in Line 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, from Line 5 to Line 7, one corona node within each corona pedigree in 𝐺𝑘 is assigned weights through 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝐼𝑚𝑝𝑎𝑐𝑡 algorithm which takes the time in the order of O(𝐶𝑆𝑘\\𝑘+1·|𝐶𝑜𝑟𝑜𝑛𝑎𝑠|) where𝐶𝑆𝑘\\𝑘+1 = � 𝑣∈𝑉𝑘 \\𝑉𝑘+1 𝐶𝑆(𝑣,𝐺) |𝑉𝑘\\𝑉𝑘+1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, considering the worst condition, 𝐶𝑆(𝑖,𝐺) iterations are executed with the total time complexity in the order of O(𝐶𝑆(𝑖,𝐺) · (|𝑉 | + 𝐶𝑆𝑘\\𝑘+1 · |𝐶𝑜𝑟𝑜𝑛𝑎𝑠|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2 ATNC Algorithm In the previous part, we give the introduction of TNC algorithm which iteratively removes one edge that connected to the corona pedigree whose detachment could cause the greatest impact on the target node for addressing the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, in each iteration, TNC algorithm needs to traverse all nodes within 𝐺′ to extract the corona nodes of 𝐺′ 𝑘 and then visit each corona pedigree through CalculateImpact algorithm to filter out the most impacted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Clearly, the process is highly time-consuming for large-scale networks which pushes the expectation of a heuristic algorithm with less time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' we offer Adjacent Targeted 𝑘-Node Collapse (ATNC) improved from TNC which actually takes the strategy of adjacent search to exploit the local information of 6 Delete one edge from 𝑬(𝟏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑮) 𝑷 Re-extract corona nodes (No) Extract corona nodes 1 3 5 6 9 8 2 4 7 1 3 5 6 9 8 2 4 7 Take node 1 as input for example If target node has collapsed (Yes) Original Graph Adversarial Graph Target Node 1 3 9 8 4 Node F[·] I[·] 4 5 4 5 2 3 2 3 1 2 1 3 8 9 4 1 3 5 6 9 8 2 4 7 Get F[·],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' I[·] by CalculateImpact Initial State \uf049={},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={1} 𝒖=1 \uf049={1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3} 𝒖=2 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3} 𝒖=5 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={3} Terminated State \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={} 𝒖=3 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2} 𝒖=5 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2} 𝒖=2 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6} 𝒖=4 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6} 𝒖=7 \uf049={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf046={1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' \uf053={6} Influenced Node Followed Node UnInfluenced Node Sort corona nodes by F[·] and I[·] Figure 3: The framework of TNC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Given a target node 5 with 𝐶𝑆(5,𝐺) = 3, we first extract the corona nodes from 𝐺3, and then evaluate the impact of each corona node on the target node through CalculateImpact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, the most impacted node 1 is filtered out and one edge existing in 𝐸𝑃 (1,𝐺) is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' If the target node has collapsed, the adversarial graph will be output and the removed edges will be returned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' otherwise, we re-extract the corona nodes and repeat the above process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The contents shown in the dotted box display the detailed operations of CalculateImpact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The detailed descriptions will be presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Algorithm 1: TNC input :the given graph 𝐺, the target node 𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' output:the removed edges 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 1 𝑒 ← empty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 𝐺′ ← 𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 𝑘 ← 𝐶(𝑖,𝐺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 2 𝐹, 𝐼 ← dictionaries with default value of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3 while |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 do 4 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 ← {𝑢|𝑢 ∈ 𝐺′ 𝑘, N(𝑢,𝐺′ 𝑘) = 𝑘};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 5 foreach 𝑢 ∈ 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 do 6 𝐹 [𝑢], 𝐼 [𝑢] ← CalculateImpact(𝐺′,𝑖,𝑢);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 7 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 ← 𝐶𝑜𝑟𝑜𝑛𝑎𝑠 \\ 𝑃(𝑢,𝐺′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 8 𝑣 ← The top corona node sorted according to 𝐹 [·] (first priority) and 𝐼 [·] (second priority);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 9 if 𝐹 [𝑣] = 0 then 10 𝑚 ← The supportive neighbor of 𝑖 in 𝐺′ with the lowest core strength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 11 𝑒 ← 𝑒 ∪ {(𝑖,𝑚)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 12 else 13 𝑒 ← 𝑒 ∪ {∀(𝑚,𝑛) ∈ 𝐸𝑃 (𝑣,𝐺′)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 14 𝐺′ ← 𝐺 \\ 𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 15 return 𝑒 the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The details of ATNC are shown in Algorithm 3 along with its descriptions as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As shown in Algorithm 3, Line 3, instead of extracting all corona nodes within 𝐺′ 𝑘 by TNC algorithm, ATNC only exploits those corona nodes adjacent to the target node which are named as corona neighbors𝐶𝑜𝑟𝑁𝑏𝑟𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Next, in Lines 5-8, through the same operations Algorithm 2: CalculateImpact input :the given graph 𝐺, the target node 𝑖, the input node 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' output:the number of followed nodes in N(𝑖,𝐺), the number of influenced nodes in N(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 1 S ← empty stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' F ← empty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' I ← empty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 2 T ← a dictionary with default value of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑝𝑢𝑠ℎ(𝑛);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 4 while S is not empty do 5 𝑢 ← S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑝𝑜𝑝();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 6 T [𝑢] ← T [𝑢] + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 7 I ← I ∪ {𝑢};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 8 if 𝐶𝑆(𝑢,𝐺) ≤ T [𝑢] then 9 F ← F ∪ {𝑢};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 10 V ← {𝑣|𝑣 ∈ � 𝑆𝑁 (𝑢,𝐺),𝐶𝑆(𝑣,𝐺) > T [𝑣]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 11 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='𝑝𝑢𝑠ℎ(V);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 12 S ← S \\ F ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 13 return |N(𝑖,𝐺) ∩ F |, |N(𝑖,𝐺) ∩ I| as those of TNC, the top corona node 𝑣 is filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, we add edge (𝑖, 𝑣) into 𝑒 with the update of 𝐺′ and the re-extraction of 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 in Lines 9-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The above process will continue until 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 is empty or there is the violation of Theorem 1 for the target node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Note that if the above loop quits with 𝑆𝑁(𝑖,𝑘,𝐺′) > 𝑘 which means that |𝐶𝑜𝑟𝑁𝑏𝑟𝑠| = 0 and the target node still remains in 𝐺𝑘, then we will randomly sample 𝐶𝑆(𝑖,𝐺′) supportive neighbors 7 from 𝑆𝑁(𝑖,𝑘,𝐺′) and make the target node 𝑖 disconnected with them in Lines 12-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Time Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Similar to the time complexity of TNC, since only the corona nodes existing in the one-hop neighbors of the target node will be selected as candidates, the time for collecting the candidates is in the order of O(|N(𝑖,𝐺) |) in Algorithm 3, Line 3 at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, from Line 5 to Line 7, each corona pedigree contained in 𝐺𝑘 is traversed with the quantification of their impact to the target node which takes the time in the order of O(𝐶𝑆 (𝑘\\𝑘+1) · |𝐶𝑜𝑟𝑁𝑏𝑟𝑠|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, considering the worst condition, 𝐶𝑆(𝑖,𝐺) iterations are executed with the total time complexity in the order of O(𝐶𝑆(𝑖,𝐺) · (|𝑉 | + 𝐶𝑆𝑘\\𝑘+1 · |𝐶𝑜𝑟𝑁𝑏𝑟𝑠|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Algorithm 3: ATNC Input: the given graph 𝐺, the target node 𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Output: the removed edges 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 1 𝑒 ← empty set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 𝐺′ ← 𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 𝑘 ← 𝐶(𝑖,𝐺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 2 𝐹, 𝐼 ← dictionaries with default value of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 3 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 ← {𝑗|𝑗 ∈ � 𝑆𝑁 (𝑖,𝐺′),𝐶𝑆(𝑗,𝐺′) = 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 4 while |𝐶𝑜𝑟𝑁𝑏𝑟𝑠| > 0 and |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 do 5 foreach 𝑢 ∈ 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 do 6 𝐹 [𝑢], 𝐼 [𝑢] ← CalculateImpact(𝐺′,𝑖,𝑢);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 7 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 ← 𝐶𝑜𝑟𝑁𝑏𝑟𝑠 \\ 𝑃(𝑢,𝐺′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 8 𝑣 ← The top corona node sorted according to 𝐹 [·] (first priority) and 𝐼 [·] (second priority);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 9 𝑒 ← 𝑒 ∪ {(𝑖, 𝑣)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 10 𝐺′ ← 𝐺 \\ 𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 11 Re-extract 𝐶𝑜𝑟𝑁𝑏𝑟𝑠;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 12 if |𝑆𝑁(𝑖,𝑘,𝐺′) | ≥ 𝑘 then 13 𝑒′ ← Sample({(𝑖, 𝑗)|𝑗 ∈ 𝑆𝑁(𝑖,𝐺′)},𝐶𝑆(𝑖,𝐺′));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 14 𝑒 ← 𝑒 ∪ 𝑒′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 15 return 𝑒 5 EXPERIMENTS In this section, our experiments will be conducted on 16 real-world network datasets collected from various domains to demonstrate the performance of TNC and ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We also include 4 baseline methods for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' All of our experiments are deployed on a server with Intel(R) Xeon(R) Gold 5218R CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='10GHz and 377GB RAM, which installs Linux Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 Datasets The basic properties of 16 real-world networks from various do- mains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', Social Network (SN), Collaboration Network (CN), In- frastructure Network (IN) and Web Network (WN), are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Different labels are exploited to distinguish the different public platforms where networks are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, those marked with stars are collected from https://networkrepository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' com/ [38] and those marked with circles are collected from http: //snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='edu/ [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Please note that all networks used in the following experiments are converted to undirected and unweighted graphs, with no self-loops or isolated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Due to the space limita- tion, more detailed information of these networks could be achieved on the mentioned websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Table 2: Basic properties of mentioned networks containing the number of nodes |𝑉 |, the number of edges |𝐸|, the maxi- mal value of 𝑘-core 𝑘𝑚𝑎𝑥 and the average degree 𝑑𝑎𝑣𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Network |𝑉 | |𝐸| 𝑘𝑚𝑎𝑥 𝑑𝑎𝑣𝑔 SN TVShow★ 3892 17239 56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='8587 LastFM◦ 7624 27806 20 7.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='5462 Citeseer★ 227320 814134 86 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1629 IN USAir★ 332 2126 26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='8072 USPower★ 4941 6594 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6691 RoadNet★ 1965206 2766607 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='8156 WN EDU★ 3031 6474 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2719 Indo★ 11358 47606 49 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3828 Arabic★ 163598 1747269 101 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3605 Google◦ 875713 4322051 44 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='8709 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2 Baselines Given that we are the first work to study the TNCP problem, there is no ready-made method that can be used as a comparison ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For this reason, we design two random-based baseline methods and adjust two existing algorithms which are originally proposed to solve the 𝑘-core minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Their details are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Random Edge Deletion (RED) arbitrarily selects an edge within 𝐸 to remove and then updates the core values of nodes within 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' These two steps will be performed iteratively until the target node collapses successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Random Neighbor Disconnection (RND) arbitrarily removes an edge connected to the target node and then updates the core values of nodes within 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' These two steps will be performed iteratively until the target node collapses successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' KNM was proposed by [8] as a solution to the 𝑘-core mini- mization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It works by iteratively removing the edge whose detachment will lead to the maximal number of nodes who collapse from 𝐺𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' This process continues until the pertur- bation budget is reached or 𝐺𝑘 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this paper, we adapt the termination condition of KNM algorithm to the collapse of the target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' SV was proposed by [34] for covering the 𝑘-core minimization problem which exploits the shapley value, a cooperative game- theoretic concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It assigns weights to the candidate edges and then chooses the top 𝑏 edges to remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In this paper, considering the consumption of time, we set 𝐸𝑘\\𝑘+1 as the candidate edges instead of 𝐸𝑘 which is originally used by [34], and we set the hyperparameter 𝜖2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then we remove 8 candidate edges one by one according to their weights until the target node collapses without the budget limitation of 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In order to evaluate the transferability not only among various networks but also among various individual nodes, we will apply all baseline methods as well as our proposed algorithms on each node within every network to achieve its node robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then we will evaluate the effectiveness of these algorithms by several global metrics which will be introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, it is necessary to be noted that both of RED and RND will be performed 10 times independently on each node in order to reduce the randomness and the mean value is recorded as the robustness of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3 Metrics We propose the following metrics, Number of Bubble Nodes (NBN), Sum of Reduced Cost (SRC), Weighted Average Reduction (WAR), and Reduction Proportion (RP) to evaluate the effectiveness of various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NBN: Through a particular algorithm, we are interested in how many bubble nodes can be explored from 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Thus, the total number of explored bubble nodes is recorded as NBN which is formulated as below: NBN = |𝐵𝑁 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (4) The higher NBN is, the more transferable the algorithm is among various nodes in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' SRC: For a bubble node𝑖, the decrease between its core strength and node robustness is named as Reduced Cost which is quan- tified as 𝑅𝐶(𝑖,𝐺) = 𝐶𝑆(𝑖,𝐺) − 𝑁𝑅(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Therefore, the sum of reduced cost of all explored bubble nodes in 𝐺 could be formu- lated as below: SRC = ∑︁ 𝑖 ∈𝐵𝑁 𝑅𝐶(𝑖,𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (5) WAR: In order to illustrate the average cost reduction of ex- plored bubble nodes in a network through some algorithm, we propose WAR which is formulated as below: WAR = � 𝑟 ∈U 𝑝−1 𝑟 𝑟 � 𝑟 ∈U 𝑝−1 𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (6) where U contains the unique elements of {𝑅𝐶(𝑖,𝐺) |𝑖 ∈ 𝐵𝑁 } and 𝑝𝑟 = |{𝑖 |𝑖 ∈𝐵𝑁,𝑅𝐶(𝑖,𝐺)=𝑟 }| |𝐵𝑁 | denotes the probability of those nodes whose reduced cost equal to 𝑟 appearing in 𝐵𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' And the reason why we do not use arithmetic average will be explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 with specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' RP: We are also interested in the reduction proportion of node robustness relative to core strength on all nodes in 𝐺 and pro- pose RP for measuring, which is formulated as below: RP = SRC � 𝑖 ∈𝑉 𝐶𝑆(𝑖,𝐺) × 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' (7) In addition, RP can be used to describe the redundancy of core strength with respect to node robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The higher the RP, the more redundant the core strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Table 3: The experimental results of RED and RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Those datasets that could not be covered within 105 seconds are marked as /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Attention that for RED and RND, the robust- ness of each node is assigned as the average of the results achieved by 10 independent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Network RED RND NBN SRC 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='9982 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='42 Google / / / / 53222 50569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2091 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 6 RESULTS AND ANALYSES The experimental results are exhibited in Table 3 and Table 4, which contrastively shows the performance of TNC and ATNC, compared with 4 baseline methods on 16 real-world networks mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Meanwhile, the detailed comparisons and analyses are pre- sented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1 Performance Evaluation Comparisons Among Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Let us concentrate on Table 3 in which the experimental results of RED and RND are exhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Notice that the robustness of each node in networks is achieved by the average of 10 independent experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From the table, it is easy to find that the NBN of RED is far fewer than that of RND on all networks which represents that RED is unable to explore bubble nodes and fails to cover the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' On the contrary, RND performs much better than RED on all used metrics which demonstrates that the strategy of adjacent search for candidate reduction is helpful for covering the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Additionally, we can realize that RED is not able to complete the search missions on the 4 networks whose number of nodes is more than 105, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Gowalla, Citeseer, RoadNet and Google, while RND only fails on RoadNet, a network with millions of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It also proves that the strategy of adjacent search can effectively reduce the time complexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After that, let us turn our sights to the experimental results achieved by KNM and SV which are illustrated in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Neither KNM nor SV displays powerful transferability among different nodes in a network compared to RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For instance, RND detects 2439 bubble nodes on CondMat network, whereas this number is 259 and 184 induced by KNM and SV, respectively, which reveals 9 Table 4: The experiment results of KNM, SV, TNC and ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Those datasets that could not be completed by the method within 105 seconds are marked as /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The best results are bolded and the second-best results are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Network KNM SV TNC ATNC NBN SRC WAR RP(%) NBN SRC WAR RP(%) NBN SRC WAR RP(%) NBN SRC WAR RP(%) TVShow 188 502 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='8894 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='73 142 403 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='0841 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='40 543 1104 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='4244 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='63 TVShow LastFM Facebook DeezerEU HepPh AstroPh CondMat USAir USPower EDU Indo Arabic 10 0 10 1 10 2 10 3 10 4 Time Consumption (seconds) TNC KNM SV ATNC Figure 4: Comparisons of the running time of TNC, KNM, SV and ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The subfigure exhibits the running time of SV and ATNC on those large-scale networks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We can see that ATNC is significantly more efficient than the other methods on the time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' a difference of almost 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Similarly, RND is able to filter out 4510 bubble nodes on Arabic network, while KNM and SV could only find 354 and 214 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, the other metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', SRC, WAR and RP, are much higher for KNM and SV compared to those for RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, on Facebook network, the SRC of KNM is 5 times larger than that of RND and on AstroPh network, the WAR of KNM is 3 times larger than that of RND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' These results tell us that the heuristic methods enable the target node to collapse at a lower budget compared to the random-based methods, although they can only work on part of bubble nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Analysis from the principle of these two algorithms, neither of them exploits the information associated with the target node to guide the removal of edges which leads to the unsatisfied performance on solving the TNCP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Benefits of Our Proposed Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Next, turning to the re- sults generated by TNC and ATNC shown in Table 4, TNC and ATNC achieve the best and second-best performance on majority of the datasets with significant benefits over KNM and SV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For ex- ample, on CondMat network, only 259 and 184 bubble nodes could be detected through KNM and SV, respectively, while there are 3008 and 2801 bubble nodes found by TNC and ATNC, respectively, resulting in a difference of more than 10-fold between the two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' This definitely demonstrates that in the comparison to KNM and SV, TNC and ATNC have stronger transferability across different nodes and different networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, our proposed algorithms also perform better than KNM and SV considering SRC, WAR and RP metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' we notice that the WAR of SV is a little larger 10 10 10 Google Gowalla Citeseer RoadNet0 10 20 Number of Deleted Edges 10 20 30 Number of Support Neighbors k=11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=17 (a) LastFM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 6101 0 10 20 Number of Deleted Edges 10 20 30 Number of Support Neighbors k=11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=14 (b) LastFM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 3103 0 5 10 15 20 Number of Deleted Edges 10 15 20 25 30 Number of Support Neighbors k=10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=17 (c) DeezerEU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 17963 0 10 20 30 40 Number of Deleted Edges 10 20 30 40 Number of Support Neighbors k=8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=12 (d) DeezerEU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 24062 0 5 10 15 Number of Deleted Edges 5 10 15 20 25 Number of Support Neighbors k=9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=6 (e) CondMat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 1233 0 5 10 15 Number of Deleted Edges 10 15 20 25 Number of Support Neighbors k=8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=8 (f) CondMat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 13621 0 1 2 3 4 Number of Deleted Edges 4 6 8 10 12 Number of Support Neighbors k=9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=1 (g) Indo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 2721 0 5 10 15 Number of Deleted Edges 5 10 15 20 Number of Support Neighbors k=5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' CS=18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' NR=10 (h) Indo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Node 4712 TNC KNM SV ATNC Figure 5: Case study on individual nodes from 4 mentioned networks operated by TNC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' KNM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' SV and ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Each method is marked with a unique label and the red dotted line in each subfigure indicates the critical value of the number of supportive neighbors for current target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' than that of ATNC on RoadNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' After the observation of the bubble nodes found by SV and ATNC, there exists the situation that among the 19 bubble nodes detected by SV, 1 node has reduced cost of 3 and 18 nodes has reduced cost of 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' while for the 4105 bubble nodes detected by ATNC, there are 8 nodes with reduced cost of 3, 54 nodes with 2 and even 4043 nodes with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In the calculation of WAR for ATNC, the bubble nodes with reduced cost of 1, which make up nearly 98% of the total, surely have a significant diluting impact on the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Actually, the existence of bubble nodes with low reduced cost is common in the other networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For instance, on Facebook network, about 65% of the bubble nodes detected by ATNC have their reduced cost less than 4 while there are 30 nodes with reduced cost larger than 30, and on Indo network, 90% of the bubble nodes detected by ATNC have their reduced cost less than 3 with 4 nodes whose reduced cost larger than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' This is why we use weighted averaging instead of arithmetic averaging to quantify the average reduced cost of each bubble node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Comparison between TNC and ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Reviewing what is discussed in Section 4, it is easy to be realized that the candidates waiting to be filtered of ATNC is a subset of those of TNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Unsur- prisingly, considering the comparison between TNC and ATNC in Table 4, the performance of TNC is better than that of ATNC on the majority of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We also notice that on Indo and Arabic, the WAR of ATNC is slightly higher than that of TNC while the other metrics of ATNC are less than those of TNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Taking Indo as example for analysis, we find that there are 3 nodes with reduced cost of 8 among the 779 bubble detected by TNC while none of these nodes with reduced cost of 8 explored by ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' This situ- ation leads to an unfair weighting process of TNC compared to ATNC in the calculation of WAR and causes the slight difference in the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For the similar reason, the slight variations in the number of bubble nodes with high reduced cost lead to the differ- ence in the final result of WAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, the performance of TNC is completely superior to that of ATNC on the whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, it is easy to find that TNC is not suitable for those large-scale networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', Gowalla, Citeseer, RoadNet and Google, due to the huge size of the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' On the contrary, ATNC is able to complete these tasks and receives appreciable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The detailed comparisons of efficiency will be discussed in the following contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Redundancy of Core Strength Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As introduced before, the RP metric measures the redundancy of core strength with re- spect to node robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1, we have shown that the core strength metric does not accurately quantify the number of necessarily removed edges for making the target 𝑘-node collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From the results of ATNC in Table 4, there are more than half of the networks whose RP is larger than 10% and even part of them owning RP larger than 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, the RP of Facebook is nearby 20% and the RP of AstroPh is more than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' These results undoubtedly demonstrate that the core strength metric is not suitable for measuring the least number of edges to remove for leading the collapse to a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Efficiency of Different Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The visualization for the time consumption of implementing KNM, SV, TNC and ATNC across all the mentioned networks is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Overall, we can find that TNC and KNM have similar performance since they both traverse all corona nodes for edge removal in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Then, we can find that SV performs better than KNM and TNC on most of the networks except for HepPh network and USAir 11 0 PRC KNM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' SV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='APRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For HepPh network, its maximal core value 𝑘𝑚𝑎𝑥 = 238 is much higher than that of the other networks which is up to 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For USAir network, its size if much smaller than the others and causes the operations of SV are much more time-consuming than those of KNM and TNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, ATNC occupies the best efficiency with significant time-consumption reduction compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For example, on DeezerEU network, the time consumption of SV method is about 10 times larger than that of ARPC and the time consumption of TNC is even more than 100 times larger than that of ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' And for large-scale networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', Gowalla, Citeseer, RoadNet and Google, neither TNC nor KNM can calculate the robustness for each node in those networks in the limitation of 105 seconds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', TNC even fails to complete the calculation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='1% of total nodes on Google network within 105 seconds, while ATNC is able to cover the task in an appreciable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In a word, our proposed methods TNC and ATNC have signifi- cant advantages over the other baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' And considering the much lower time complexity of ATNC compared to TNC, ATNC is more suitable to be deployed on large-scale networks for solving TNCP problem, although the effect of TNC is slightly better than that of ATNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='2 Case Study In the previous section, we provide the performance of different methods from a macroscopic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Here, in this part, we offer a microscopic point of view as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We visualize the variation in the number of supportive neighbors of the target node when the implementation is processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' As illustrated in Figure 5, 8 individual target nodes collected from 4 of the mentioned net- works are visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In each subfigure, the horizontal coordinate indicates the number of removed edges during the process, the vertical coordinate indicates the number of remaining supportive neighbors of the target node after the removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Different imple- mented methods are marked with different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Meanwhile, the red dotted line in each subfigure represents the critical number of supportive neighbors for the target node which is equal to its core value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The collapse of the target node happens when the curve drops below the red dotted line since the violation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From the examples, it is clear that fewer removed edges is needed through TNC and ATNC compared to those of KNM and SV, and TNC is able to remove fewer edges than ATNC in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 7 APPLICATION Currently, 𝑘-core has been widely used in numerous downstream tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', anomaly detection [41, 42], community detection [37], detection of influential spreaders [6, 19, 28, 29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Laishram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' [23] demonstrated that the performance of those downstream tasks is highly relative to the resilience of the 𝑘-core structure in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' They proposed a heuristic metric named CIS whose cal- culation is based on the core strength metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' They indicated that the resilience of 𝑘-core is positively correlated with CIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' However, as mentioned before, we have demonstrated that the core strength metric is highly redundant for measuring the robustness of indi- vidual 𝑘-nodes in real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Thus, the CIS calculated from core strength, named as CS-based CIS, probably overestimates TVShow LastFM Facebook DeezerEU Gowalla HepPh AstroPh CondMat Citeseer USAir USPower RoadNet EDU Indo Arabic Google 0 2 4 6 8 CIS CS-based NR-based Figure 6: Comparisons of CS-based CIS and NR-based CIS on all mentioned networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It is clear that, on most net- works, NR-based CIS is able to measure the resilience of 𝑘- core more precisely than CS-based CIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' the resilience of 𝑘-core structures in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' For the above rea- sons, we replace core strength with the node robustness achieved by ATNC algorithm in the calculation of CIS, which is named as NR-based CIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' The results of CS-based CIS and NR-based CIS on real-world networks are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' It is clear that on most networks, NR-based CIS is much smaller than CS-based CIS and is able to precisely measure the resilience of the 𝑘-core in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Besides, combining the information illustrated in Table 4, we can find that the difference between NR-based CIS and CS-based CIS is proportional to the RP metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=', on Facebook network, ATNC provides RP=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='59% and there is a two-fold difference between CS- based CIS and NR-based CIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' while the difference on EDU network who receives RP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content='89% by ATNC is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' From this, it is clear that the node robustness metric has better performance, compared to the core strength metric, in precisely describing the resilience of 𝑘-core structures in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 8 CONCLUSION In this paper, we engage in the first work on studying the robustness of individual nodes within 𝑘-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' We propose the TNCP problem, which aims to remove the minimal number of edges for making the target node collapse, and we also provide a proof of its NP-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' In order to solve TNCP problem, we propose two heuristic algo- rithms including TNC algorithm which exploits corona nodes to improve search efficiency, and ATNC algorithm which introduces adjacent-search strategy to further lower down computational com- plexity on large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Extensive experimental results on various real-world networks, together with thorough analyses, demonstrate the superiority of our proposed methods over the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Meanwhile, we offer the detailed processes of different algorithms being implemented on various target nodes for case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' Finally, we demonstrate that studying TNCP problem is helpful for precisely estimating the resilience of 𝑘-core in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by the Key R&D Program of Zhejiang under Grant 2022C01018, by the National Natural Science Foundation of China under Grants 61973273 and U21B2001, by the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfTffD/content/2301.00108v1.pdf'} diff --git a/JdFLT4oBgHgl3EQfKC80/content/tmp_files/2301.12006v1.pdf.txt b/JdFLT4oBgHgl3EQfKC80/content/tmp_files/2301.12006v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..193269d5d45548d37961ae6f678f8b146d6538ef --- /dev/null +++ b/JdFLT4oBgHgl3EQfKC80/content/tmp_files/2301.12006v1.pdf.txt @@ -0,0 +1,853 @@ +Improved knowledge distillation by utilizing +backward pass knowledge in neural networks +Aref Jafari∗ +University of Waterloo +aref.jafari@uwaterloo.ca +Mehdi Rezagholizadeh +Huawei Noah’s Ark Lab +mehdi.rezagholizadeh@huawei.com +Ali Ghodsi +University of Waterloo +ali.ghodsi@uwaterloo.ca +Abstract +Knowledge distillation (KD) is one of the prominent techniques for model com- +pression. In this method, the knowledge of a large network (teacher) is distilled +into a model (student) with usually significantly fewer parameters. KD tries to +better-match the output of the student model to that of the teacher model based on +the knowledge extracts from the forward pass of the teacher network. Although +conventional KD is effective for matching the two networks over the given data +points, there is no guarantee that these models would match in other areas for which +we do not have enough training samples. In this work, we address that problem +by generating new auxiliary training samples based on extracting knowledge from +the backward pass of the teacher in the areas where the student diverges greatly +from the teacher. We compute the difference between the teacher and the student +and generate new data samples that maximize the divergence. This is done by +perturbing data samples in the direction of the gradient of the difference between +the student and the teacher. Augmenting the training set by adding this auxiliary +improves the performance of KD significantly and leads to a closer match between +the student and the teacher. Using this approach, when data samples come from a +discrete domain, such as applications of natural language processing (NLP) and +language understanding, is not trivial. However, we show how this technique can +be used successfully in such applications. We evaluated the performance of our +method on various tasks in computer vision and NLP domains and got promising +results. +1 +Introduction +During the last few years, we faced the emerge of a huge number of cumbersome state-of-the-art deep +neural network models in different fields of machine learning, including computer vision [27, 10], +natural language processing [16, 12, 13, 3] and speech processing [1, 7]. We need powerful servers +to be able to deploy such large models. Running such large models on edge devices would be +infeasible due to the limited memory and computational power of edge devices [22]. On the other +hand, considering users’ privacy concerns, network reliability issues, and network delays increase +the demand for offline machine learning solutions on edge devices. The field of neural model +compression focuses on providing compression solutions such as quantization [11], pruning [26], +tensor decomposition [24] and knowledge distillation (KD) [9] for large neural networks. +∗This work was done while doing internship at Huawei Noah’s Ark Lab +Preprint. Under review. +arXiv:2301.12006v1 [cs.LG] 27 Jan 2023 + +(a) +(b) +Figure 1: (a) Minimization Step: Using the teacher model knowledge for training the student in KD +(utilizing forward knowledge) (b) Maximization Step: Augmenting the input dataset x with auxiliary +data samples x′ which is generated by the back propagation of gradient through both networks +(utilizing backward knowledge) +Knowledge distillation (KD) is one of the most prominent compression techniques in the literature. +As its name implies, KD tries to transfer the learned knowledge from a large teacher network to a +small student. The idea of KD was proposed by Rich Caruana et al. [4] for the first time and later this +idea generalized by Hinton et al. 2015 [9] for deep neural nets. The original KD method concerns +transferring knowledge from a teacher to a student network only by matching their forward pass +outputs. Later on, several works in the literature suggested other sources of knowledge in the teacher +network besides the logit outputs of the last layer. This includes using intermediate layer feature +maps [20, 21, 12], gradients of the network outputs w.r.t the inputs [5, 19]), and matching decision +boundaries for classification tasks [8]. using this additional information might be useful to get the +student network performance closer to that of the teacher. +In this work, we focus on identifying regions of the input space of the teacher and student networks in +which the two functions diverge the most from each other. Moreover, we highlight the importance of +incorporating backward knowledge of the teacher and student networks in the knowledge distillation +process. Our proposed iterative backward KD approach is comprised of: first, a maximization +step in which a new set of auxiliary training samples is generated by pushing training samples +towards maximum divergence regions of the two functions; second, a minimization step in which +the student network is trained using the regular KD approach over its training data together with the +generated auxiliary samples from the first step. We show the success of our backward KD technique +in improving KD on both classification and regression tasks over the image and textual data and also +in the few-sample KD scenario. We summarize the main contributions of this paper in the following: +• Our technique extracts knowledge from both the forward and backward passes of the teacher +and student networks in order to identify the maximum divergence regions between the two +functions and generate auxiliary data samples around those regions. +• We provide a solution on how to address the non-differentiability of discrete tokens in NLP +applications. +• Our approach is generic and is applicable to any improved KD approach. +• The results of our experiments, show 4% improvement on MNIST with a student network +that is 160 times smaller, 1% improvement on the CIFAR-10 dataset with a student that +is 9 times smaller, and an average 1.5% improvement on the GLUE benchmark with a +distilroBERTa-base student. +2 + +LKD(c) +T(oackpropagation +oackpropagation +VLBKD(c) +VαLBKD(α) +LBKD(α) +T(i)2 +Related Works +2.1 +Knowledge Distillation +In the original KD, the process of transferring knowledge from a teacher to a student model accom- +plishes by minimizing a loss function between the logits of student and teacher networks. This loss +function has been used in addition to the regular training loss function for the student network. In +other words, we have an additional loss term in the KD loss function between the softmax outputs of +teacher and student networks which is softened by a temperature term. +LKD = αL +� +Softmax +� +S(x) +� +, y +� ++ (1 − α)L +� +Softmax +�S(x) +τ +� +, Softmax +�T(x) +τ +�� +(1) +where S(x) and T(x) are student and teacher networks respectively. τ is the temperature parameter +and α is a coefficient between [0, 1]. This loss function is a linear combination of two loss functions. +The first loss function minimizes the difference between the output of the student model and the given +true label. The second loss function minimizes the difference between the outputs of the student +model and the teacher model. Therefore the whole loss function minimizes the distance between the +student and both underlying and teacher functions. Since the teacher network is assumed to be a good +approximation of the underlying function, it should be close enough to the underlying function of +data samples. Fig. 2-(a) shows a simple example with three data points, an underlying function, a +trained teacher and a potential student function that satisfies the KD loss function in eq. 1. However, +the problem is, even though the student satisfies the KD objective function and intersects the teacher +function close to the training data samples, there is no guarantee that it would fit the teacher network +in other regions of the input space as well. In this work, we try to address this problem by deploying +the backward gradient information w.r.t the input (we refer to as backward knowledge) in the two +networks. +2.2 +Sobolev Training for KD +As we mentioned in 2.1 (see Fig. 2.), the KD loss cannot guarantee the student and teacher functions +to match over the entire input space. The reason is training two networks based on the original KD +loss function would only match their output values on the training samples and not their gradients. +There are some work in the literature to address this issue by matching the gradients of the two +networks at given training samples during training [5, 19]. However, since we usually deal with +networks with multidimensional inputs and outputs, the gradients of output vectors w.r.t input vectors +give rise to large Jacobin matrices. Matching these Jacobian matrices is not computationally efficient +and is not practical in real-world problems. +Sobolev training [5] proposes a solution to avoid large Jacobian matrices: instead of directly matching +the gradients of the two networks, one can match the projection of the gradients onto a random +vector v which is sampled uniformly from the unit sphere. Although this approach can reduce the +computational complexity of matching gradients during the training, still computing Jacobian matrices +before this projection can be very computationally expensive (especially for NLP applications that +deal with large vocabulary sizes). To tackle this problem in our work, we define a new scalar loss +function based on an l2 norm to measure the distance between the teacher and student networks (see +Fig. 2-(c)). Gradients of this scalar loss function is a vector with the same size as the input vector x +and can be used as a proxy for the network gradients introduced in [5, 19]. +3 +Methodology: Improving Knowledge Distillation using Backward Pass +Knowledge +In this section, we propose our improved KD method based on generating new out of sample points +around the areas of the input domain where the student output diverges greatly from the teacher. This +approach identifies the areas of the input space X around which the two functions have maximum +distance. Then we generate out of sample points X′ ⊂ X from the existing training set X ⊂ X over +those regions. These new generated samples X′ can be labelled by the teacher and then X ← X ∪X′ +be deployed in the KD’s training process to match the student better to the teacher over a broader +3 + +Figure 2: Visualizing the data insufficiency issue for the original KD algorithm. (a) behaviour of +the teacher and the student function when training with KD loss. (b) divergence areas between the +teacher and the student networks. (c) behaviour of l2 − norm loss function between teacher and the +student and the way of obtaining auxiliary data samples. +range in the input space (see Fig. 2). We show that augmenting the training set by adding this auxiliary +set improves the performance of KD significantly and leads to a closer match between the student +and teacher. Our improved KD approach follows a procedure similar to the minimax principle [2] : +first, in the maximization step we generate auxiliary data samples; second, in the minimization step +we apply regular KD on the union of existing X and generated auxiliary data X′. +To have a better understanding of how this can be cast as an instance of minimax estimator, assume +that we are given the data samples {xi, T(xi))}N +i=1. The goal is to estimate T(x) by S(x). We +may seek an estimator S(x) attaining the minimax principle. In minimax principle, where θ is an +estimand and δ is an estimator, we evaluate all estimators according to its maximum risk R(θ, δ). An +estimator δ0 , then, is said to be minimax if: +sup +θ +R(θ, δ0) = inf +δ∈C sup +θ∈Θ +R(θ, δ) +(2) +That is we chose the estimator for the situation that the worst divergence between θ and δ is smallest. +We follow a similar insight: i.e. the maximization step computes X′, where there is the worst +divergence between the teacher and the student. The minimization step finds the weights of the +student network such that the difference between the student and teacher for this worst scenario is the +smallest. +min +w max +x +R(Tx, Sx,w) +(3) +3.1 +Maximization Step: Generating Auxilary Data based on Backward-KD Loss +In the maximization step of our technique, we define a new loss function (we refer to as the backward +KD loss or BKD throughout this paper) to measure the distance between the output of the teacher +and the student networks: +LBKD = ||S(x) − T(x)||2 +2 +(4) +Here the main idea is that by taking the gradient of LBKD loss function in eq. 4 w.r.t the input samples, +we can perturb the training samples along the directions of their gradients to increase the loss between +two networks. Using this process, we can generate new auxiliary training samples for which the +student and the teacher networks are in maximum distance. To obtain these auxiliary data samples, +we can consider the following optimization problem. +x′ = max +x∈X ||S(x) − T(x)||2 +2 +(5) +We can solve this problem using stochastic gradient ascent method. Therefore our perturbation +formula for each data sample will be: +xi+1 = xi + η ∇x ||S(x) − T(x)||2 +2 +(6) +4 + +(b) +y. +X1 +X2 +X3 +a +L(αx) = II S(x) -T(x) /2 +y=f(x)[underlying function +r = T(x) [Teacher Network] +ys = S(x) [Student Network] +Data Samples +Logits +Auxiliary Samples +22where in this formula η is the perturbation rate. This is an iterative algorithm and i is the iteration +index. xi is a training sample at ith iteration. Each time, we perturb xi by adding a portion of the +gradient of loss to this sample. For more detail about this algorithm consider algorithm 1 in the +Appendix. +Fig. 2 demonstrates our idea using a simple example more clearly. Fig. 2-(a) shows a trained teacher +and student functions given the training samples (x1,y1), (x2,y2), (x3,y3). Fig. 2-(c) constructs the +LBKD between these two networks. LBKD shows where the two networks diverge in the original +space. Bear in mind that LBKD gives a scalar for each input. Hence, the gradient of LBKD with +respect to input variable x will be a vector with the same size as the variable x. Therefore, it does not +need to deal with the large dimensionality issue of the Jacobian matrix as described in [5]. Fig. 2-(c) +also illustrates an example of generating one auxiliary sample from the training sample x2. If we +apply eq. 6 to this sample, after several iterations, we will reach to a new auxiliary data point (x′ +2). It +is evident in Fig. 2-(a) that, as expected, there is a large divergence between the teacher and student +networks in (x′ +2) point. +3.2 +Minimization Step: Improving KD with Generated Auxiliary Data +We can apply the maximization step to all given training data to generate their corresponding auxiliary +samples. Then by adding the auxiliary samples X′ into the training dataset X ← X′ ∪ X, we can +train the student network again based on the original KD algorithm over the updated training set in +order to obtain a better output match between the student and teacher networks. Inspired by [15], we +have used the following KD loss function in our work: +LKD = (1 − λ) H +� +σ +� +S(x) +� +, y +� ++ τ 2 λ KL +� +σ +�S(x) +τ +� +, σ +�T(x) +τ +�� +(7) +where σ(.) is the softmax function, H(.) is the cross-entropy loss function, KL(.) is the Kullback +Leibler divergence, λ is a hyper parameter, τ is the temperature parameter, and y is the true labels. +The intuition behind expecting to get a better KD performance using the updated training data is as +follows. Now given the auxiliary data samples which point toward the regions of the input space +where the student and teacher have maximum divergence, these regions of input space are not dark +for the original KD algorithm anymore. Therefore, it is expected from the KD algorithm to be able to +match the student to the teacher network over a larger input space (see Fig. 4). Moreover, it is worth +mentioning that the maximization and minimization steps can be taken multiple times. In this regard, +for each maximization step, we need to construct the auxiliary set X′ from scratch and we do not +need the previously generated auxiliary sets. However, in our few-sample training scenarios where +the number of data samples is small, we can keep the auxiliary samples. +3.3 +Backward KD for NLP Applications +It is not trivial how to deploy the introduced backward KD approach (i.e. calculating ∇xLBKD for +discrete inputs) when data samples come from a discrete domain, such as NLP applications. Here, +we propose a solution to how this technique can be adapted for the NLP domain. For neural NLP +models, first, we pass the one-hot vectors of the input tokens to the so-called embedding layer of +neural networks. Then, these one-hot vectors are converted into embedding vectors (see Fig. 3). The +key for our solution is that embedding vectors of input tokens are not discrete and we can take the +gradient of loss function w.r.t the embedding vectors z. But the problem is that, in the KD algorithm, +we have two networks with different embedding sizes (see Fig. 3). To address this issue, we can take +the gradient of the loss function w.r.t one of the embedding vectors (here student embedding vector +zS). However, then we need a transformation matrix like Q to be able to derive the corresponding +embedding vector zT for the teacher network form zS. +zT = QzS +(8) +We can show that the transform matrix Q is equal to the following equation: +Q = WT W T +S (WSW T +S )−1 +(9) +where in this equation W T +S (WSW T +S )−1 is the pseudo inverse of WS embedding matrix. We refer +you to the Appendix to see the proof of this derivation. Therefore, to obtain the auxiliary samples, +5 + +Figure 3: General procedure of utilizing auxiliary samples in NLP models. Here x is the one-hot +vector of input tokens, W is the embedding matrix, and z is the embedding vector of x. +we can take the gradient of the LBKD loss function w.r.t the student embedding vector zS. Then by +using equations 10 and 9, we can re-construct zT during the steps of data perturbation as following. +zi+1 +S += zi +S + η∇zSLBKD +zi+1 +T += WT W T +S (WSW T +S )−1zi+1 +S +(10) +4 +Experiments and Results +We designed five experiments to evaluate our proposed method.The first experiment is on synthetic +data in order to visualize the idea behind our technique. The second and third experiments are on the +image classification tasks and the last two experiments are in NLP. For all of these experiments, we +followed the general procedure illustrated in algorithm 1 in the Appendix. For NLP experiments, we +applied the method explained in section 3.3 (see algorithm 2 in the Appendix for more details). We +summarize the procedure of our experiments in the following. +Pre-training Step: We train the student network based on the original KD procedure for a few +epochs (e epochs). In this step, the student network will get close to the teacher network around the +given training samples and will diverge from the teacher in some other areas. +Iterative Min Max Step: We do the following two steps iteratively for several epochs (h epochs) : +1) Using the pre-trained student network and the trained teacher network, we use the proposed +maximization step in 3.2 for generating an auxiliary dataset. +2) Combine the auxiliary data with the training dataset and train the student network based on the +augmented dataset using the original KD procedure for e epochs again. +Fine-tuning Step: Finally, fine-tune the student network using original KD only based on the train +samples for e epochs again. The reason for this step is that, although during the previous step the +student network has been got close to the teacher network in general since the student has a limited +amount of parameter, it might not be able to completely converge to the teacher network using all +augmented data samples. On the other hand, since the given data points are more important than the +auxiliary points, then during the last step, we only train the student based on the given dataset in order +to have the maximum match between student and teacher over the given data samples in the end. +4.1 +Synthetic data experiment +For visualizing our technique and showing the intuition behind it, we designed a very simple +experiment to show how the proposed method works over a synthetic setting. In this experiment, we +consider a polynomial function of degree 20 as the trained teacher function. Then, we considered +8 data points on its surface as our data samples to train a student network which is a polynomial +function from degree 15 (see Fig. 4-(a)). As it is depicted in this figure, although the student model +perfectly fits the given data points, it diverges from the teacher model in some areas between the +given points. After applying our backward KD method, we can generate some auxiliary samples +6 + +Backpropagation +Vzs La(α) +La(a) +Teacher +Student +Inner +Layers +ZT = QZs +Embedding +Vectors +Embedding +WT +Ws +Matrix(a) +(b) +(c) +Figure 4: Visualizing the generation of auxiliary samples and their utilization in training the student +model. +in the diverged areas between the teacher and student models in Fig. 4-(b). Then, we augmented +the training data samples with the generated auxiliary samples and trained the student model based +on this new augmented dataset. The resulting student model has achieved a much better fit on the +teacher model as it is evident in Fig. 4-(c). +4.2 +MNIST classification: +In this experiment, one of our goals was testing the performance of the proposed method in the +scenario of extremely small student networks. Because of that, we considered two fully connected +neural networks as student and teacher networks for the MNIST dataset classification task. The +teacher network consists of only one hidden layer with 800 neurons which leads in 636010 trainable +parameters. The student network was an extremely simplified version of the same network with +only 5 neurons in the hidden layer. This network has only 3985 trainable parameters, which is 160x +smaller than the teacher network. The student network is trained in three different ways: a) from +scratch with only training data, b) based on the original KD approach with training data samples +augmented by random noise, and c) based on the proposed method. As it is illustrated in table 1, +the student network which is trained by using the proposed method achieves much better results in +comparison with two other trained networks. +Table 1: Results of experiment on the MNIST dataset +Model +method +#parameters +accuracy on test set +teacher +from scratch +636010 +98.14 +student +from scratch +3985 +87.62 +student +original KD +3985 +88.04 +student +proposed method +3985 +91.45 +4.3 +CIFAR-10 classification +The second experiment is conducted on the CIFAR10 dataset with two popular network structures as +the teacher and the student networks. In this experiment, we used the inception v3 [23] network as +the teacher and mobileNet v2 [17] as the student. The teacher is approximately 9 times bigger than +the student. We repeated the previous experiment on CIFAR10 by using these two networks. Table 2 +shows the results of this experiment. +Table 2: Results of experiment on CIFAR10 dataset +Model +method +#parameters +accuracy on test set +inception v3 (teacher) +from scratch +21638954 +95.41% +mobilenet (student) +from scratch +2236682 +91.17% +mobilenet (student) +original KD +2236682 +91.74% +mobilenet (student) +proposed method +2236682 +92.60% +7 + +student and tezdher models +teacher +8 +student +data +6- +2 +0transferbetweendatasamplestoauxiliarydata +samples +Teacher +8 +student +data +auxiliarydata +2 +0 +0 +6result of proposed method +Teacher +8 +student +student trained by augmented data +6 +data +auxiliary data +2 +04.4 +GLUE tasks +The third experiment is designed based on General Language Understanding Evaluation (GLUE) +benchmark [25] and roBERTa family language models [14, 18]. The GLUE benchmark is a set of nine +language understanding tasks, which are designed to evaluate the performance of natural language +understanding systems. roBERTa models (roBERTa-large, roBERTa-base, and distilroBERTa) are +BERT [6] based language understanding pre-trained models where roBERTa-large and roBERTa-base +are the cumbersome versions which are proposed in [14] and have 24 and 12 transformer layers +respectively. distilroBERTa is the compressed version of these models with 6 transformer layers +and has been trained based on KD procedure proposed in [18] with utilizing the roBERTa-base +as the teacher. The general procedure in GLUE tasks is fine-tuning the pre-trained models for its +down-stream tasks and the average performance score. Here, we fine-tuned the distilroBERTa model +based on the proposed method by utilizing the fine-tuned roBERTa-large teacher for each of these +tasks. As it is shown in table 3, the proposed method could improve the distilroBERTa performance +on most of these tasks. +Table 3: Results of experiment on GLUE tasks +Model (Network) +ColA +SST-2 +MRPC +STS-B +QQP +MNLI +QNLI +RTE +WNLI +Score +roBERTa-large (Teacher) +60.56 +96.33 +89.95 +91.75 +91.01 +89.11 +93.08 +79.06 +56.33 +85.82 +DistilroBERTa (Student) +56.61 +92.77 +84.06 +87.28 +90.8 +84.14 +91.36 +65.70 +56.33 +78.78 +Our DistilroBERTa (Student) +60.49 +92.51 +87.25 +87.56 +91.21 +85.1 +91.19 +71.11 +56.33 +80.30 +4.5 +GLUE tasks with few sample points +In this experiment, we modified the previous experiment slightly to investigate the performance of +the proposed method in the few data sample scenario. Here we randomly select a small portion of +samples in each data set and fine-tuned the distilroBERTa based on these samples. For CoLA, MRPC, +STS-B, QNLI, RTE, and WNLI, 10% of data samples and for SST-2, QQP, and MNLI 5% of them in +the dataset are used for fine-tuning the student model. +Table 4: Results of few sample experiment on GLUE tasks +Model (Network) +ColA +SST-2 +MRPC +STS-B +QQP +MNLI +QNLI +RTE +WNLI +Score +roBERTa-large (Teacher) +60.56 +96.33 +89.95 +91.75 +91.01 +89.11 +93.08 +79.06 +56.33 +85.82 +DistilroBERTa (Student) +43.82 +91.05 +76.96 +81.51 +84.92 +75.88 +83.94 +52.07 +56.33 +71.90 +Our DistilroBERTa (Student) +44.11 +91.74 +77.20 +82.82 +85.32 +76.75 +84.34 +56.31 +56.33 +72.76 +5 +Conclusion +In this paper, we have introduced the backward KD method and showed how we can use the backward +knowledge of teacher model to train the student model. Based on this method, we could easily locate +the diverge areas between teacher and student model in order to acquire auxiliary samples at those +areas with utilizing the gradient of the networks and use these samples in the training procedure of +the student model. We showed that our proposal can be efficiently applied to the KD procedure to +improve its performance. Also, we introduced an efficient way to apply backward KD on discrete +domain applications such as NLP tasks. In addition to the synthetic experiment which is performed +to visualize the mechanism of our method, we tested its performance on several image and NLP +tasks. Also, we examined the extremely small student and the few sample scenarios in two of +these experiments. We showed that the backward KD can improve the performance of the trained +student network in all of these practices. We believe that all auxiliary samples do not have the same +contribution to improving the performance of the student model. Also perturbing all data samples +can be computationally expensive in large datasets. +Broader Impact +This research provides a simple but efficient method for model compression and knowledge distillation +which is easily applicable on a variety of domains in machine learning from computer vision to +8 + +natural language processing with the hope of achieving better results. The proposed procedure in this +work is a general procedure which can be used beside the other KD methods in order to improve their +results. Since the main idea just deals with the data samples and generate more samples for better +training, without any major changes in the body of other algorithms, they can use this procedure in +their methods easily. It is applicable in different scenarios like extremely small student models, few +data sample regimes, and zero-shot KD. +Acknowledgments +We thank Mindspore2 for the partial support of this work. We thank the anonymous reviewers for +their insightful comments. +References +[1] BIE, A., VENKITESH, B., MONTEIRO, J., HAIDAR, M., REZAGHOLIZADEH, M., ET AL. Fully +quantizing a simplified transformer for end-to-end speech recognition. arXiv preprint arXiv:1911.03604 +(2019). +[2] BRATKO, I., AND GAMS, M. Error analysis of the minimax principle. In Advances in computer chess. +Elsevier, 1982, pp. 1–15. +[3] BROWN, T. B., MANN, B., RYDER, N., SUBBIAH, M., KAPLAN, J., DHARIWAL, P., NEELAKANTAN, +A., SHYAM, P., SASTRY, G., ASKELL, A., ET AL. Language models are few-shot learners. arXiv preprint +arXiv:2005.14165 (2020). +[4] BUCILU ˇA, C., CARUANA, R., AND NICULESCU-MIZIL, A. Model compression. In Proceedings +of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining (2006), +pp. 535–541. +[5] CZARNECKI, W. M., OSINDERO, S., JADERBERG, M., ´SWIRSZCZ, G., AND PASCANU, R. 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G., ZHU, M., CHEN, B., KALENICHENKO, D., WANG, W., WEYAND, T., ANDREETTO, +M., AND ADAM, H. Mobilenets: Efficient convolutional neural networks for mobile vision applications. +arXiv preprint arXiv:1704.04861 (2017). +[11] JACOB, B., KLIGYS, S., CHEN, B., ZHU, M., TANG, M., HOWARD, A., ADAM, H., AND +KALENICHENKO, D. Quantization and training of neural networks for efficient integer-arithmetic-only +inference. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2018), +pp. 2704–2713. +[12] JIAO, X., YIN, Y., SHANG, L., JIANG, X., CHEN, X., LI, L., WANG, F., AND LIU, Q. Tinybert: +Distilling bert for natural language understanding. arXiv preprint arXiv:1909.10351 (2019). +[13] LAN, Z., CHEN, M., GOODMAN, S., GIMPEL, K., SHARMA, P., AND SORICUT, R. 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Yolo nano: a +highly compact you only look once convolutional neural network for object detection. arXiv preprint +arXiv:1910.01271 (2019). +Supplementary Materials +6 +Transform matrix between student and teacher embedding +If WS ∈ Rd1×|V | be the embedding matrix of the student network and WT ∈ Rd2×|V | be the embedding +matrix of the teacher network, where |V | is the vocabulary size and d1 and d2 are the embedding vector size +of the student and the teacher networks respectively. If x ∈ {0, 1}|v| be the one-hot vector of a token in a text +document and if zS = WSx and zT = WT x be the student and teacher embedding vectors of x, then there +exists a transform matrix Q ∈ Rd2×d1 such that: +zT = QzS +(11) +10 + +Proof: +zT = WT x +(12) +zS = WSx +(13) +We want to find a transform matrix Q such that: +WT = QWS +(14) +For this purpose we can solve the following optimization problem by using list square method: +min +Q ||WT − QWS||2 +(15) +By solving the above optimization problem using the least squares method, we achieves the following solution +for Q: +Q = WT W T +s (WsW T +s )−1 +(16) +Now, from Eq. 14 we have: +WT = QWs +(17) +WT x = QWsx +(18) +zT = Qzs +(19) +7 +Algorithm 1 +Algorithm 1 explains the details of the proposed method in section 3 of the paper. The input variables of our +proposed KD function are the student network S(.), the teacher network T(.), the input dataset X, the number of +training epochs e, and the number of hyper epochs h. In this algorithm, we assume that the teacher network T(.) +has trained and the student network S(.) has not trained yet. Also, we assume X′ is the set of the augmented +data samples. We first initialize X′ with data set X in line 3 of the algorithm. The basic idea is that each time +we train the student network using the Vanilla-KD function for a few training epochs e in the outer loop of +line 4. Then, in line 6 first, we re-initialize X′ with dataset X and in lines 7 to 11 we perturb data samples +in X′ using the gradient of the loss between teacher and student iteratively in order to generate new auxiliary +samples. Then in line 12 we replace X with the union of X and X′ sets. In the next iteration of the loop in +line 4, Vanilla-KD function will be fed with the augmented data samples X′. Note that just in the first iteration, +Vanilla-KD function is fed with original data set X. +Algorithm 1 +1: function PROPOSED-KD(S,T,X, e, h) +2: +▷ S is the student network, T is the teacher network, X is input dataset, e is #training epochs, +h is #hyper epochs +3: +X′ ← X +4: +for i = 1 to h do +5: +VANILLA-KD(S,T,X′,e) +6: +X′ ← X +7: +for x′ in X′ do +8: +while converge do +9: +x′ ← x′ + η∇x||S(x′) − T(x′)||2 +2 +10: +end while +11: +end for +12: +X′ ← X′ ∪ X +13: +end for +14: +VANILLA-KD(S,T,X,e) +15: +return S +16: end function +11 + +8 +Algorithm 2 +Algorithm 2, explains how to apply the proposed method in NLP tasks. This algorithm is almost similar to +algorithm 1. The only main difference is in the way we feed the networks. Here instead of considering the +one-hot index vectors of tokens in the text documents, we consider the embedding vectors zS and zT of the +input vector x (see lines 5 and 6 in the algorithm). Then we fed each of the teacher and the student networks +separately using their own embedding vectors. Only in line 16 we use the transform method which is proposed +in section 3.2 of the paper to transform student’s perturbed embedding vectors into teacher’s embedding vectors. +Algorithm 2 +1: function PROPOSED-KD(S,T,X, e, h) +2: +▷ S is the student network, T is the teacher network, X is input dataset, e is #training epochs, +h is #hyper epochs +3: +WT ← EMBEDDING-MATRIX(T) +4: +WS ← EMBEDDING-MATRIX(S) +5: +ZT ← WT X +6: +ZS ← WSX +7: +Z′ +T ← ZT +8: +Z′ +S ← ZS +9: +for i = 1 to h do +10: +VANILLA-KD(S,T,Z′ +T , Z′ +S,e) +11: +Z′ +T ← ZT +12: +Z′ +S ← ZS +13: +for (z′ +S, z′ +T ) in (Z′ +S, Z′ +T ) do +14: +while converge do +15: +z′ +S ← z′ +S + η∇zS||S(z′ +S) − T(z′ +S)||2 +2 +16: +z′ +T ← WT WS(WSW T +S )−1z′ +S +17: +end while +18: +end for +19: +Z′ +S ← Z′ +S ∪ ZS +20: +Z′ +T ← Z′ +T ∪ ZT +21: +end for +22: +VANILLA-KD(S,T,ZT , ZS,e) +23: +return S +24: end function +12 + diff --git a/JdFLT4oBgHgl3EQfKC80/content/tmp_files/load_file.txt b/JdFLT4oBgHgl3EQfKC80/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31dd1cd6b9ec5bc11929e47aef4041b83f72076b --- /dev/null +++ b/JdFLT4oBgHgl3EQfKC80/content/tmp_files/load_file.txt @@ -0,0 +1,613 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf,len=612 +page_content='Improved knowledge distillation by utilizing backward pass knowledge in neural networks Aref Jafari∗ University of Waterloo aref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='jafari@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='ca Mehdi Rezagholizadeh Huawei Noah’s Ark Lab mehdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='rezagholizadeh@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='com Ali Ghodsi University of Waterloo ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='ghodsi@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='ca Abstract Knowledge distillation (KD) is one of the prominent techniques for model com- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this method, the knowledge of a large network (teacher) is distilled into a model (student) with usually significantly fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' KD tries to better-match the output of the student model to that of the teacher model based on the knowledge extracts from the forward pass of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Although conventional KD is effective for matching the two networks over the given data points, there is no guarantee that these models would match in other areas for which we do not have enough training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this work, we address that problem by generating new auxiliary training samples based on extracting knowledge from the backward pass of the teacher in the areas where the student diverges greatly from the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We compute the difference between the teacher and the student and generate new data samples that maximize the divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This is done by perturbing data samples in the direction of the gradient of the difference between the student and the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Augmenting the training set by adding this auxiliary improves the performance of KD significantly and leads to a closer match between the student and the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Using this approach, when data samples come from a discrete domain, such as applications of natural language processing (NLP) and language understanding, is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' However, we show how this technique can be used successfully in such applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We evaluated the performance of our method on various tasks in computer vision and NLP domains and got promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 1 Introduction During the last few years, we faced the emerge of a huge number of cumbersome state-of-the-art deep neural network models in different fields of machine learning, including computer vision [27, 10], natural language processing [16, 12, 13, 3] and speech processing [1, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We need powerful servers to be able to deploy such large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Running such large models on edge devices would be infeasible due to the limited memory and computational power of edge devices [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' On the other hand, considering users’ privacy concerns, network reliability issues, and network delays increase the demand for offline machine learning solutions on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The field of neural model compression focuses on providing compression solutions such as quantization [11], pruning [26], tensor decomposition [24] and knowledge distillation (KD) [9] for large neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' ∗This work was done while doing internship at Huawei Noah’s Ark Lab Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='12006v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='LG] 27 Jan 2023 (a) (b) Figure 1: (a) Minimization Step: Using the teacher model knowledge for training the student in KD (utilizing forward knowledge) (b) Maximization Step: Augmenting the input dataset x with auxiliary data samples x′ which is generated by the back propagation of gradient through both networks (utilizing backward knowledge) Knowledge distillation (KD) is one of the most prominent compression techniques in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' As its name implies, KD tries to transfer the learned knowledge from a large teacher network to a small student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The idea of KD was proposed by Rich Caruana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' [4] for the first time and later this idea generalized by Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2015 [9] for deep neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The original KD method concerns transferring knowledge from a teacher to a student network only by matching their forward pass outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Later on, several works in the literature suggested other sources of knowledge in the teacher network besides the logit outputs of the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This includes using intermediate layer feature maps [20, 21, 12], gradients of the network outputs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t the inputs [5, 19]), and matching decision boundaries for classification tasks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' using this additional information might be useful to get the student network performance closer to that of the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this work, we focus on identifying regions of the input space of the teacher and student networks in which the two functions diverge the most from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Moreover, we highlight the importance of incorporating backward knowledge of the teacher and student networks in the knowledge distillation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Our proposed iterative backward KD approach is comprised of: first, a maximization step in which a new set of auxiliary training samples is generated by pushing training samples towards maximum divergence regions of the two functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' second, a minimization step in which the student network is trained using the regular KD approach over its training data together with the generated auxiliary samples from the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We show the success of our backward KD technique in improving KD on both classification and regression tasks over the image and textual data and also in the few-sample KD scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We summarize the main contributions of this paper in the following: Our technique extracts knowledge from both the forward and backward passes of the teacher and student networks in order to identify the maximum divergence regions between the two functions and generate auxiliary data samples around those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We provide a solution on how to address the non-differentiability of discrete tokens in NLP applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Our approach is generic and is applicable to any improved KD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The results of our experiments, show 4% improvement on MNIST with a student network that is 160 times smaller, 1% improvement on the CIFAR-10 dataset with a student that is 9 times smaller, and an average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='5% improvement on the GLUE benchmark with a distilroBERTa-base student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2 LKD(c) T(oackpropagation oackpropagation VLBKD(c) VαLBKD(α) LBKD(α) T(i)2 Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='1 Knowledge Distillation In the original KD, the process of transferring knowledge from a teacher to a student model accom- plishes by minimizing a loss function between the logits of student and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This loss function has been used in addition to the regular training loss function for the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In other words, we have an additional loss term in the KD loss function between the softmax outputs of teacher and student networks which is softened by a temperature term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' LKD = αL � Softmax � S(x) � , y � + (1 − α)L � Softmax �S(x) τ � , Softmax �T(x) τ �� (1) where S(x) and T(x) are student and teacher networks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' τ is the temperature parameter and α is a coefficient between [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This loss function is a linear combination of two loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The first loss function minimizes the difference between the output of the student model and the given true label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The second loss function minimizes the difference between the outputs of the student model and the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Therefore the whole loss function minimizes the distance between the student and both underlying and teacher functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Since the teacher network is assumed to be a good approximation of the underlying function, it should be close enough to the underlying function of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(a) shows a simple example with three data points, an underlying function, a trained teacher and a potential student function that satisfies the KD loss function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' However, the problem is, even though the student satisfies the KD objective function and intersects the teacher function close to the training data samples, there is no guarantee that it would fit the teacher network in other regions of the input space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this work, we try to address this problem by deploying the backward gradient information w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t the input (we refer to as backward knowledge) in the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='2 Sobolev Training for KD As we mentioned in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' ), the KD loss cannot guarantee the student and teacher functions to match over the entire input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The reason is training two networks based on the original KD loss function would only match their output values on the training samples and not their gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' There are some work in the literature to address this issue by matching the gradients of the two networks at given training samples during training [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' However, since we usually deal with networks with multidimensional inputs and outputs, the gradients of output vectors w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t input vectors give rise to large Jacobin matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Matching these Jacobian matrices is not computationally efficient and is not practical in real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Sobolev training [5] proposes a solution to avoid large Jacobian matrices: instead of directly matching the gradients of the two networks, one can match the projection of the gradients onto a random vector v which is sampled uniformly from the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Although this approach can reduce the computational complexity of matching gradients during the training, still computing Jacobian matrices before this projection can be very computationally expensive (especially for NLP applications that deal with large vocabulary sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' To tackle this problem in our work, we define a new scalar loss function based on an l2 norm to measure the distance between the teacher and student networks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Gradients of this scalar loss function is a vector with the same size as the input vector x and can be used as a proxy for the network gradients introduced in [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 3 Methodology: Improving Knowledge Distillation using Backward Pass Knowledge In this section, we propose our improved KD method based on generating new out of sample points around the areas of the input domain where the student output diverges greatly from the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This approach identifies the areas of the input space X around which the two functions have maximum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then we generate out of sample points X′ ⊂ X from the existing training set X ⊂ X over those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' These new generated samples X′ can be labelled by the teacher and then X ← X ∪X′ be deployed in the KD’s training process to match the student better to the teacher over a broader 3 Figure 2: Visualizing the data insufficiency issue for the original KD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' (a) behaviour of the teacher and the student function when training with KD loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' (b) divergence areas between the teacher and the student networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' (c) behaviour of l2 − norm loss function between teacher and the student and the way of obtaining auxiliary data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' range in the input space (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We show that augmenting the training set by adding this auxiliary set improves the performance of KD significantly and leads to a closer match between the student and teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Our improved KD approach follows a procedure similar to the minimax principle [2] : first, in the maximization step we generate auxiliary data samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' second, in the minimization step we apply regular KD on the union of existing X and generated auxiliary data X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' To have a better understanding of how this can be cast as an instance of minimax estimator, assume that we are given the data samples {xi, T(xi))}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The goal is to estimate T(x) by S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We may seek an estimator S(x) attaining the minimax principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In minimax principle, where θ is an estimand and δ is an estimator, we evaluate all estimators according to its maximum risk R(θ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' An estimator δ0 , then, is said to be minimax if: sup θ R(θ, δ0) = inf δ∈C sup θ∈Θ R(θ, δ) (2) That is we chose the estimator for the situation that the worst divergence between θ and δ is smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We follow a similar insight: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' the maximization step computes X′, where there is the worst divergence between the teacher and the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The minimization step finds the weights of the student network such that the difference between the student and teacher for this worst scenario is the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' min w max x R(Tx, Sx,w) (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='1 Maximization Step: Generating Auxilary Data based on Backward-KD Loss In the maximization step of our technique, we define a new loss function (we refer to as the backward KD loss or BKD throughout this paper) to measure the distance between the output of the teacher and the student networks: LBKD = ||S(x) − T(x)||2 2 (4) Here the main idea is that by taking the gradient of LBKD loss function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t the input samples, we can perturb the training samples along the directions of their gradients to increase the loss between two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Using this process, we can generate new auxiliary training samples for which the student and the teacher networks are in maximum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' To obtain these auxiliary data samples, we can consider the following optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' x′ = max x∈X ||S(x) − T(x)||2 2 (5) We can solve this problem using stochastic gradient ascent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Therefore our perturbation formula for each data sample will be: xi+1 = xi + η ∇x ||S(x) − T(x)||2 2 (6) 4 (b) y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' X1 X2 X3 a L(αx) = II S(x) -T(x) /2 y=f(x)[underlying function r = T(x) [Teacher Network] ys = S(x) [Student Network] Data Samples Logits Auxiliary Samples 22where in this formula η is the perturbation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This is an iterative algorithm and i is the iteration index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' xi is a training sample at ith iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Each time, we perturb xi by adding a portion of the gradient of loss to this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' For more detail about this algorithm consider algorithm 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2 demonstrates our idea using a simple example more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(a) shows a trained teacher and student functions given the training samples (x1,y1), (x2,y2), (x3,y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(c) constructs the LBKD between these two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' LBKD shows where the two networks diverge in the original space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Bear in mind that LBKD gives a scalar for each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Hence, the gradient of LBKD with respect to input variable x will be a vector with the same size as the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Therefore, it does not need to deal with the large dimensionality issue of the Jacobian matrix as described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(c) also illustrates an example of generating one auxiliary sample from the training sample x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' If we apply eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 6 to this sample, after several iterations, we will reach to a new auxiliary data point (x′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' It is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2-(a) that, as expected, there is a large divergence between the teacher and student networks in (x′ 2) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='2 Minimization Step: Improving KD with Generated Auxiliary Data We can apply the maximization step to all given training data to generate their corresponding auxiliary samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then by adding the auxiliary samples X′ into the training dataset X ← X′ ∪ X, we can train the student network again based on the original KD algorithm over the updated training set in order to obtain a better output match between the student and teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Inspired by [15], we have used the following KD loss function in our work: LKD = (1 − λ) H � σ � S(x) � , y � + τ 2 λ KL � σ �S(x) τ � , σ �T(x) τ �� (7) where σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=') is the softmax function, H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=') is the cross-entropy loss function, KL(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=') is the Kullback Leibler divergence, λ is a hyper parameter, τ is the temperature parameter, and y is the true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The intuition behind expecting to get a better KD performance using the updated training data is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Now given the auxiliary data samples which point toward the regions of the input space where the student and teacher have maximum divergence, these regions of input space are not dark for the original KD algorithm anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Therefore, it is expected from the KD algorithm to be able to match the student to the teacher network over a larger input space (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Moreover, it is worth mentioning that the maximization and minimization steps can be taken multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this regard, for each maximization step, we need to construct the auxiliary set X′ from scratch and we do not need the previously generated auxiliary sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' However, in our few-sample training scenarios where the number of data samples is small, we can keep the auxiliary samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='3 Backward KD for NLP Applications It is not trivial how to deploy the introduced backward KD approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' calculating ∇xLBKD for discrete inputs) when data samples come from a discrete domain, such as NLP applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Here, we propose a solution to how this technique can be adapted for the NLP domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' For neural NLP models, first, we pass the one-hot vectors of the input tokens to the so-called embedding layer of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then, these one-hot vectors are converted into embedding vectors (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The key for our solution is that embedding vectors of input tokens are not discrete and we can take the gradient of loss function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t the embedding vectors z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' But the problem is that, in the KD algorithm, we have two networks with different embedding sizes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' To address this issue, we can take the gradient of the loss function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t one of the embedding vectors (here student embedding vector zS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' However, then we need a transformation matrix like Q to be able to derive the corresponding embedding vector zT for the teacher network form zS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' zT = QzS (8) We can show that the transform matrix Q is equal to the following equation: Q = WT W T S (WSW T S )−1 (9) where in this equation W T S (WSW T S )−1 is the pseudo inverse of WS embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We refer you to the Appendix to see the proof of this derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Therefore, to obtain the auxiliary samples, 5 Figure 3: General procedure of utilizing auxiliary samples in NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Here x is the one-hot vector of input tokens, W is the embedding matrix, and z is the embedding vector of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' we can take the gradient of the LBKD loss function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='t the student embedding vector zS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then by using equations 10 and 9, we can re-construct zT during the steps of data perturbation as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' zi+1 S = zi S + η∇zSLBKD zi+1 T = WT W T S (WSW T S )−1zi+1 S (10) 4 Experiments and Results We designed five experiments to evaluate our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='The first experiment is on synthetic data in order to visualize the idea behind our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The second and third experiments are on the image classification tasks and the last two experiments are in NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' For all of these experiments, we followed the general procedure illustrated in algorithm 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' For NLP experiments, we applied the method explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='3 (see algorithm 2 in the Appendix for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We summarize the procedure of our experiments in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Pre-training Step: We train the student network based on the original KD procedure for a few epochs (e epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this step, the student network will get close to the teacher network around the given training samples and will diverge from the teacher in some other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Iterative Min Max Step: We do the following two steps iteratively for several epochs (h epochs) : 1) Using the pre-trained student network and the trained teacher network, we use the proposed maximization step in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='2 for generating an auxiliary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 2) Combine the auxiliary data with the training dataset and train the student network based on the augmented dataset using the original KD procedure for e epochs again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Fine-tuning Step: Finally, fine-tune the student network using original KD only based on the train samples for e epochs again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The reason for this step is that, although during the previous step the student network has been got close to the teacher network in general since the student has a limited amount of parameter, it might not be able to completely converge to the teacher network using all augmented data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' On the other hand, since the given data points are more important than the auxiliary points, then during the last step, we only train the student based on the given dataset in order to have the maximum match between student and teacher over the given data samples in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='1 Synthetic data experiment For visualizing our technique and showing the intuition behind it, we designed a very simple experiment to show how the proposed method works over a synthetic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this experiment, we consider a polynomial function of degree 20 as the trained teacher function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then, we considered 8 data points on its surface as our data samples to train a student network which is a polynomial function from degree 15 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4-(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' As it is depicted in this figure, although the student model perfectly fits the given data points, it diverges from the teacher model in some areas between the given points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' After applying our backward KD method, we can generate some auxiliary samples 6 Backpropagation Vzs La(α) La(a) Teacher Student Inner Layers ZT = QZs Embedding Vectors Embedding WT Ws Matrix(a) (b) (c) Figure 4: Visualizing the generation of auxiliary samples and their utilization in training the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' in the diverged areas between the teacher and student models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then, we augmented the training data samples with the generated auxiliary samples and trained the student model based on this new augmented dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The resulting student model has achieved a much better fit on the teacher model as it is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='2 MNIST classification: In this experiment, one of our goals was testing the performance of the proposed method in the scenario of extremely small student networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Because of that, we considered two fully connected neural networks as student and teacher networks for the MNIST dataset classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The teacher network consists of only one hidden layer with 800 neurons which leads in 636010 trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The student network was an extremely simplified version of the same network with only 5 neurons in the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This network has only 3985 trainable parameters, which is 160x smaller than the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The student network is trained in three different ways: a) from scratch with only training data, b) based on the original KD approach with training data samples augmented by random noise, and c) based on the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' As it is illustrated in table 1, the student network which is trained by using the proposed method achieves much better results in comparison with two other trained networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Table 1: Results of experiment on the MNIST dataset Model method #parameters accuracy on test set teacher from scratch 636010 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='14 student from scratch 3985 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='62 student original KD 3985 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='04 student proposed method 3985 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='3 CIFAR-10 classification The second experiment is conducted on the CIFAR10 dataset with two popular network structures as the teacher and the student networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this experiment, we used the inception v3 [23] network as the teacher and mobileNet v2 [17] as the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The teacher is approximately 9 times bigger than the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We repeated the previous experiment on CIFAR10 by using these two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Table 2 shows the results of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Table 2: Results of experiment on CIFAR10 dataset Model method #parameters accuracy on test set inception v3 (teacher) from scratch 21638954 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='41% mobilenet (student) from scratch 2236682 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='17% mobilenet (student) original KD 2236682 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='74% mobilenet (student) proposed method 2236682 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='60% 7 student and tezdher models teacher 8 student data 6- 2 0transferbetweendatasamplestoauxiliarydata samples Teacher 8 student data auxiliarydata 2 0 0 6result of proposed method Teacher 8 student student trained by augmented data 6 data auxiliary data 2 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='4 GLUE tasks The third experiment is designed based on General Language Understanding Evaluation (GLUE) benchmark [25] and roBERTa family language models [14, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The GLUE benchmark is a set of nine language understanding tasks, which are designed to evaluate the performance of natural language understanding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' roBERTa models (roBERTa-large, roBERTa-base, and distilroBERTa) are BERT [6] based language understanding pre-trained models where roBERTa-large and roBERTa-base are the cumbersome versions which are proposed in [14] and have 24 and 12 transformer layers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' distilroBERTa is the compressed version of these models with 6 transformer layers and has been trained based on KD procedure proposed in [18] with utilizing the roBERTa-base as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The general procedure in GLUE tasks is fine-tuning the pre-trained models for its down-stream tasks and the average performance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Here, we fine-tuned the distilroBERTa model based on the proposed method by utilizing the fine-tuned roBERTa-large teacher for each of these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' As it is shown in table 3, the proposed method could improve the distilroBERTa performance on most of these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Table 3: Results of experiment on GLUE tasks Model (Network) ColA SST-2 MRPC STS-B QQP MNLI QNLI RTE WNLI Score roBERTa-large (Teacher) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='56 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='95 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='75 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='01 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='11 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='08 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='06 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='82 DistilroBERTa (Student) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='61 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='77 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='06 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='28 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='14 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='36 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='70 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='78 Our DistilroBERTa (Student) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='49 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='51 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='25 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='56 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='21 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='19 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='11 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='5 GLUE tasks with few sample points In this experiment, we modified the previous experiment slightly to investigate the performance of the proposed method in the few data sample scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Here we randomly select a small portion of samples in each data set and fine-tuned the distilroBERTa based on these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' For CoLA, MRPC, STS-B, QNLI, RTE, and WNLI, 10% of data samples and for SST-2, QQP, and MNLI 5% of them in the dataset are used for fine-tuning the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Table 4: Results of few sample experiment on GLUE tasks Model (Network) ColA SST-2 MRPC STS-B QQP MNLI QNLI RTE WNLI Score roBERTa-large (Teacher) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='56 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='95 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='75 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='01 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='11 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='08 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='06 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='82 DistilroBERTa (Student) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='82 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='05 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='96 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='51 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='92 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='88 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='94 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='07 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='90 Our DistilroBERTa (Student) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='11 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='74 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='20 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='82 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='32 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='75 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='34 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='31 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='33 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='76 5 Conclusion In this paper, we have introduced the backward KD method and showed how we can use the backward knowledge of teacher model to train the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Based on this method, we could easily locate the diverge areas between teacher and student model in order to acquire auxiliary samples at those areas with utilizing the gradient of the networks and use these samples in the training procedure of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We showed that our proposal can be efficiently applied to the KD procedure to improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Also, we introduced an efficient way to apply backward KD on discrete domain applications such as NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In addition to the synthetic experiment which is performed to visualize the mechanism of our method, we tested its performance on several image and NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Also, we examined the extremely small student and the few sample scenarios in two of these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We showed that the backward KD can improve the performance of the trained student network in all of these practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We believe that all auxiliary samples do not have the same contribution to improving the performance of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Also perturbing all data samples can be computationally expensive in large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Broader Impact This research provides a simple but efficient method for model compression and knowledge distillation which is easily applicable on a variety of domains in machine learning from computer vision to 8 natural language processing with the hope of achieving better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The proposed procedure in this work is a general procedure which can be used beside the other KD methods in order to improve their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Since the main idea just deals with the data samples and generate more samples for better training, without any major changes in the body of other algorithms, they can use this procedure in their methods easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' It is applicable in different scenarios like extremely small student models, few data sample regimes, and zero-shot KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Acknowledgments We thank Mindspore2 for the partial support of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We thank the anonymous reviewers for their insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' References [1] BIE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='12579 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' [27] WONG, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=', FAMUORI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=', SHAFIEE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=', LI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=', CHWYL, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=', AND CHUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Yolo nano: a highly compact you only look once convolutional neural network for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='01271 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Supplementary Materials 6 Transform matrix between student and teacher embedding If WS ∈ Rd1×|V | be the embedding matrix of the student network and WT ∈ Rd2×|V | be the embedding matrix of the teacher network, where |V | is the vocabulary size and d1 and d2 are the embedding vector size of the student and the teacher networks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' If x ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 1}|v| be the one-hot vector of a token in a text document and if zS = WSx and zT = WT x be the student and teacher embedding vectors of x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' then there exists a transform matrix Q ∈ Rd2×d1 such that: zT = QzS (11) 10 Proof: zT = WT x (12) zS = WSx (13) We want to find a transform matrix Q such that: WT = QWS (14) For this purpose we can solve the following optimization problem by using list square method: min Q ||WT − QWS||2 (15) By solving the above optimization problem using the least squares method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' we achieves the following solution for Q: Q = WT W T s (WsW T s )−1 (16) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' 14 we have: WT = QWs (17) WT x = QWsx (18) zT = Qzs (19) 7 Algorithm 1 Algorithm 1 explains the details of the proposed method in section 3 of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The input variables of our proposed KD function are the student network S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' ), the teacher network T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' ), the input dataset X, the number of training epochs e, and the number of hyper epochs h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In this algorithm, we assume that the teacher network T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=') has trained and the student network S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=') has not trained yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Also, we assume X′ is the set of the augmented data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' We first initialize X′ with data set X in line 3 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The basic idea is that each time we train the student network using the Vanilla-KD function for a few training epochs e in the outer loop of line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then, in line 6 first, we re-initialize X′ with dataset X and in lines 7 to 11 we perturb data samples in X′ using the gradient of the loss between teacher and student iteratively in order to generate new auxiliary samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then in line 12 we replace X with the union of X and X′ sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' In the next iteration of the loop in line 4, Vanilla-KD function will be fed with the augmented data samples X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Note that just in the first iteration, Vanilla-KD function is fed with original data set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Algorithm 1 1: function PROPOSED-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' h) 2: ▷ S is the student network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' T is the teacher network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' X is input dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' e is #training epochs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' h is #hyper epochs 3: X′ ← X 4: for i = 1 to h do 5: VANILLA-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e) 6: X′ ← X 7: for x′ in X′ do 8: while converge do 9: x′ ← x′ + η∇x||S(x′) − T(x′)||2 2 10: end while 11: end for 12: X′ ← X′ ∪ X 13: end for 14: VANILLA-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e) 15: return S 16: end function 11 8 Algorithm 2 Algorithm 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' explains how to apply the proposed method in NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' This algorithm is almost similar to algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' The only main difference is in the way we feed the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Here instead of considering the one-hot index vectors of tokens in the text documents, we consider the embedding vectors zS and zT of the input vector x (see lines 5 and 6 in the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Then we fed each of the teacher and the student networks separately using their own embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Only in line 16 we use the transform method which is proposed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='2 of the paper to transform student’s perturbed embedding vectors into teacher’s embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Algorithm 2 1: function PROPOSED-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' h) 2: ▷ S is the student network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' T is the teacher network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' X is input dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' e is #training epochs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' h is #hyper epochs 3: WT ← EMBEDDING-MATRIX(T) 4: WS ← EMBEDDING-MATRIX(S) 5: ZT ← WT X 6: ZS ← WSX 7: Z′ T ← ZT 8: Z′ S ← ZS 9: for i = 1 to h do 10: VANILLA-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='Z′ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Z′ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e) 11: Z′ T ← ZT 12: Z′ S ← ZS 13: for (z′ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' z′ T ) in (Z′ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' Z′ T ) do 14: while converge do 15: z′ S ← z′ S + η∇zS||S(z′ S) − T(z′ S)||2 2 16: z′ T ← WT WS(WSW T S )−1z′ S 17: end while 18: end for 19: Z′ S ← Z′ S ∪ ZS 20: Z′ T ← Z′ T ∪ ZT 21: end for 22: VANILLA-KD(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='ZT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content=' ZS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} +page_content='e) 23: return S 24: end function 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFLT4oBgHgl3EQfKC80/content/2301.12006v1.pdf'} diff --git a/N9E1T4oBgHgl3EQfZwR9/content/tmp_files/2301.03154v1.pdf.txt b/N9E1T4oBgHgl3EQfZwR9/content/tmp_files/2301.03154v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..05d44b0bc3bc88bb8924d90e2a09bcbc916398d0 --- /dev/null +++ b/N9E1T4oBgHgl3EQfZwR9/content/tmp_files/2301.03154v1.pdf.txt @@ -0,0 +1,1151 @@ +arXiv:2301.03154v1 [physics.atom-ph] 9 Jan 2023 +Calculation of the hyperfine structure of Dy, Ho, Cf, and Es +Saleh O. Allehabi, V. A. Dzuba, and V. V. Flambaum +School of Physics, University of New South Wales, Sydney 2052, Australia +(Dated: January 10, 2023) +A recently developed version of the configuration interaction (CI) method for open shells with +a large number of valence electrons has been used to study two heavy atoms, californium (Cf, +Z= 98) and einsteinium (Es, Z= 99). Motivated by experimental work to measure the hyperfine +structure (HFS) for these atoms, we perform the calculations of the magnetic dipole HFS constants +A and electric quadrupole HFS constant B for the sake of interpretation of the measurements in +terms of nuclear magnetic moment µ and electric quadrupole moment Q. For verification of our +computations, we have also carried out similar calculations for the lighter homologs dysprosium (Dy, +Z= 66) and holmium (Ho, Z= 67), whose electronic structures are similar to Cf and Es, respectively. +We have conducted a revision of the nuclear moments of some isotopes of Es leading to an improved +value of the magnetic moment of 253Es [µ(253Es) = 4.20(13)µN ]. +I. +INTRODUCTION +The study of atomic properties of heavy actinides has +gained growing interest [1–8]. Transition frequencies and +hyperfine structure (HFS) are being measured. Measur- +ing HFS is motivated by obtaining data on the nuclear +momenta of heavy nuclei. This would advance our knowl- +edge about the nuclear structure of superheavy nuclei +benefiting the search for the hypothetical stability island. +In light of this, we focus on theoretically studying of the +hyperfine structure for heavy actinides, californium (Cf, +Z= 98) and einsteinium (Es, Z= 99). +Combining the +calculations with the measurements would allow the ex- +traction of the nuclear magnetic moment µ and electric +quadrupole moments Q of the studied isotopes. +HFS constants of some states of odd isotopes of Cf +(249Cf,251Cf,253Cf) were recently measured and nuclear +moments µ and Q were extracted using our calcula- +tions [8]. This work presents a detailed account of these +calculations as well as similar calculations for Es. In the +case of Es, there are no theoretical results currently avail- +able, whereas several experimental papers have been pub- +lished. Using different empirical techniques, Refs. [1–3] +studied the HFS of Es for three isotopes with non-zero +nuclear spins, 253,254,255Es. +Heavy actinides like Cf and Es, are atoms with an open +5f subshell. The number of electrons on open shells is +twelve for Cf and thirteen for Es (including the 7s elec- +trons).This presents a challenge for the calculations. We +use the configuration interaction with perturbation the- +ory (CIPT) [9] method, which has been developed for +such systems. To check the applicability of the method +and the expected accuracy of the results we performed +similar calculations for lanthanides dysprosium (Dy, Z= +66) and holmium (Ho, Z= 67), whose electronic struc- +tures are similar to Cf and Es, respectively. Both, Dy +and Ho were extensively studied experimentally and the- +oretically (see, e.g. [10–18]). Here we compare our results +to experimental data, Refs. [10, 16] for Dy and Refs. [16– +18] for Ho, to check the accuracy of the method we use. +II. +METHOD OF CALCULATION +A. +Calculation of energy levels +As it was mentioned in the introduction the Dy and Cf +atoms have twelve valence electrons, the Ho and Es atoms +have thirteen valence electrons. It is well known that as +the number of valence electrons increases, the size of the +CI matrix increases dramatically, making the standard +CI calculations practically impossible for such systems. +In this work we use the CIPT method [9] which has been +especially developed for such systems. It reduces the size +of the CI matrix by neglecting the off-diagonal matrix el- +ements between high-energy states and reducing the con- +tribution of these states to the perturbation theory-like +corrections to the matrix elements between low-energy +states. The size of the resulting CI matrix is equal to the +number of low-energy states. +The CI Hamiltonian can be written as follows +ˆHCI = +Nv +� +i=1 +ˆHHF +i ++ +Nv +� +i 0. In this case we have equation (21) has be form +z′ +� +z + b +2 +�2 − 1 +4(b2 − 4c) += −1, +⇒ +z′ +� +z + b +2 +�2 − +� +1 +2 +√ +b2 − 4c +�2 = −1, +⇒ +1 +21 +2 +√ +b2 − 4c +log +����� +z + b +2 − 1 +2 +√ +b2 − 4c +z + b +2 + 1 +2 +√ +b2 − 4c +����� = −x + C1,1. +(22) +Let in the equation (22): z + b +2 = ξ, 1 +2 +√ +b2 − 4c = γ. Then we obtain +1 +√ +b2 − 4c +log +���� +ξ − γ +ξ + γ +���� = −x + C1,1, +⇒ +ξ − γ +ξ + γ = C1,2e− +√ +b2−4cx, +C1,2 = eC1,1, +⇒ +1 − +2γ +ξ + γ = C1,2e− +√ +b2−4cx, +⇒ +1 +ξ + γ = 1 +2γ + C1,3e− +√ +b2−4cx, +C1,3 = − 1 +2γ C1,2, +⇒ +ξ + γ = +1 +1 +2γ + C1,3e− +√ +b2−4cx . +(23) +Returning to the change of variables z + b +2 = ξ, 1 +2 +√ +b2 − 4c = γ in the equation (23), we obtain +z + b +2 + 1 +2 +� +b2 − 4c = +1 +1 +√ +b2−4c + C1,3e− +√ +b2−4cx , +⇒ +z = − +�b +2 + 1 +2 +� +b2 − 4c +� ++ +1 +1 +√ +b2−4c + C1,3e− +√ +b2−4cx, +⇒ +6 + +z = − +�b +2 + 1 +2 +� +b2 − 4c +� ++ +√ +b2 − 4c +1 + C1,4e− +√ +b2−4cx , +C1,4 = +� +b2 − 4cC1,3, +⇒ +z = − +�b +2 + 1 +2 +� +b2 − 4c +� ++ e +√ +b2−4cx√ +b2 − 4c +e +√ +b2−4cx + C1,4 +. +(24) +Because z = (log |y|)′, then we have in the equation (24) +(log |y|)′ = − +�b +2 + 1 +2 +� +b2 − 4c +� ++ e +√ +b2−4cx√ +b2 − 4c +e +√ +b2−4cx + C1,4 +, +⇒ +log |y| = +� e +√ +b2−4cx√ +b2 − 4c +e +√ +b2−4cx + C1,4 +dx − +�b +2 + 1 +2 +� +b2 − 4c +� +x + log |C2,1| = += log +���e +√ +b2−4cx + C1,4 +��� − +� b +2 + 1 +2 +� +b2 − 4c +� +x + log |C2,1|, +⇒ +y = +� +e +√ +b2−4cx + C1,4 +� +e−( b +2+ 1 +2 +√ +b2−4c)xC2,1, +⇒ +y = C1e(− b +2 − 1 +2 +√ +b2−4c)x + C2e(− b +2+ 1 +2 +√ +b2−4c)x, +(25) +where C1 = C2,1C1,4, C2 = C2,1 is an integration constant. +Case 2. b2 − 4c = 0. In this case we have equation (21) has be form +z′ +� +z + b +2 +�2 = −1. +Step by step from the last equation we obtain +� +− +1 +z + b +2 +�′ += −1, +⇒ +1 +z + b +2 += x + C1,5, +⇒ +z + b +2 = +1 +x + C1,5 +, +⇒ +z = −b +2 + +1 +x + C1,5 +. +(26) +Because z = (log |y|)′, then we have in the equation (26) +(log |y|)′ = −b +2 + +1 +x + C1,5 +, +⇒ +log |y| = −b +2x + log |x + C1,5| + log |C2| , +⇒ +y = e− b +2 x (x + C1,5) C2 = C1e− b +2 x + C2xe− b +2 x, +(27) +7 + +where C1 = C1,5C2 is an integration constant. +Case 3. b2 − 4c < 0. In this case we have equation (21) has be form +z′ +� +z + b +2 +�2 + 1 +4(4c − b2) += −1, +⇒ +z′ +� +z + b +2 +�2 + +� +1 +2 +√ +4c − b2 +�2 = −1, +⇒ +1 +1 +2 +√ +4c − b2 arctan +z + b +2 +1 +2 +√ +4c − b2 = −x + C1,6 +⇒ +arctan +z + b +2 +1 +2 +√ +4c − b2 = −1 +2 +� +4c − b2x + C1,7, +C1,7 = C1,6 +1 +2 +� +4c − b2, +⇒ +z + b +2 +1 +2 +√ +4c − b2 = tan +� +−1 +2 +� +4c − b2x + C1,7 +� +, +⇒ +z = −b +2 + 1 +2 +� +4c − b2 tan +� +−1 +2 +� +4c − b2x + C1,7 +� +. +(28) +Because z = (log |y|)′, then we have in the equation (28) +(log |y|)′ = −b +2 + 1 +2 +� +4c − b2 tan +� +−1 +2 +� +4c − b2x + C1,7 +� +, +⇒ +log |y| = −b +2x + 1 +2 +� +4c − b2 +� +tan +� +−1 +2 +� +4c − b2x + C1,7 +� +dx + log |C2,2| = += −b +2x + log +����cos +� +−1 +2 +� +4c − b2x + C1,7 +����� + log |C2,2| , +⇒ +y = e− b +2x cos +� +−1 +2 +� +4c − b2x + C1,7 +� +C2,2 = += C2,2e− b +2x +� +cos +� +−1 +2 +� +4c − b2x +� +cos (C1,7) − sin +� +−1 +2 +� +4c − b2x +� +sin (C1,7) +� += += e− b +2 x +� +C1 cos +�1 +2 +� +4c − b2x +� ++ C2 sin +�1 +2 +� +4c − b2x +�� +, +(29) +where C1 = C2,2 cos(C1,7), C2 = C2,2 sin(C1,7) is an integration constant. +The formulas (25), (27), (29), solve the equation (18) in the respective cases 1,2,3. +Conclusion. This method in chapter 2 makes it possible to obtain these solutions without +applying a complex analysis and finding a solution in the form y = ψ(x)eζx. Also, we got exact +solutions for many kinds of first-order differential equations in chapter 1. +8 + +References +[1] C. H. Edwards, D. E. Penny, D. Calvis. Differential equations and boundary value prob- +lems. Firth Edition, (2014) - 797 p. +[2] C. H. Edwards, D. E. Penny, D. Calvis. Elementary differential equations, - 632 p. +[3] Charles Roberts, Jr., Elementary Differential Equations, Second Edition, A Chapman and +Hall Book, 2016, 380 p. +9 + diff --git a/QNAzT4oBgHgl3EQflf3i/content/tmp_files/load_file.txt b/QNAzT4oBgHgl3EQflf3i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b23e829cc21de5fe0785d3ee5e2e2defe3ac459a --- /dev/null +++ b/QNAzT4oBgHgl3EQflf3i/content/tmp_files/load_file.txt @@ -0,0 +1,181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf,len=180 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='01550v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='GM] 4 Jan 2023 Logarithmic Integration Method for Solving of First and Second Order Differential Equations A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Ponomarenko In this article we present logarithmic methods for solving first order and second order ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The essence of the method is that we apply the basic properties derivatives and logarithms to reduce the number of terms in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Here we carry out this only for equations of the first and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Similar methods can also be used to obtain solutions to higher order equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Keywords: differential equations, Riemann integrable functions, logarithmic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The main methods for solving ordinary differential equations have long been studied and are known to everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In [1], [2], [3] presents some basic methods for integrating simple ordinary differential equations (ODEs) and focuses on real solutions of ODEs with real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' It describes homogeneous linear equations with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In [1] shows that the general solution of nonhomogeneous linear equations with constant coefficients is the sum of the com- plementary function (the general solution of the corresponding homogeneous equation) and a particular integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' This article discusses a new approach to solving ordinary differential equa- tions using the simplest elementary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The calculations can be cumbersome, but we do not lose particular solutions to differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Let f(x), g(x) be Riemann integrable functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' y = y(x), y′(x) = dy(x) dx , y′′(x) = d2y(x) dx2 , log y = ln y = loge y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' C, C1, C2, C1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' , C1,7, C2,1, C2,2 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The symbol ⇒ between two formulas will mean that the second formula follows from the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' 1 First order differential equations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Linear inhomogeneous first order differential equation: y′(x) + f(x)y(x) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (1) Logarithmic integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In equation (1) the function g(x) is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then y(x) be not identifically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then with equations (1) we get y′(x) y(x) + f(x) = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y(x)|)′ + f(x) = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y(x)|)′ + �� f(x)dx �′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (2) (log |y(x)|)′ + � log e � f(x)dx�′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � log |y(x)| + log e � f(x)dx�′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ 1 � log � |y(x)|e � f(x)dx��′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � y(x)e � f(x)dx�′ y(x)e � f(x)dx = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � y(x)e � f(x)dx�′ = g(x)e � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y(x)e � f(x)dx = � g(x)e � f(x)dxdx + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y(x) = e− � f(x)dx �� g(x)e � f(x)dxdx + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (3) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' A similar method can be obtain solution the equation (1) in the Cauchy form: y(x) = e− � x x0 f(t)dt �� x x0 g(τ)e � τ x0 f(σ)dσdτ + y(x0) � , (4) where x0 is a given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Indeed, the equation (2) is equivalent to the equation (log |y(x)|)′ + �� f(x)dx + C1 �′ = g(x) y(x), (5) where C1 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Let C1 = −F(x0), where F(x) is a function that has property F ′(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then the equation (5) can be represented as (log |y(x)|)′ + �� x x0 f(t)dt �′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y(x)|)′ + � log e � x x0 f(t)dt�′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � log |y(x)| + log e � x x0 f(t)dt�′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � log � |y(x)|e � x x0 f(t)dt��′ = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � y(x)e � x x0 f(t)dt�′ y(x)e � x x0 f(t)dt = g(x) y(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � y(x)e � x x0 f(t)dt�′ = g(x)e � x x0 f(σ)dσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y(x)e � x x0 f(t)dt = � x x0 g(τ)e � τ x0 f(σ)dσdτ + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y(x) = e− � x x0 f(t)dt �� x x0 g(τ)e � τ x0 f(σ)dσdτ + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (6) If in the equation (6) we let C = y(x0), then we have the formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Bernoulli Differential equation: y′ + f(x)y = g(x)yα, (7) where α ∈ R\\{0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Logarithmic integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Let y is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then from the equations (7) we obtain y′ y + f(x) = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y|)′ + f(x) = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y|)′ + �� f(x)dx �′ = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (log |y|)′ + � log e � f(x)dx�′ = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � log |y| + log e � f(x)dx�′ = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � log � |y|e � f(x)dx��′ = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (8) � ye � f(x)dx�′ ye � f(x)dx = g(x) y yα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � ye � f(x)dx�′ = g(x)e � f(x)dxyα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � ye � f(x)dx�′ = g(x)e(1−α) � f(x)dxyαeα � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � ye � f(x)dx�′ = g(x)e(1−α) � f(x)dx � ye � f(x)dx�α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � ye � f(x)dx�′ � ye � f(x)dx�α = g(x)e(1−α) � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ � 1 1 − α � ye � f(x)dx�1−α�′ = g(x)e(1−α) � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ 1 1 − α �� ye � f(x)dx�1−α�′ = g(x)e(1−α) � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ �� ye � f(x)dx�1−α�′ = (1 − α) g(x)e(1−α) � f(x)dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ (9) � ye � f(x)dx�1−α = (1 − α) � g(x)e(1−α) � f(x)dxdx + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ ye � f(x)dx = � (1 − α) � g(x)e(1−α) � f(x)dxdx + C � 1 1−α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ 3 y = e− � f(x)dx � (1 − α) � g(x)e(1−α) � f(x)dxdx + C � 1 1−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (10) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' At the beginning of the course of the method, we assumed that y be not identically zero 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' It follows that the equation (7) has a particular solution y = 0, if α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (The second version of the logarithmic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=') In the equation (7) we obtain y′ y + f(x) = g(x) y yα, ⇒ (log |y|)′ + f(x) = g(x)yα−1, ⇒ (1 − α)(log |y|)′ + (1 − α)f(x) = (1 − α)g(x)yα−1, ⇒ ((1 − α) log |y|)′ + (1 − α)f(x) = (1 − α)g(x)yα−1, ⇒ (log |y|1−α)′ + (1 − α)f(x) = (1 − α)g(x)yα−1, ⇒ � y1−α�′ y1−α + (1 − α)f(x) = (1 − α)g(x)yα−1, ⇒ � y1−α�′ + (1 − α)f(x)y1−α = (1 − α)g(x)yα−1y1−α, ⇒ � y1−α�′ + (1 − α)f(x)y1−α = (1 − α)g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (11) The equation (11) is a linear inhomogeneous first order differential equation, with respect to the function y1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Its solution by the with formula (3), has the form y1−α = e−(1−α) � f(x)dx � (1 − α) � g(x)e(1−α) � f(x)dxdx + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (12) The formula (12) implies the solution (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (The third version of the logarithmic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=') In the equation (8) we obtain (1 − α) � log � |y|e � f(x)dx��′ = (1 − α)g(x)yα−1, ⇒ � (1 − α) log � |y|e � f(x)dx��′ = (1 − α)g(x)yα−1, ⇒ � log � |y|e � f(x)dx�1−α�′ = (1 − α)g(x)yα−1, ⇒ �� ye � f(x)dx�1−α�′ � ye � f(x)dx�1−α = (1 − α)g(x)yα−1, ⇒ �� ye � f(x)dx�1−α�′ = (1 − α)g(x)yα−1 � ye � f(x)dx�1−α = (1 − α)g(x)e(1−α) � f(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (13) The equation (13) is similar to the equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The equation of the form: y′ + f(x)eβy = g(x), (14) 4 where β ∈ R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Logarithmic integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In the equation (14) we get (log (ey))′ + f(x)eβy = g(x), ⇒ −β (log (ey))′ − βf(x)eβy = −βg(x), ⇒ (−β log (ey))′ − βf(x)eβy = −βg(x), ⇒ � log � e−βy��′ − βf(x)eβy = −βg(x), ⇒ � e−βy�′ e−βy − βf(x)eβy = −βg(x), ⇒ � e−βy�′ − βf(x)eβye−βy = −βg(x)e−βy, ⇒ � e−βy�′ − βf(x) = −βg(x)e−βy, ⇒ � e−βy�′ + βg(x)e−βy = βf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (15) The equation (15) is a linear inhomogeneous first order differential equation, with respect to the function e−βy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Its solution, by the formula (3), has the form e−βy = e−β � g(x)dx � β � f(x)eβ � g(x)dxdx + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (16) Solving the equation (16), with respect to y, we have y = − 1 β log � e−β � g(x)dx � β � f(x)eβ � g(x)dxdx + C �� , ⇒ y = � g(x)dx − 1 β log � β � f(x)eβ � g(x)dxdx + C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (17) 2 Second order differential equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Linear homogeneous second order differential equation: y′′ + by′ + cy = 0, (18) where b ∈ R, c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Let y is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then from the equation (18) we obtain y′′ y + by′ y + c = 0, ⇒ (log |y|)′′ + ((log |y|)′)2 + b(log |y|)′ + c = 0, (19) 5 because y′ y = (log |y|)′, (log |y|)′′ = � y′ y �′ = y′′ y − � y′ y �2 = y′′ y − ((log |y|)′)2, ⇒ y′′ y = (log |y|)′′ + ((log |y|)′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Let in the equation (19): (log |y|)′ = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then we have equation (19) in the form z′ + z2 + bz + c = 0, ⇒ (20) z′ z2 + bz + c = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (21) Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' b2 − 4c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In this case we have equation (21) has be form z′ � z + b 2 �2 − 1 4(b2 − 4c) = −1, ⇒ z′ � z + b 2 �2 − � 1 2 √ b2 − 4c �2 = −1, ⇒ 1 21 2 √ b2 − 4c log ����� z + b 2 − 1 2 √ b2 − 4c z + b 2 + 1 2 √ b2 − 4c ����� = −x + C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (22) Let in the equation (22): z + b 2 = ξ, 1 2 √ b2 − 4c = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Then we obtain 1 √ b2 − 4c log ���� ξ − γ ξ + γ ���� = −x + C1,1, ⇒ ξ − γ ξ + γ = C1,2e− √ b2−4cx, C1,2 = eC1,1, ⇒ 1 − 2γ ξ + γ = C1,2e− √ b2−4cx, ⇒ 1 ξ + γ = 1 2γ + C1,3e− √ b2−4cx, C1,3 = − 1 2γ C1,2, ⇒ ξ + γ = 1 1 2γ + C1,3e− √ b2−4cx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (23) Returning to the change of variables z + b 2 = ξ, 1 2 √ b2 − 4c = γ in the equation (23), we obtain z + b 2 + 1 2 � b2 − 4c = 1 1 √ b2−4c + C1,3e− √ b2−4cx , ⇒ z = − �b 2 + 1 2 � b2 − 4c � + 1 1 √ b2−4c + C1,3e− √ b2−4cx, ⇒ 6 z = − �b 2 + 1 2 � b2 − 4c � + √ b2 − 4c 1 + C1,4e− √ b2−4cx , C1,4 = � b2 − 4cC1,3, ⇒ z = − �b 2 + 1 2 � b2 − 4c � + e √ b2−4cx√ b2 − 4c e √ b2−4cx + C1,4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (24) Because z = (log |y|)′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' then we have in the equation (24) (log |y|)′ = − �b 2 + 1 2 � b2 − 4c � + e √ b2−4cx√ b2 − 4c e √ b2−4cx + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ log |y| = � e √ b2−4cx√ b2 − 4c e √ b2−4cx + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='4 dx − �b 2 + 1 2 � b2 − 4c � x + log |C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1| = = log ���e √ b2−4cx + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='4 ��� − � b 2 + 1 2 � b2 − 4c � x + log |C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y = � e √ b2−4cx + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='4 � e−( b 2+ 1 2 √ b2−4c)xC2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y = C1e(− b 2 − 1 2 √ b2−4c)x + C2e(− b 2+ 1 2 √ b2−4c)x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (25) where C1 = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' C2 = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='1 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' b2 − 4c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In this case we have equation (21) has be form z′ � z + b 2 �2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Step by step from the last equation we obtain � − 1 z + b 2 �′ = −1, ⇒ 1 z + b 2 = x + C1,5, ⇒ z + b 2 = 1 x + C1,5 , ⇒ z = −b 2 + 1 x + C1,5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (26) Because z = (log |y|)′, then we have in the equation (26) (log |y|)′ = −b 2 + 1 x + C1,5 , ⇒ log |y| = −b 2x + log |x + C1,5| + log |C2| , ⇒ y = e− b 2 x (x + C1,5) C2 = C1e− b 2 x + C2xe− b 2 x, (27) 7 where C1 = C1,5C2 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' b2 − 4c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' In this case we have equation (21) has be form z′ � z + b 2 �2 + 1 4(4c − b2) = −1, ⇒ z′ � z + b 2 �2 + � 1 2 √ 4c − b2 �2 = −1, ⇒ 1 1 2 √ 4c − b2 arctan z + b 2 1 2 √ 4c − b2 = −x + C1,6 ⇒ arctan z + b 2 1 2 √ 4c − b2 = −1 2 � 4c − b2x + C1,7, C1,7 = C1,6 1 2 � 4c − b2, ⇒ z + b 2 1 2 √ 4c − b2 = tan � −1 2 � 4c − b2x + C1,7 � , ⇒ z = −b 2 + 1 2 � 4c − b2 tan � −1 2 � 4c − b2x + C1,7 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (28) Because z = (log |y|)′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' then we have in the equation (28) (log |y|)′ = −b 2 + 1 2 � 4c − b2 tan � −1 2 � 4c − b2x + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ log |y| = −b 2x + 1 2 � 4c − b2 � tan � −1 2 � 4c − b2x + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7 � dx + log |C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2| = = −b 2x + log ����cos � −1 2 � 4c − b2x + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7 ����� + log |C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' ⇒ y = e− b 2x cos � −1 2 � 4c − b2x + C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7 � C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2 = = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2e− b 2x � cos � −1 2 � 4c − b2x � cos (C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7) − sin � −1 2 � 4c − b2x � sin (C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7) � = = e− b 2 x � C1 cos �1 2 � 4c − b2x � + C2 sin �1 2 � 4c − b2x �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' (29) where C1 = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2 cos(C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' C2 = C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='2 sin(C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content='7) is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' The formulas (25), (27), (29), solve the equation (18) in the respective cases 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' This method in chapter 2 makes it possible to obtain these solutions without applying a complex analysis and finding a solution in the form y = ψ(x)eζx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Also, we got exact solutions for many kinds of first-order differential equations in chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' 8 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Penny, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Calvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Differential equations and boundary value prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Firth Edition, (2014) - 797 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Penny, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Calvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' Elementary differential equations, - 632 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' [3] Charles Roberts, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=', Elementary Differential Equations, Second Edition, A Chapman and Hall Book, 2016, 380 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQflf3i/content/2301.01550v1.pdf'} diff --git a/StE3T4oBgHgl3EQfDwk0/content/tmp_files/2301.04289v1.pdf.txt b/StE3T4oBgHgl3EQfDwk0/content/tmp_files/2301.04289v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cce5259c59f42206beff804e8908e1b08e9e971 --- /dev/null +++ b/StE3T4oBgHgl3EQfDwk0/content/tmp_files/2301.04289v1.pdf.txt @@ -0,0 +1,1169 @@ +arXiv:2301.04289v1 [eess.SY] 11 Jan 2023 +Balancing a Stick with Eyes Shut: Inverted Pendulum on a Cart without Angle +Measurement +Bidhayak Goswami∗ +Anindya Chatterjee† +Mechanical Engineering, IIT Kanpur +January 12, 2023 +A shorter version of this paper is due to appear in ASME JDSMC. +Abstract +We consider linear time-invariant dynamic systems in the +single-input, single-output (SISO) framework. In particular, +we consider stabilization of an inverted pendulum on a cart +using a force on the cart. +This system is easy to stabilize +with pendulum angle feedback. However, with cart position +feedback it cannot be stabilized with stable and proper com- +pensators. Here we demonstrate that with an additional com- +pensator in a parallel feedforward loop, stabilization is possible +with such compensators. Sensitivity to noise seems to be about +3 times worse than for the situation with angle feedback. For +completeness, discussion is presented of compensator parame- +ter choices, robustness, fragility and comparison with another +control approach. +Keywords: +Stabilization, Stable Compensator, Inverted +Pendulum, Cart. +1 +Introduction +Stabilization of an inverted pendulum on a cart is a familiar +problem in control theory, and also one that is interesting to a +broad audience. The control input is a horizontal force on the +cart; and it is desired to use feedback to stabilize the cart at a +given location and the pendulum in the inverted (or standing +vertical) position. Formally, the linearized system is control- +lable. In the classical control theory framework, with a single +input and a single output (SISO), it is important whether the +output is the pendulum angle or the cart position. Due to the +obvious resemblance to balancing a stick on one’s palm, we +refer to the latter case (i.e., cart position known and pendu- +lum angle unknown) as balancing a stick with eyes shut. This +problem, although simple to state, is interesting to a broad +audience because a few attempts to balance a stick on one’s +palm with eyes shut will convince the reader that the task is +difficult if not impossible. Technically, the problem is also in- +teresting within the usual classical single-loop feedback control +framework because, in the case of solely position feedback, it +turns out that the system is not stabilizable with a stable and +proper controller [1]. +As a control problem, the inverted pendulum is a famil- +iar favorite. It has been studied by several researchers in the +past [2–5]. It has been used as a teaching example for many +decades [6, 7]. +Linear control theory has been used for the +angular position feedback case, and that case does not rep- +∗bidhayak1728@gmail.com,bidhayak@iitk.ac.in +†anindya100@gmail.com, anindya@iitk.ac.in +resent significant challenges any more. Control in the nonlin- +ear regime, including swing-up dynamics from a hanging-down +position, has been studied [8–12] from various viewpoints in- +cluding energy-based control as well as control input shap- +ing. Some researchers have studied the effectiveness of simple +feedback laws with delays [13], again with angular position +feedback. Lee et al. [14] have considered system uncertainties, +feedback with multi-timescale structure and an extended high- +gain observer. An optimal control approach has been used as +well, even for harder variants, e.g., a double inverted pendulum +on a moving cart [15]. With advances in control theory, more +modern techniques like robust control, fuzzy logic control, etc. +have been considered as well [16–18]. +In spite of the abovementioned works with modern ap- +proaches, in this paper we remain within the classical SISO +linear regime for two reasons. Firstly, a large number of in- +dustrial control systems are still linear in their thinking, and +often close to just PID control or variations thereof. Secondly, +in the absence of angle feedback, for the inverted pendulum +on a cart, we obtain a nonminimum phase system with an odd +number of real poles on the right of an RHP real zero which, +it is known, cannot be stabilized by a stable and proper com- +pensator in the usual single feedback loop configuration. +Of course, not all control is based solely on feedback. Feed- +forward compensators [19,20] have been used earlier for stabi- +lizing nonminimum phase systems. Kim et al. [21] used feed- +forward compensation for the synchronization of a multi-agent +system to achieve faster convergence. Golovin and Palis [22,23] +used feedforward compensation for stabilizing an electrome- +chanical system with friction-induced instabilities. Here, we +seek suitable feedforward and feedback compensators, both sta- +ble and proper, to stabilize the inverted pendulum using only +the cart position as output. +In what follows, we describe the system in section 2, present +the control approach and show some relevant results in sec- +tion 3, and discuss the effect of noise in section 4. A detailed +derivation of the equations of motion is given in appendix A, +a description of the controller design methodology is given in +appendix B, the modern control approach in terms of controlla- +bility and observability is discussed in appendix C, robustness +and fragility of the stable compensators have been examined +in appendix D, and finally, in appendix E, the effect of noise +on the system has been discussed. +2 +The inverted pendulum +Consider an inverted pendulum on a cart (Fig. 1). A control +force u acts on the cart as shown. The pendulum consists of a +1 + +massless rigid rod of length L with a point mass m at the tip. +The cart’s mass is M. There is gravity g. The coordinates used +are θ for the pendulum angle and x for the cart displacement. +Figure 1: An inverted pendulum on a cart. +The equations of motion (see appendix A for details) for +small θ and ˙θ are +M ¨x + m +� +¨x + L¨θ +� += u, +(1a) +¨x + L¨θ = gθ. +(1b) +By choice of units of mass, length and time, we can set m, L +and g to unity. This leaves +M ¨x + +� +¨x + ¨θ +� += u, +(2a) +¨x + ¨θ = θ, +(2b) +where M should henceforth be thought of as dimensionless. +In the SISO framework within elementary classical control, +we have a single output: this will often be either the pendulum +angle or the cart position. See the block diagram in Fig. 2 +for the basic control structure we will consider in this paper. +Here, U is the input, K is a gain, G is the plant, C0, C1 and C2 +are compensators, P is an optional feedforward compensator +placed in parallel, Y is the actual plant output, and Z is the +quantity that is fed back. +Obviously, in the absence of P, +or with P = 0, we will have Z = Y and obtain the usual +single-loop feedback control design. The parallel feedforward +compensator P is the novelty we will consider here. +A feedforward compensator before the input signal U is rou- +tinely used in physical systems where the input from the user +may be, e.g., a desired angle and the input to the measure- +ment and control system is, typically, a voltage. +However, +subsequent analysis of the control system is independent of +that compensator. So, we have not included it in this paper. +Returning to the inverted pendulum on a cart, when we +balance a stick on our palm, we always look at the pendulum +angle θ. The corresponding transfer function of the plant is +F = +1 +1 + M − M s2 , +(3) +which has a right half plane or RHP pole but is easily sta- +bilizable with a stable controller without any P in a parallel +feedforward path. For example, with M = 0.3, C0 = C1 = 1, +and K > 4.33, +F = +1 +1.3 − 0.3s2 +(4) +Figure 2: A feedfoward-feedback control system: basic layout. +We take C0 = 1 for simplicity (see section 3 for justification). +can be stabilized by the stable compensator +C2 = − s + 3 +s + 10. +(5) +However, if our output variable is x, which loosely corre- +sponds to trying to balance the stick on our palm with our +eyes shut, then the plant becomes +G = +s2 − 1 +s2 (0.3s2 − 1.3), +(6) +which has a real RHP zero at s = 1 and a single real RHP pole +to its right, at s = +� +1.3/0.3. This is more interesting. +3 +Stabilization +In classical control theory with a single control loop [24], +P = 0. Then, although C0, C1 and C2 can in principle all +be present, stability is affected only by the product C0C1C2, +and so we can for stabilization purposes take any two of them +to be unity. The gain K, too, can be included within C1 if +we wish. In this paper, we take C0 = 1 for simplicity. From +a design viewpoint, non unity C0 can be thought as a system +modification and we leave it for more challenging problems. +In simple control systems, the compensators C1 and C2 may +be physically realized using simple circuits made with resistors, +capacitors, and op-amps. In such cases we want the compen- +sator transfer functions themselves to be stable, i.e., C0, C1 +and C2 should not have RHP poles. Here we assume that P +has no RHP poles either. Thus we are interested in stabilizing +G with stable controllers1. +In the absence of P, a fundamental fact has been known +for almost 50 years [1]. If G has one or more real RHP zeros; +and if G also has an odd number of real RHP poles that lie +to the right of any positive real zero; then in the absence of +P in Fig. 2, G is not stabilizable [1] with stable and proper +compensators C0, C1 and C2. +The mathematical aim of this paper can now be clearly +stated. +We will take the troublesome G of Eq. (6), set +K = C1 = 1 (as well as C0 = 1 as stated above), and find +a stable and proper C2 = C along with a stable and proper P +such that the plant output Y is stabilized. +Let +C = nC +dC +, +G = nG +dG +, +and P = nP +dP +, +1The motivation is that the analog control card should not saturate +and lose linearity before the actual system dynamics is established, e.g., +during a warm-up phase. + +where the n’s stand for numerator polynomials in s and the +d’s stand for denominator polynomials in s of equal or greater +order. Routine manipulations show that +Y = +GU +1 + C(P + G) = HU, +(7) +i.e., the controlled transfer function is +H = +nGdCdP +dCdGdP + nCnP dG + nCnGdP +. +(8) +Thus, our controller design for stabilization reduces to +choosing polynomials nC, dC, nP and dP such that the d- +polynomials are stable (i.e., they have only LHP roots), and +the denominator polynomial +dCdGdP + nCnP dG + nCnGdP +is stable as well (has only LHP roots). +We are not aware of systematic and guaranteed ways of ob- +taining such polynomials. We have used trial and error based +on a simple optimization routine. Some details of the opti- +mization are given in appendix B. Our main aim here is to +demonstrate and assess specific numerical solutions. +Two stabilizing solutions, out of many that we found, are +shown below in Eqs. (9) and (10), labeled “a” and “b” respec- +tively. +Pa = 0.05s3 + 7s2 − 0.1s − 1.9 +11s3 + 21.7s2 + 5.4s + 1 , +Ca = −10.1s3 + 2s2 + 0.9s + 0.09 +0.002s3 + 4.2s2 + 10.2s + 1 , +(9) +and +Pb = 0.2s3 + 1.4s2 − 2s − 0.8 +4.1s3 + 10.8s2 + 5.3s + 1, +Cb = −6.9s3 + 1.6s2 + 1.1s + 0.3 +0.08s3 + 0.4s2 + 9.3s + 1 . +(10) +The corresponding unit-step responses of the controlled sys- +tems are shown in Fig. 3. +0 +10 +20 +30 +40 +0 +5 +10 +15 +20 +0 +5 +10 +15 +-5 +0 +5 +10 +15 +20 +Figure 3: Unit-step responses of G of Eq. (6), with P and C +as given by Eq. (9) and Eq. (10). +At this point we can check to see, for the same controlled +system (with only position feedback), the angle response of the +inverted pendulum. Recalling F from Eq. (4), we write +nF = 1, +and +dF = 1.3 − 0.3 s2. +0 +10 +20 +30 +-3 +-2 +-1 +0 +1 +2 +3 +0 +5 +10 +15 +-4 +-2 +0 +2 +4 +Figure 4: Angular response of the pendulum. +Further, recalling Eqs. (6), (7) and (8), we find that the angular +response of the inverted pendulum must be +HF +G U, +with +HF +G += − +nF dCdP +dCdGdP + nCnP dG + nCnGdP +s2, +(11) +where we have used +dG = −s2 dF . +The angular response of the pendulum is given by the step +response of Eq. (11): see figure 4. +It is also interesting to ask how other control strategies might +work for this same system. A discussion of the textbook ap- +proach of modern control theory, with state estimation and +full state feedback, is given in appendix C. +Finally, we must address two more issues: (i) the robust- +ness of the controller, i.e., its ability to stabilize plants with +slightly different plant-parameter values, and (ii) the fragility +of the controller, i.e., its tendency to lose effectiveness under +small perturbations of the controller-parameter values. Both +robustness and fragility are good, as shown in appendix D. +4 +Effect of noise +The system has now been mathematically stabilized. We must +also study the effect of noise on the stabilized system. Since +the basic control problem is difficult at least in some ways +(as in balancing a stick on one’s palm with eyes shut), we may +expect increased sensitivity to noise. The system now has more +than one input (the control force u along with noise inputs +ek, k ∈ {1, 2, . . ., 6}), but still only one output (y), as shown +in Fig. 5. In the Laplace domain, elementary calculations give +us individual transfer functions for each input, with the net +output given as +Y (s) = H(s)U(s) + +6 +� +k=1 +Hek(s)Ek(s). +(12) +In the above the seven transfer functions H and Hek, k ∈ +{1, 2, . . ., 6}, share a common denominator. +If H is stable, +then so is each Hek. The Bode magnitude plots for Hek with +P = Pb and C = Cb are shown in Fig. 6. For Pa and Ca, the +maximum magnitude is higher. It is seen in Fig. 6 that for + +Figure 5: +The control system with added noise inputs +ek(t), k ∈ {1, 2, . . ., 6}. +nondimensional frequencies on the order of unity, the magni- +tudes of the transfer functions take their highest values, which +are around 30 dB. This corresponds to amplification by roughly +a factor of 30, which is quite large. +10-2 +100 +102 +-80 +-60 +-40 +-20 +0 +20 +10-2 +100 +102 +-80 +-60 +-40 +-20 +0 +20 +40 +10-2 +100 +102 +-80 +-60 +-40 +-20 +0 +20 +40 +10-2 +100 +102 +-40 +-20 +0 +20 +40 +10-2 +100 +102 +-100 +-50 +0 +50 +10-2 +100 +102 +-40 +-20 +0 +20 +40 +Figure 6: Bode magnitude plots of Hek, k ∈ {1, 2, . . ., 6}, for +the system shown in Fig. 5 with P = Pb and C = Cb given by +Eq. (10). +We mention that in separate calculations with G as in Eq. +(3), K > 4.33, P = 0 (i.e., no parallel feedforward compen- +sator), and C = C2 of Eq. (5), maximum amplification fac- +tors about 10 dB lower were easily obtained (details omitted). +This is not intuitively surprising because balancing a stick with +one’s eyes open is easier than with one’s eyes shut; and corre- +spondingly, stabilizing the inverted pendulum on a cart in the +classical SISO setting with pendulum angle feedback is easier +than with cart position feedback. The sensitivity to noise in +the eyes-shut case, for our control design, seems to be greater +by about a factor of 3. +Some numerical examples of the response to noise inputs, +where the “noise” is the sum of a large number of small sinu- +soidal inputs with randomly chosen amplitudes and frequen- +cies, are given in appendix E. The results there are consistent +with the above estimate of 30 dB. +5 +Concluding remarks +In some applications such as low cost consumer products or +toys, there may be simple analog control cards which, if oper- +ated within the linear domain, produce desirable behavior in +the controlled device. In such situations, stabilization with a +stable controller has practical utility. Additionally, there may +be sophisticated scientific or technical instruments wherein +simple controllers are implemented with some parameters ad- +justable by the user. +In such cases, too, while the system +is warming up, or is outside the operational position range, +an unstable compensator may lead to overly large actuator +commands that cause saturation, deviation from linearity, or +possibly specimen damage. In such cases, also, stabilization +with a stable controller may make the system more foolproof. +With the above motivation, we appreciate the classic pa- +per [1] which lays down the mathematical conditions under +which, in the single loop control structure, stabilization is not +possible with a stable and proper compensator. One of the +most familiar control problems, namely balancing an inverted +pendulum on a cart, presents this situation when position feed- +back is used. For this system, using a parallel stable feedfor- +ward compensator, we have demonstrated using numerical ex- +amples that it is possible to achieve stabilization with stable +and proper compensators. +Finding such stable and stabilizing compensators is not a +familiar and routine control design problem within classical +control theory. Others have studied this control approach be- +fore as well [21,22], but not widely and not for such a popular +problem as balancing a stick. We hope that the community of +industrial and academic control systems practitioners and re- +searchers will find this class of problems sufficiently interesting +as to develop this kind of control design further, and possibly +even seek rational design criteria for when such controllers ex- +ist and how they may be found easily. +Moreover, once such a block diagram framework is adopted, +it can also be used for design of controllers for nonlinear sys- +tems. Such work [25] has begun to appear. +A +Equations of motion +We draw free-body diagrams for both cart and pendulum (Fig. +7). The pendulum’s pivot point P (see Fig. 7(a)) experiences +a reaction force from the cart. This force has components Rx +and Ry along unit vectors ˆe1 and ˆe2 respectively. The cart ex- +periences equal and opposite reactions. There are two ground +forces on the cart wheels, named N1 and N2 (see Fig. 7(b)). + +Friction has not been included. The weight of the pendulum +and cart are mg and Mg respectively. A horizontal force u +acts on the cart. +Figure 7: Free body diagrams for the pendulum and the cart. +Moving on to the kinematics, P has an acceleration ¨x ˆe1. +The acceleration of G is +aG = aP + aG/P += ¨x ˆe1 + +� +L ¨θ ˆeθ − L ˙θ2 ˆen +� +, +(13) +where ˆeθ and ˆen are unit vectors shown in Fig. 7(a). +For the pendulum, linear momentum balance gives +Rx = m (aG · ˆe1) = m +� +¨x + L ¨θ cos(θ) − L ˙θ2 sin(θ) +� +, +(14) +and for the cart, it gives +− Rx + u = M ¨x. +(15) +Substituting Rx from Eq. (14) in Eq. (15) +M ¨x + m +� +¨x + L ¨θ cos(θ) +� +− m L ˙θ2 sin(θ) = u. +(16) +Now, for the pendulum, the moment about spatial point P +(coincides with the pivot instantaneously) is +τ P = IG · α + rG/P × m aG, +(17) +where IG is zero and rG/P is the position vector of from P to +G. This yields +− m g L sin(θ) = −m L +� +¨x cos(θ) + L¨θ +� +. +(18) +or +¨x cos(θ) + L¨θ = g sin(θ). +(19) +Linearizing Eqs. (16) and (19), for small θ and ˙θ, we obtain +M ¨x + m +� +¨x + L¨θ +� += u, +(20a) +¨x + L¨θ = gθ. +(20b) +B +Methodology used for obtaining C +and P +For both compensators C and P, we choose nth order polyno- +mials with unknown coefficients for both numerator and de- +nominator, where n is a positive integer to be chosen by trial +and error. The compensators’ transfer functions are taken as +C = +a0 + a1 s + . . . an sn +1 + an+1s + · · · + a2nsn , P = +b0 + b1 s + · · · + bn sn +1 + bn+1 s + · · · + b2n sn , +where ak, bk constitute 4n + 2 unknown coefficients. +Now we construct an objective function F as follows. +(i) F takes 4n+2 numbers as a vector input q and first forms +the candidate C and P. +(ii) From the numerators and the denominators of C and P, +it calculates H (Eq. (8)). +(iii) It calculates the poles of C, P and H. +(iv) From the poles of C and P, the right-most real part is +saved as p1. +(v) From the poles zHi of H, the right-most real part is saved +as p2. +(vi) A preliminary function value f0 is defined as +f0 = +� +p2 + 6 p1 +if p1 ≥ 0, +p2 +otherwise, +where the “6” is a penalty parameter, found to be big +enough by trial and error (unnecessarily large penalty pa- +rameters are best avoided). +(vii) For better behavior, the actual objective function used +was +F = f0 + ε1 ||q|| + ε2 max{|zHi|}, +i = 1, 2, . . . , 2 n + 4, +where ε1 = 10−5 and ε2 = 10−4. +If we can find a q such that +F(q) < 0, +then our goal is accomplished. +We can now use any optimization techniques we like. We +used a simple in-house genetic algorithm. The code is available +on request. +C +Controllability and observability +So far, we have studied the system from the viewpoint of classi- +cal control theory. In the modern control approach, the system +state consists of x, θ, ˙x and ˙θ given as a column matrix +x = + + + + + +x +θ +˙x +˙θ + + + + + +. +(21) +Writing Eqs. (1a) and (1b) in state space form, we obtain +˙x = A x + B u, +(22) + +where, for M = 0.3, +A = + + +0 +0 +1 +0 +0 +0 +0 +1 +0 +− 10 +3 +0 +0 +0 +13 +3 +0 +0 + + +and +B = + + + + + + + + + + + +0 +0 +10 +3 +− 10 +3 + + + + + + + + + + + +. +(23) +In this problem, only the measurement of the cart displace- +ment is available. So, the measured quantity +y = x = C x, +where C = [1 0 0 0]. +(24) +Taking Laplace transforms of both sides of Eq. (22), we obtain +for zero initial conditions +X(s) = (s I − A)−1 B U(s), +where X(s) = L[x(t)]. +(25) +Using the symbolic algebra package Maple, we have verified +that +C (s I − A)−1 B = G(s) = +s2 − 1 +s2 (0.3 s2 − 1.3). +(26) +The controllability matrix [24] is +PC = +� +A3B | A2B | AB | B +� += + + +100 +9 +0 +10 +3 +0 +− 130 +9 +0 +− 10 +3 +0 +0 +100 +9 +0 +10 +3 +0 +− 130 +9 +0 +− 10 +3 + + +, +(27) +which has full rank. The observability matrix [24] +PO = + + +CA3 +CA2 +CA +C + + = + + +0 +0 +0 +− 10 +3 +0 +− 10 +3 +0 +0 +0 +0 +1 +0 +1 +0 +0 +0 + + +(28) +also has full rank. The system is both controllable and ob- +servable. A controller can be designed by constructing a state +estimator and then using full state feedback. Let us consider +the following system +˙x = A x − B K ˜x + B u +(29a) +˙˜x = A ˜x − B K ˜x + G C (x − ˜x) + B u +(29b) +where ˜x is the estimated state and the gain matrices K and +G are found by placing the system poles (arbitrarily) at +− 1 ± i and − 2 ± i, +(30) +and the estimator poles (also arbitrarily) at +− 1, −2, and − 3 ± i +(31) +on the complex plane. These numbers are chosen for demon- +stration only. +Combining Eqs. (29a) and (29b), we obtain +˙˜Z = ˜A ˜Z + ˜B u, +(32) +where, +˜Z = +� +x +˜x +� +, ˜A = +� +A +−B K +G C +A − B K − G C +� +, and ˜B = +� +B +B +� +. +(33) +The output +y = C x = ˜C ˜Z, +where ˜C = [C, 0, 0, 0, 0]. +(34) +To interpret these result in light of the main paper, we can +now think of an implied feedback controller, with closed loop +transfer function +Q(s) = ˜C +� +sI − ˜A +�−1 ˜B. +(35) +Using Maple, we obtain +Q(s) = +10 s2 − 10 +3 s4 + 18 s3 + 45 s2 + 54 s + 30. +(36) +We observe that Q and G share the same zeros, and the poles of +Q are the same as the system poles chosen for placement (Eq. +(30)). We may think of a feedback control system equivalent to +Figure 8: An equivalent single loop feedback control system. +the implied control system as shown in Fig. 8, where plant G +is assigned compensators Kb and Kf in forward and feedback +loops respectively. Hence +Q = +Kf G +1 + Kb Kf G. +(37) +From algebraic manipulations, we obtain; +Kf = Q +G +1 +1 − Kb Q +and +Kb = 1 +Q − +1 +Kf G. +(38) +Clearly, there are infinitely many solutions for Kf and Kb. We +examine two limiting cases for better understanding. +(i) The system shown in Fig. 8 has a compensator only in the +feedback loop, i.e., Kf = 1. In this case, +Kb = 1 +Q − 1 +G = 9 s3 + 29 s2 + 27 s + 15 +5 s2 − 5 +, +(39) +which is unacceptable (both improper and unstable). +(ii) The system shown in Fig. 8 has a compensator only in the +forward loop, i.e., Kb = 1. Now we have +Kf = Q +G +1 +1 − Q = +dG +dQ − nQ += +s2 � +3 s2 − 13 +� +3 s4 + 18 s3 + 35 s2 + 54 s + 40, +(40) + +where dQ and nQ are the denominator and numerator of +Q respectively. +The compensator Kf is stable, but relies on pole zero +cancellation which is not allowed in classical control. A +commonly stated reason for not allowing pole zero can- +cellation is that the slightest inaccuracy in the controller +will destroy the cancellation and instability will reappear. +Youla et al. [1] also point out that exact pole zero cancel- +lation may represent nonobservable modes which remain +unstable. In any case, we cannot accept this Kf. +We already know that the controller obtained in this ap- +pendix cannot be realized (Youla et al. [1]) with stable and +proper compensators in the classical single loop configuration. +Equations (39) and (40) merely provide two examples of the +difficulties encountered if such an attempt is made. +D +Robustness and fragility +Having found a stable closed loop transfer function H as ex- +plained in appendix B, we can check its sensitivity to small +changes in plant and compensator parameters. +Robustness, for a control system, is its ability to retain sta- +bility under small changes in the plant parameters. Here, the +plant parameters depend on the system parameters: L, m, g +and M. Of them, the first three were eliminated from the gov- +erning equations by introducing nondimensional displacement, +time and mass ˜x, ˜t and ˜m respectively where +˜x = x +L, +˜t = t +� g +L, +˜m = M +m , +(41) +which is analogous to setting the values of m, L and g to unity +and treating M as the only free parameter. So far we have +considered M = 0.3. To investigate the effect of small changes +in parameter values on the system behavior, we rewrite the +plant transfer function as +G = +A0 s2 − A1 +s2 (0.3 A2 s2 − 1.3 A3), +(42) +where the parameters A0, A1, A2 and A3 are notionally equal +to unity along with M = 0.3. For a large number of random +calculations (1000 times), we perturb the A’s by normally dis- +tributed iid random variables ri, i = 0, 1, 2, 3, with zero mean +and standard deviation 0.02 (99.7% of the points are within ± +6%) in the following way +Ai �→ Ai (1 + ri), +i = 0, 1, 2, 3, . +We then plot the poles (zH) of the respective closed loop trans- +fer functions (CLTF). Results are shown in Fig. 9. +For compensators Ca and Pa, the entire cloud (Fig. 9(a)) of +poles remains in the left half plane. For compensators Cb and +Pb, a significant part of the cloud (Fig. 9(b)) remains in the +left half plane. In 44 out of 1000 cases, the CLTF has poles in +the right half plane. Thus, the compensators are fairly robust; +and Ca and Pa are more robust than Cb and Pb. +Some robust control systems perform poorly under small +perturbations in the compensator parameters. This is called +the fragility [26] of the system. To check fragility, we perturb +the compensator parameters, again 1000 times, by normally +distributed iid random variables si, i = 0, 1, . . ., 4 n+ 1. Here, +Figure 9: Robustness under small changes in plant parameters +(1000 random perturbations). +Figure 10: Performance of the system under small changes in +compensator parameters (1000 random perturbations). +n is degree of the polynomials in the numerator and denomi- +nator of the compensators. The random variables si have zero +mean and standard deviation 0.02 (99.7% of them are within +± 6%). We perturb the a’s and b’s as follows: +ai �→ ai (1 + si), +bi �→ bi (1 + s2 n+1+i), +i = 0, 1, . . . , 2 n. +We then calculate the poles (zH) of the respective closed loop +transfer functions. The results are shown in Fig. 10. For the +compensators Ca and Pa, a large portion of the cloud of poles +again remains in the left half plane. In 9 out of 1000 cases, the +CLTF has poles in the right half plane. For the compensators +Cb and Pb, in 32 out of 1000 cases, the CLTF has poles in the +right half plane. +We conclude with the following observation. Implementabil- +ity, albeit implicitly discussed, has motivated this entire paper. +Finding stable compensators (which we have now shown are +fairly robust and not fragile) indicates that the compensators +are implementable. +E +Response to noise +In section 4, we examined the system’s sensitivity to six noise +inputs ei(t), i = 1, 2, . . ., 6 by using Bode plots. To demon- +strate the effect of noise on the time response of the system, +we use the following input +ei(t) = +N +� +k=1 +ck sin (ωk t) , +i = 1, 2, . . ., 6, +(43) +where the ω’s are randomly chosen numbers uniformly dis- +tributed in the interval [0.5, 1.5]. +The amplitudes ck, k = + +41类 +2 +米 +Im(ZH) +0 +米 +米 +-2 +米 +-4 +米 +-2 +-1 +0 +Re(ZH)21 +Im(ZH) +0 +-1 +-2 +-1 +-0.5 +0 +0.5 +Re(ZH)41米 +■**米 +2 +米 +来米 +Im(ZH) +0 +米 +-2 +米 +-4 +-2 +-1 +0 +Re(ZH)211 +(Hz) +0 +Im ( +米瓣 +-1 +-2 +-1 +-0.5 +0 +0.5 +Re +(ZH)80 +100 120 140 160 +200 +-0.1 +0 +0.05 +0.1 +0.15 +0 +20 +40 +60 +180 +-0.05 +-0.15 +80 +100 120 140 160 +180 200 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0 +20 +40 +60 +100 120 140 160 +180 200 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0 +20 +40 +60 +80 +80 +100 120 140 160 180 200 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0 +20 +40 +60 +Figure 11: Time responses to noise inputs e1(t) and e3(t). +1, . . . , N are random numbers where the norm of the vector +c = [c1, c2, . . . , cN]⊤ is set to 0.01. For calculations, we have +used N = 4000. +The response, with phase randomized, is +taken as +xei(t) = +N +� +k=1 +ck|Hei(i ωk )| sin (ωk t + arg (Hei(i ωk) ) + φk) , +i = 1, 2, . . ., 6, +(44) +where i = √−1, and the φk are random numbers uniformly +distributed in the interval [0, 2 π]. +In Fig. 11, the amplification factor is consistent with the +Bode plots of section 4. +References +[1] Youla, D. C., Bongiorno Jr, J. J., and Lu, C. N., Single-loop +feedback-stabilization of linear multivariable dynamical plants. +Automatica, 10(2): 159-173, (1974). +[2] Blitzer, L., Inverted pendulum. American Journal of Physics, +33(12): 1076-1078, (1965). +[3] Phelps III, F. M., and Hunter Jr, J. H., An analytical solution +of the inverted pendulum. American Journal of Physics, 33(4): +285-295, (1965). +[4] Kalmus, H. P., The inverted pendulum. 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Journal of Process Control, 24(3): 33-46, (2014). +[21] Kim, H., Kim, S., Back, J., Shim, H., and Seo, J. H., Design +of stable parallel feedforward compensator and its application +to synchronization problem. Automatica, 64: 208-216, (2016). +[22] Golovin, I., and Palis, S., Design of parallel feed-forward com- +pensator and its application to electromechanical system with +friction load. IFAC-PapersOnLine, 50(1): 15524-15529, (2017). +[23] Golovin, I., and Palis, S., PFC-Based Control of Friction- +Induced Instabilities in Drive Systems. Machines, 9(7): 134, +(2021). +[24] Ogata, K., Modern Control Engineering (Vol. 5), Upper Saddle +River, NJ: Prentice Hall, (2010). +[25] R¨obenack, K., Palis, S., On the control of non-minimum phase +systems using a parallel compensator. International Conference +on System Theory, Control and Computing (ICSTCC),IEEE, +308-313, (2019). +[26] Keel, L. H., and Bhattacharyya, S. P., Robust, fragile, or op- +timal? IEEE Transactions on Automatic Control, 42(8): 1098- +1105, (1997). + diff --git a/StE3T4oBgHgl3EQfDwk0/content/tmp_files/load_file.txt b/StE3T4oBgHgl3EQfDwk0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fe633f116cb14687125469d05cbf9447282d6f3 --- /dev/null +++ b/StE3T4oBgHgl3EQfDwk0/content/tmp_files/load_file.txt @@ -0,0 +1,567 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf,len=566 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='04289v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='SY] 11 Jan 2023 Balancing a Stick with Eyes Shut: Inverted Pendulum on a Cart without Angle Measurement Bidhayak Goswami∗ Anindya Chatterjee† Mechanical Engineering, IIT Kanpur January 12, 2023 A shorter version of this paper is due to appear in ASME JDSMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Abstract We consider linear time-invariant dynamic systems in the single-input, single-output (SISO) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In particular, we consider stabilization of an inverted pendulum on a cart using a force on the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This system is easy to stabilize with pendulum angle feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' However, with cart position feedback it cannot be stabilized with stable and proper com- pensators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here we demonstrate that with an additional com- pensator in a parallel feedforward loop, stabilization is possible with such compensators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Sensitivity to noise seems to be about 3 times worse than for the situation with angle feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For completeness, discussion is presented of compensator parame- ter choices, robustness, fragility and comparison with another control approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Keywords: Stabilization, Stable Compensator, Inverted Pendulum, Cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 1 Introduction Stabilization of an inverted pendulum on a cart is a familiar problem in control theory, and also one that is interesting to a broad audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The control input is a horizontal force on the cart;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' and it is desired to use feedback to stabilize the cart at a given location and the pendulum in the inverted (or standing vertical) position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Formally, the linearized system is control- lable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In the classical control theory framework, with a single input and a single output (SISO), it is important whether the output is the pendulum angle or the cart position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Due to the obvious resemblance to balancing a stick on one’s palm, we refer to the latter case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', cart position known and pendu- lum angle unknown) as balancing a stick with eyes shut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This problem, although simple to state, is interesting to a broad audience because a few attempts to balance a stick on one’s palm with eyes shut will convince the reader that the task is difficult if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Technically, the problem is also in- teresting within the usual classical single-loop feedback control framework because, in the case of solely position feedback, it turns out that the system is not stabilizable with a stable and proper controller [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' As a control problem, the inverted pendulum is a famil- iar favorite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' It has been studied by several researchers in the past [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' It has been used as a teaching example for many decades [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Linear control theory has been used for the angular position feedback case, and that case does not rep- ∗bidhayak1728@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='com,bidhayak@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='in †anindya100@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='com, anindya@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='in resent significant challenges any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Control in the nonlin- ear regime, including swing-up dynamics from a hanging-down position, has been studied [8–12] from various viewpoints in- cluding energy-based control as well as control input shap- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Some researchers have studied the effectiveness of simple feedback laws with delays [13], again with angular position feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' [14] have considered system uncertainties, feedback with multi-timescale structure and an extended high- gain observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' An optimal control approach has been used as well, even for harder variants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', a double inverted pendulum on a moving cart [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' With advances in control theory, more modern techniques like robust control, fuzzy logic control, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' have been considered as well [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In spite of the abovementioned works with modern ap- proaches, in this paper we remain within the classical SISO linear regime for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Firstly, a large number of in- dustrial control systems are still linear in their thinking, and often close to just PID control or variations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Secondly, in the absence of angle feedback, for the inverted pendulum on a cart, we obtain a nonminimum phase system with an odd number of real poles on the right of an RHP real zero which, it is known, cannot be stabilized by a stable and proper com- pensator in the usual single feedback loop configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Of course, not all control is based solely on feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Feed- forward compensators [19,20] have been used earlier for stabi- lizing nonminimum phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' [21] used feed- forward compensation for the synchronization of a multi-agent system to achieve faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Golovin and Palis [22,23] used feedforward compensation for stabilizing an electrome- chanical system with friction-induced instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here, we seek suitable feedforward and feedback compensators, both sta- ble and proper, to stabilize the inverted pendulum using only the cart position as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In what follows, we describe the system in section 2, present the control approach and show some relevant results in sec- tion 3, and discuss the effect of noise in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A detailed derivation of the equations of motion is given in appendix A, a description of the controller design methodology is given in appendix B, the modern control approach in terms of controlla- bility and observability is discussed in appendix C, robustness and fragility of the stable compensators have been examined in appendix D, and finally, in appendix E, the effect of noise on the system has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 2 The inverted pendulum Consider an inverted pendulum on a cart (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A control force u acts on the cart as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The pendulum consists of a 1 massless rigid rod of length L with a point mass m at the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The cart’s mass is M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' There is gravity g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The coordinates used are θ for the pendulum angle and x for the cart displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Figure 1: An inverted pendulum on a cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The equations of motion (see appendix A for details) for small θ and ˙θ are M ¨x + m � ¨x + L¨θ � = u, (1a) ¨x + L¨θ = gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (1b) By choice of units of mass, length and time, we can set m, L and g to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This leaves M ¨x + � ¨x + ¨θ � = u, (2a) ¨x + ¨θ = θ, (2b) where M should henceforth be thought of as dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In the SISO framework within elementary classical control, we have a single output: this will often be either the pendulum angle or the cart position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' See the block diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 2 for the basic control structure we will consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here, U is the input, K is a gain, G is the plant, C0, C1 and C2 are compensators, P is an optional feedforward compensator placed in parallel, Y is the actual plant output, and Z is the quantity that is fed back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Obviously, in the absence of P, or with P = 0, we will have Z = Y and obtain the usual single-loop feedback control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The parallel feedforward compensator P is the novelty we will consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A feedforward compensator before the input signal U is rou- tinely used in physical systems where the input from the user may be, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', a desired angle and the input to the measure- ment and control system is, typically, a voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' However, subsequent analysis of the control system is independent of that compensator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' So, we have not included it in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Returning to the inverted pendulum on a cart, when we balance a stick on our palm, we always look at the pendulum angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The corresponding transfer function of the plant is F = 1 1 + M − M s2 , (3) which has a right half plane or RHP pole but is easily sta- bilizable with a stable controller without any P in a parallel feedforward path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For example, with M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3, C0 = C1 = 1, and K > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='33, F = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3s2 (4) Figure 2: A feedfoward-feedback control system: basic layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We take C0 = 1 for simplicity (see section 3 for justification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' can be stabilized by the stable compensator C2 = − s + 3 s + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (5) However, if our output variable is x, which loosely corre- sponds to trying to balance the stick on our palm with our eyes shut, then the plant becomes G = s2 − 1 s2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3s2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3), (6) which has a real RHP zero at s = 1 and a single real RHP pole to its right, at s = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This is more interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 3 Stabilization In classical control theory with a single control loop [24], P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Then, although C0, C1 and C2 can in principle all be present, stability is affected only by the product C0C1C2, and so we can for stabilization purposes take any two of them to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The gain K, too, can be included within C1 if we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In this paper, we take C0 = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' From a design viewpoint, non unity C0 can be thought as a system modification and we leave it for more challenging problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In simple control systems, the compensators C1 and C2 may be physically realized using simple circuits made with resistors, capacitors, and op-amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In such cases we want the compen- sator transfer functions themselves to be stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', C0, C1 and C2 should not have RHP poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here we assume that P has no RHP poles either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Thus we are interested in stabilizing G with stable controllers1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In the absence of P, a fundamental fact has been known for almost 50 years [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' If G has one or more real RHP zeros;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' and if G also has an odd number of real RHP poles that lie to the right of any positive real zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' then in the absence of P in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 2, G is not stabilizable [1] with stable and proper compensators C0, C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The mathematical aim of this paper can now be clearly stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We will take the troublesome G of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (6), set K = C1 = 1 (as well as C0 = 1 as stated above), and find a stable and proper C2 = C along with a stable and proper P such that the plant output Y is stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Let C = nC dC , G = nG dG , and P = nP dP , 1The motivation is that the analog control card should not saturate and lose linearity before the actual system dynamics is established, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', during a warm-up phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' where the n’s stand for numerator polynomials in s and the d’s stand for denominator polynomials in s of equal or greater order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Routine manipulations show that Y = GU 1 + C(P + G) = HU, (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', the controlled transfer function is H = nGdCdP dCdGdP + nCnP dG + nCnGdP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (8) Thus, our controller design for stabilization reduces to choosing polynomials nC, dC, nP and dP such that the d- polynomials are stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', they have only LHP roots), and the denominator polynomial dCdGdP + nCnP dG + nCnGdP is stable as well (has only LHP roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We are not aware of systematic and guaranteed ways of ob- taining such polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We have used trial and error based on a simple optimization routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Some details of the opti- mization are given in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Our main aim here is to demonstrate and assess specific numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Two stabilizing solutions, out of many that we found, are shown below in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (9) and (10), labeled “a” and “b” respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Pa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05s3 + 7s2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='9 11s3 + 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='7s2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='4s + 1 , Ca = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1s3 + 2s2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='9s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='002s3 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='2s2 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='2s + 1 , (9) and Pb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='2s3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='4s2 − 2s − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1s3 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='8s2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3s + 1, Cb = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='9s3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='6s2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='08s3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='4s2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3s + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (10) The corresponding unit-step responses of the controlled sys- tems are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 0 10 20 30 40 0 5 10 15 20 0 5 10 15 5 0 5 10 15 20 Figure 3: Unit-step responses of G of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (6), with P and C as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' At this point we can check to see, for the same controlled system (with only position feedback), the angle response of the inverted pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Recalling F from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (4), we write nF = 1, and dF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 0 10 20 30 3 2 1 0 1 2 3 0 5 10 15 4 2 0 2 4 Figure 4: Angular response of the pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Further, recalling Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (6), (7) and (8), we find that the angular response of the inverted pendulum must be HF G U, with HF G = − nF dCdP dCdGdP + nCnP dG + nCnGdP s2, (11) where we have used dG = −s2 dF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The angular response of the pendulum is given by the step response of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (11): see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' It is also interesting to ask how other control strategies might work for this same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A discussion of the textbook ap- proach of modern control theory, with state estimation and full state feedback, is given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Finally, we must address two more issues: (i) the robust- ness of the controller, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', its ability to stabilize plants with slightly different plant-parameter values, and (ii) the fragility of the controller, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', its tendency to lose effectiveness under small perturbations of the controller-parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Both robustness and fragility are good, as shown in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 4 Effect of noise The system has now been mathematically stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We must also study the effect of noise on the stabilized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Since the basic control problem is difficult at least in some ways (as in balancing a stick on one’s palm with eyes shut), we may expect increased sensitivity to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The system now has more than one input (the control force u along with noise inputs ek, k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6}), but still only one output (y), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In the Laplace domain, elementary calculations give us individual transfer functions for each input, with the net output given as Y (s) = H(s)U(s) + 6 � k=1 Hek(s)Ek(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (12) In the above the seven transfer functions H and Hek, k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6}, share a common denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' If H is stable, then so is each Hek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The Bode magnitude plots for Hek with P = Pb and C = Cb are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For Pa and Ca, the maximum magnitude is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' It is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 6 that for Figure 5: The control system with added noise inputs ek(t), k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' nondimensional frequencies on the order of unity, the magni- tudes of the transfer functions take their highest values, which are around 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This corresponds to amplification by roughly a factor of 30, which is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 10-2 100 102 80 60 40 20 0 20 10-2 100 102 80 60 40 20 0 20 40 10-2 100 102 80 60 40 20 0 20 40 10-2 100 102 40 20 0 20 40 10-2 100 102 100 50 0 50 10-2 100 102 40 20 0 20 40 Figure 6: Bode magnitude plots of Hek, k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6}, for the system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 5 with P = Pb and C = Cb given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We mention that in separate calculations with G as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (3), K > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='33, P = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', no parallel feedforward compen- sator), and C = C2 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (5), maximum amplification fac- tors about 10 dB lower were easily obtained (details omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This is not intuitively surprising because balancing a stick with one’s eyes open is easier than with one’s eyes shut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' and corre- spondingly, stabilizing the inverted pendulum on a cart in the classical SISO setting with pendulum angle feedback is easier than with cart position feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The sensitivity to noise in the eyes-shut case, for our control design, seems to be greater by about a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Some numerical examples of the response to noise inputs, where the “noise” is the sum of a large number of small sinu- soidal inputs with randomly chosen amplitudes and frequen- cies, are given in appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The results there are consistent with the above estimate of 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 5 Concluding remarks In some applications such as low cost consumer products or toys, there may be simple analog control cards which, if oper- ated within the linear domain, produce desirable behavior in the controlled device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In such situations, stabilization with a stable controller has practical utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Additionally, there may be sophisticated scientific or technical instruments wherein simple controllers are implemented with some parameters ad- justable by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In such cases, too, while the system is warming up, or is outside the operational position range, an unstable compensator may lead to overly large actuator commands that cause saturation, deviation from linearity, or possibly specimen damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In such cases, also, stabilization with a stable controller may make the system more foolproof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' With the above motivation, we appreciate the classic pa- per [1] which lays down the mathematical conditions under which, in the single loop control structure, stabilization is not possible with a stable and proper compensator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' One of the most familiar control problems, namely balancing an inverted pendulum on a cart, presents this situation when position feed- back is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For this system, using a parallel stable feedfor- ward compensator, we have demonstrated using numerical ex- amples that it is possible to achieve stabilization with stable and proper compensators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Finding such stable and stabilizing compensators is not a familiar and routine control design problem within classical control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Others have studied this control approach be- fore as well [21,22], but not widely and not for such a popular problem as balancing a stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We hope that the community of industrial and academic control systems practitioners and re- searchers will find this class of problems sufficiently interesting as to develop this kind of control design further, and possibly even seek rational design criteria for when such controllers ex- ist and how they may be found easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Moreover, once such a block diagram framework is adopted, it can also be used for design of controllers for nonlinear sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Such work [25] has begun to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A Equations of motion We draw free-body diagrams for both cart and pendulum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The pendulum’s pivot point P (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 7(a)) experiences a reaction force from the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This force has components Rx and Ry along unit vectors ˆe1 and ˆe2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The cart ex- periences equal and opposite reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' There are two ground forces on the cart wheels, named N1 and N2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Friction has not been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The weight of the pendulum and cart are mg and Mg respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A horizontal force u acts on the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Figure 7: Free body diagrams for the pendulum and the cart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Moving on to the kinematics, P has an acceleration ¨x ˆe1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The acceleration of G is aG = aP + aG/P = ¨x ˆe1 + � L ¨θ ˆeθ − L ˙θ2 ˆen � , (13) where ˆeθ and ˆen are unit vectors shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For the pendulum, linear momentum balance gives Rx = m (aG · ˆe1) = m � ¨x + L ¨θ cos(θ) − L ˙θ2 sin(θ) � , (14) and for the cart, it gives − Rx + u = M ¨x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (15) Substituting Rx from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (14) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (15) M ¨x + m � ¨x + L ¨θ cos(θ) � − m L ˙θ2 sin(θ) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (16) Now, for the pendulum, the moment about spatial point P (coincides with the pivot instantaneously) is τ P = IG · α + rG/P × m aG, (17) where IG is zero and rG/P is the position vector of from P to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This yields − m g L sin(θ) = −m L � ¨x cos(θ) + L¨θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (18) or ¨x cos(θ) + L¨θ = g sin(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (19) Linearizing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (16) and (19), for small θ and ˙θ, we obtain M ¨x + m � ¨x + L¨θ � = u, (20a) ¨x + L¨θ = gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (20b) B Methodology used for obtaining C and P For both compensators C and P, we choose nth order polyno- mials with unknown coefficients for both numerator and de- nominator, where n is a positive integer to be chosen by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The compensators’ transfer functions are taken as C = a0 + a1 s + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' an sn 1 + an+1s + · · · + a2nsn , P = b0 + b1 s + · · · + bn sn 1 + bn+1 s + · · · + b2n sn , where ak, bk constitute 4n + 2 unknown coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Now we construct an objective function F as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (i) F takes 4n+2 numbers as a vector input q and first forms the candidate C and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (ii) From the numerators and the denominators of C and P, it calculates H (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (iii) It calculates the poles of C, P and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (iv) From the poles of C and P, the right-most real part is saved as p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (v) From the poles zHi of H, the right-most real part is saved as p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (vi) A preliminary function value f0 is defined as f0 = � p2 + 6 p1 if p1 ≥ 0, p2 otherwise, where the “6” is a penalty parameter, found to be big enough by trial and error (unnecessarily large penalty pa- rameters are best avoided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (vii) For better behavior, the actual objective function used was F = f0 + ε1 ||q|| + ε2 max{|zHi|}, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' , 2 n + 4, where ε1 = 10−5 and ε2 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' If we can find a q such that F(q) < 0, then our goal is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We can now use any optimization techniques we like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We used a simple in-house genetic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The code is available on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' C Controllability and observability So far, we have studied the system from the viewpoint of classi- cal control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In the modern control approach, the system state consists of x, θ, ˙x and ˙θ given as a column matrix x = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x θ ˙x ˙θ \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (21) Writing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (1a) and (1b) in state space form, we obtain ˙x = A x + B u, (22) where, for M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3, A = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 1 0 − 10 3 0 0 0 13 3 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb and B = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 0 10 3 − 10 3 \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (23) In this problem, only the measurement of the cart displace- ment is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' So, the measured quantity y = x = C x, where C = [1 0 0 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (24) Taking Laplace transforms of both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (22), we obtain for zero initial conditions X(s) = (s I − A)−1 B U(s), where X(s) = L[x(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (25) Using the symbolic algebra package Maple, we have verified that C (s I − A)−1 B = G(s) = s2 − 1 s2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 s2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (26) The controllability matrix [24] is PC = � A3B | A2B | AB | B � = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 100 9 0 10 3 0 − 130 9 0 − 10 3 0 0 100 9 0 10 3 0 − 130 9 0 − 10 3 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , (27) which has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The observability matrix [24] PO = \uf8ee \uf8ef\uf8f0 CA3 CA2 CA C \uf8f9 \uf8fa\uf8fb = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 − 10 3 0 − 10 3 0 0 0 0 1 0 1 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (28) also has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The system is both controllable and ob- servable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A controller can be designed by constructing a state estimator and then using full state feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Let us consider the following system ˙x = A x − B K ˜x + B u (29a) ˙˜x = A ˜x − B K ˜x + G C (x − ˜x) + B u (29b) where ˜x is the estimated state and the gain matrices K and G are found by placing the system poles (arbitrarily) at − 1 ± i and − 2 ± i, (30) and the estimator poles (also arbitrarily) at − 1, −2, and − 3 ± i (31) on the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' These numbers are chosen for demon- stration only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (29a) and (29b), we obtain ˙˜Z = ˜A ˜Z + ˜B u, (32) where, ˜Z = � x ˜x � , ˜A = � A −B K G C A − B K − G C � , and ˜B = � B B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (33) The output y = C x = ˜C ˜Z, where ˜C = [C, 0, 0, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (34) To interpret these result in light of the main paper, we can now think of an implied feedback controller, with closed loop transfer function Q(s) = ˜C � sI − ˜A �−1 ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (35) Using Maple, we obtain Q(s) = 10 s2 − 10 3 s4 + 18 s3 + 45 s2 + 54 s + 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (36) We observe that Q and G share the same zeros, and the poles of Q are the same as the system poles chosen for placement (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We may think of a feedback control system equivalent to Figure 8: An equivalent single loop feedback control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' the implied control system as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 8, where plant G is assigned compensators Kb and Kf in forward and feedback loops respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Hence Q = Kf G 1 + Kb Kf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (37) From algebraic manipulations, we obtain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Kf = Q G 1 1 − Kb Q and Kb = 1 Q − 1 Kf G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (38) Clearly, there are infinitely many solutions for Kf and Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We examine two limiting cases for better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (i) The system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 8 has a compensator only in the feedback loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', Kf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In this case, Kb = 1 Q − 1 G = 9 s3 + 29 s2 + 27 s + 15 5 s2 − 5 , (39) which is unacceptable (both improper and unstable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' (ii) The system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 8 has a compensator only in the forward loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', Kb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Now we have Kf = Q G 1 1 − Q = dG dQ − nQ = s2 � 3 s2 − 13 � 3 s4 + 18 s3 + 35 s2 + 54 s + 40, (40) where dQ and nQ are the denominator and numerator of Q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The compensator Kf is stable, but relies on pole zero cancellation which is not allowed in classical control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' A commonly stated reason for not allowing pole zero can- cellation is that the slightest inaccuracy in the controller will destroy the cancellation and instability will reappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Youla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' [1] also point out that exact pole zero cancel- lation may represent nonobservable modes which remain unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In any case, we cannot accept this Kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We already know that the controller obtained in this ap- pendix cannot be realized (Youla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' [1]) with stable and proper compensators in the classical single loop configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Equations (39) and (40) merely provide two examples of the difficulties encountered if such an attempt is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' D Robustness and fragility Having found a stable closed loop transfer function H as ex- plained in appendix B, we can check its sensitivity to small changes in plant and compensator parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Robustness, for a control system, is its ability to retain sta- bility under small changes in the plant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here, the plant parameters depend on the system parameters: L, m, g and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Of them, the first three were eliminated from the gov- erning equations by introducing nondimensional displacement, time and mass ˜x, ˜t and ˜m respectively where ˜x = x L, ˜t = t � g L, ˜m = M m , (41) which is analogous to setting the values of m, L and g to unity and treating M as the only free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' So far we have considered M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' To investigate the effect of small changes in parameter values on the system behavior, we rewrite the plant transfer function as G = A0 s2 − A1 s2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 A2 s2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3 A3), (42) where the parameters A0, A1, A2 and A3 are notionally equal to unity along with M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For a large number of random calculations (1000 times), we perturb the A’s by normally dis- tributed iid random variables ri, i = 0, 1, 2, 3, with zero mean and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='02 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='7% of the points are within ± 6%) in the following way Ai �→ Ai (1 + ri), i = 0, 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We then plot the poles (zH) of the respective closed loop trans- fer functions (CLTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For compensators Ca and Pa, the entire cloud (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 9(a)) of poles remains in the left half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For compensators Cb and Pb, a significant part of the cloud (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 9(b)) remains in the left half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In 44 out of 1000 cases, the CLTF has poles in the right half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Thus, the compensators are fairly robust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' and Ca and Pa are more robust than Cb and Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Some robust control systems perform poorly under small perturbations in the compensator parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' This is called the fragility [26] of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' To check fragility, we perturb the compensator parameters, again 1000 times, by normally distributed iid random variables si, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 4 n+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Here, Figure 9: Robustness under small changes in plant parameters (1000 random perturbations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Figure 10: Performance of the system under small changes in compensator parameters (1000 random perturbations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' n is degree of the polynomials in the numerator and denomi- nator of the compensators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The random variables si have zero mean and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='02 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='7% of them are within ± 6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We perturb the a’s and b’s as follows: ai �→ ai (1 + si), bi �→ bi (1 + s2 n+1+i), i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' , 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We then calculate the poles (zH) of the respective closed loop transfer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For the compensators Ca and Pa, a large portion of the cloud of poles again remains in the left half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In 9 out of 1000 cases, the CLTF has poles in the right half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For the compensators Cb and Pb, in 32 out of 1000 cases, the CLTF has poles in the right half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' We conclude with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Implementabil- ity, albeit implicitly discussed, has motivated this entire paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' Finding stable compensators (which we have now shown are fairly robust and not fragile) indicates that the compensators are implementable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' E Response to noise In section 4, we examined the system’s sensitivity to six noise inputs ei(t), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6 by using Bode plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' To demon- strate the effect of noise on the time response of the system, we use the following input ei(t) = N � k=1 ck sin (ωk t) , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6, (43) where the ω’s are randomly chosen numbers uniformly dis- tributed in the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The amplitudes ck, k = 41类 2 米 Im(ZH) 0 米 米 2 米 4 米 2 1 0 Re(ZH)21 Im(ZH) 0 1 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5 Re(ZH)41米 ■**米 2 米 来米 Im(ZH) 0 米 2 米 4 2 1 0 Re(ZH)211 (Hz) 0 Im ( 米瓣 1 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='5 Re (ZH)80 100 120 140 160 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0 20 40 60 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 80 100 120 140 160 180 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0 20 40 60 100 120 140 160 180 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='2 0 20 40 60 80 80 100 120 140 160 180 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='2 0 20 40 60 Figure 11: Time responses to noise inputs e1(t) and e3(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' , N are random numbers where the norm of the vector c = [c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' , cN]⊤ is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' For calculations, we have used N = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' The response, with phase randomized, is taken as xei(t) = N � k=1 ck|Hei(i ωk )| sin (ωk t + arg (Hei(i ωk) ) + φk) , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', 6, (44) where i = √−1, and the φk are random numbers uniformly distributed in the interval [0, 2 π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' 11, the amplification factor is consistent with the Bode plots of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' References [1] Youla, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=', Bongiorno Jr, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 42(8): 1098- 1105, (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE3T4oBgHgl3EQfDwk0/content/2301.04289v1.pdf'} diff --git a/UNE0T4oBgHgl3EQflQHp/content/tmp_files/2301.02485v1.pdf.txt b/UNE0T4oBgHgl3EQflQHp/content/tmp_files/2301.02485v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3f6386ae8ab8882e01e5652b904b06d2bc0a3de --- /dev/null +++ b/UNE0T4oBgHgl3EQflQHp/content/tmp_files/2301.02485v1.pdf.txt @@ -0,0 +1,2668 @@ +Hard Jet Substructure in a Multi-stage Approach +Y. Tachibana,1, ∗ A. Kumar,2, 3, † A. Majumder,3 A. Angerami,4 R. Arora,5 S. A. Bass,6 S. Cao,7, 3 Y. Chen,8, 9 +T. Dai,10 L. Du,2 R. Ehlers,11, 12 H. Elfner,13, 14, 15 W. Fan,6 R. J. Fries,16, 17 C. Gale,2 Y. He,18, 19 +M. Heffernan,2 U. Heinz,20 B. V. Jacak,21, 22 P. M. Jacobs,21, 22 S. Jeon,2 Y. Ji,23 K. Kauder,24 L. Kasper,25 +W. Ke,26 M. Kelsey,3 M. Kordell II,16, 17 J. Latessa,27 Y.-J. Lee,8, 9 D. Liyanage,20 A. Lopez,28 M. Luzum,28 +S. Mak,23 A. Mankolli,25 C. Martin,11 H. Mehryar,27 T. Mengel,11 J. Mulligan,21, 22 C. Nattrass,11 +D. Oliinychenko,22, 29 J.-F. Paquet,10 J. H. Putschke,3 G. Roland,8, 9 B. Schenke,30 L. Schwiebert,27 +A. Sengupta,16, 17 C. Shen,3, 31 A. Silva,11 C. Sirimanna,3 D. Soeder,10 R. A. Soltz,3, 4 I. Soudi,3 J. Staudenmaier,14 +M. Strickland,32 J. Velkovska,25 G. Vujanovic,3, 33 X.-N. Wang,34, 21, 22 R. L. Wolpert,23 and W. Zhao3 +(The JETSCAPE Collaboration) +1Akita International University, Yuwa, Akita-city 010-1292, Japan. +2Department of Physics, McGill University, Montr´eal QC H3A 2T8, Canada. +3Department of Physics and Astronomy, Wayne State University, Detroit MI 48201. +4Lawrence Livermore National Laboratory, Livermore CA 94550. +5Research Computing Group, University Technology Solutions, +The University of Texas at San Antonio, San Antonio TX 78249. +6Department of Physics, Duke University, Durham, NC 27708, USA +7Institute of Frontier and Interdisciplinary Science, +Shandong University, Qingdao, Shandong 266237, China +8Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge MA 02139. +9Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139. +10Department of Physics, Duke University, Durham NC 27708. +11Department of Physics and Astronomy, University of Tennessee, Knoxville TN 37996. +12Physics Division, Oak Ridge National Laboratory, Oak Ridge TN 37830. +13GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany. +14Institute for Theoretical Physics, Goethe University, 60438 Frankfurt am Main, Germany. +15Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany. +16Cyclotron Institute, Texas A&M University, College Station TX 77843. +17Department of Physics and Astronomy, Texas A&M University, College Station TX 77843. +18Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, +South China Normal University, Guangzhou 510006, China. +19Guangdong-Hong Kong Joint Laboratory of Quantum Matter, +Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China. +20Department of Physics, The Ohio State University, Columbus OH 43210. +21Department of Physics, University of California, Berkeley CA 94270. +22Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley CA 94270. +23Department of Statistical Science, Duke University, Durham NC 27708. +24Department of Physics, Brookhaven National Laboratory, Upton NY 11973. +25Department of Physics and Astronomy, Vanderbilt University, Nashville TN 37235. +26Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545. +27Department of Computer Science, Wayne State University, Detroit MI 48202. +28Instituto de F`ısica, Universidade de S˜ao Paulo, C.P. 66318, 05315-970 S˜ao Paulo, SP, Brazil. +29Institute for Nuclear Theory, University of Washington, Seattle WA, 98195. +30Physics Department, Brookhaven National Laboratory, Upton NY 11973. +31RIKEN BNL Research Center, Brookhaven National Laboratory, Upton NY 11973. +32Department of Physics, Kent State University, Kent, OH 44242. +33Department of Physics, University of Regina, Regina, SK S4S 0A2, Canada. +34Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, +Central China Normal University, Wuhan 430079, China. +We present predictions and postdictions for a wide variety of hard jet-substructure observables +using a multi-stage model within the JETSCAPE framework. The details of the multi-stage model +and the various parameter choices are described in Ref. [1]. A novel feature of this model is the +presence of two stages of jet modification: a high virtuality phase (modeled using MATTER), where +coherence effects diminish medium-induced radiation, and a lower virtuality phase (modeled using +LBT), where parton splits are fully resolved by the medium as they endure multiple scattering in- +duced energy loss. Energy loss calculations are carried out on event-by-event viscous fluid dynamic +backgrounds constrained by experimental data. The uniformed and consistent descriptions of multi- +ple experimental observables demonstrate the essential role of coherence effects and the multi-stage +modeling of the jet evolution. Using the best choice of parameters from Ref. [1], and with no further +tuning, we present calculations for the medium modified jet fragmentation function, the groomed +arXiv:2301.02485v1 [hep-ph] 6 Jan 2023 + +2 +jet momentum fraction zg and angular separation rg distributions, as well as the nuclear modifi- +cation factor of groomed jets. These calculations provide accurate descriptions of published and +preliminary data from experiments at RHIC and LHC. Furthermore, we provide predictions from +the multi-stage model for future measurements at RHIC. +I. +INTRODUCTION +In high-energy heavy-ion collisions, the high-transverse +momentum (pT ) partons (pT ⪆ 10 GeV) are generated +almost at the instant at which the incoming nuclei over- +lap. Such high pT partons are generated in parton-parton +exchanges with large momentum transfers Q ≫ ΛQCD. +They are typically produced far from their mass shell +and engender multiple collinear emissions produced over +a large time range. In the case of a heavy-ion collision, +the propagation and development of these parton show- +ers are strongly affected by the produced Quark Gluon +Plasma (QGP). Studying jet modification in nucleus- +nucleus collisions relative to proton-proton collisions, to- +gether with constraints from model-to-data comparison +provides unique opportunities to probe the properties of +the QGP [2–20]. +The experimental attempts started at the Relativis- +tic Heavy Ion Collider (RHIC) with the observation of +suppression in the yield of single inclusive hadrons [21– +25] and associated hadrons (dihadrons) [26–28] produced +with high transverse momentum relative to the yield +in proton-proton collisions. Since 2010, starting at the +Large Hadron Collider (LHC) and later at RHIC, the +ability of experiments evolved from single hadrons and +dihadrons to jets [29–31]. +Over the last decade, experiments have attained the +ability to not just study the energy-momentum and cross +section of a jet but also to look at modifications of the +internal properties of the jet, often referred to as jet +substructure. +Based on current detector improvements +and accumulated high statistics data at RHIC and the +LHC, it is possible to analyze a vast variety of observ- +ables revealing different aspects of the jet-medium in- +teraction [32]. For example, the yield suppression and +internal structure of fully reconstructed jets, revealed in +observables such as the jet fragmentation function and +jet shape (respectively), provide details on the diffusion +of jet energy and momentum in momentum or angular +space due to the interaction with the medium [29–31, 33– +52]. Even the structural modification of hard partonic +branching is now potentially accessible through groomed +jet observables [53–58]. +On the theory side, many studies have attempted to +describe and understand the jet-medium interaction by +constructing models that reproduce these various observ- +ables or propose predictions and new observables [59–79]. +In particular, to obtain a universal understanding, it is +∗ Corresponding author: ytachibana@aiu.ac.jp +† Corresponding author: amit.kumar3@mail.mcgill.ca +essential to simultaneously explain multiple observables, +ultimately all observables, with a consistent theoretical +picture. Therefore, Monte Carlo calculations, which can +generate experiment-like events by a single model, are +a powerful tool for theoretical approaches because they +enable one to calculate a wide range of event-by-event +defined jet observables [80–113]. +Jets evolve dynamically, moving through the expand- +ing medium, and generating more partons from splits +and interactions with the dense medium. The original +partons start at very high virtuality, and thus, the early +splits have a small transverse size. These splittings from +the leading parton and the still highly-virtual daugh- +ters are driven by their individual virtualities, with mi- +nor medium correction via the scattering, strongly sup- +pressed due to their small transverse size. We refer to +these as Vacuum Like Emissions (VLE) [114]. To simu- +late the VLEs, taking into account the reduction in the +effective interaction rate with scale dependence, an event +generator such as MATTER [115, 116] can be employed. +With repeated splittings, the virtuality of the partons +reduces to the point that splits are widely separated in +time. With decreasing virtuality, the transverse size of +the parton becomes larger, thereby increasing the rate +of interaction with the medium, which in turn triggers +more radiation. Thus, the main mechanism causing par- +ton splittings changes dynamically in the medium. The +evolution of such partons at lower virtuality but en- +ergy still large enough to treat the medium interaction +perturbatively can be approximated by kinetic theory- +based approaches for on-shell particles, as implemented +by generators such as LBT [90, 92, 96, 97], or MAR- +TINI [84, 111, 113]. +As partons transition to energies +and virtualities close to those of the QGP, they begin to +undergo strong coupling [89] and thermalization with the +medium [117]. Thus, jets interact with the medium over +a wide range of scales, which requires incorporating mul- +tiple generators at different scales for simulations [17]. +JETSCAPE is a general-purpose framework for Monte +Carlo simulations of the complete evolution in high- +energy heavy-ion collisions [1, 118–124]. The framework +is designed to be as general and extensive as possible +while modularizing each physics element involved in a +collision event, such as the generation of geometric initial +conditions, hydrodynamic evolution of the soft sector, jet +production by hard scattering, etc. so that users can em- +ploy a module based on their favorite physical description +for each. For the in-medium parton shower evolution, the +most distinctive feature of the JETSCAPE framework is +its support for multi-stage descriptions that, by stitching +multiple models together, cover a broader range of scales. +Depending on the virtuality or energy of a parton, each +model becomes active to handle the parton shower evo- + +3 +lution interactions with the medium. +Recently, we systematically studied the energy loss of +large-transverse momentum particles, jets, and charmed +particles using a multi-stage model, combining two mod- +ules, MATTER for high-virtual parton shower and LBT +for low virtuality, developed within the JETSCAPE +framework in Refs. [1, 124]. +Our simulations indicate +that the single high-pT particle spectra are dominated +by the large virtuality phase simulated by the MATTER +module. On the other hand, to describe the suppression +of reconstructed jets and D mesons, we found that the +energy loss of soft daughter partons and heavy quarks +is governed by the low-virtuality scattering dominated +phase simulated by the LBT module. +One further important insight from our prior work is +that the reduction of the interaction with the medium +at high virtuality due to coherence effects plays a crucial +role in explaining the weak suppression of single charged +particles with pT ⪆ 10 GeV. These coherence effects oc- +cur because the partons probing the medium have a small +transverse size when the virtuality is large. A section of +QGP resolved at such a shorter distance scale appears +more dilute, resulting in fewer interactions [125].1 +Coherence effects implemented in MATTER drasti- +cally improve the description of the transverse momen- +tum dependence of the nuclear modification factor for +inclusive single-charged particles, even at the qualitative +functional behavior level. In contrast, for reconstructed +jets at the currently available collision energies, coher- +ence effects are not visible in the transverse momentum +dependence of the nuclear modification factor, which only +necessitated a readjustment of the overall medium cou- +pling parameter αfix +s . Thus, it is essential to search for +the role of coherence effects in the evolution of jet show- +ering patterns by examining further inner jet structure +modification. +In this paper, we systematically analyze the observ- +ables characterizing the internal structure of jets using +the results of the exact same numerical simulations with +MATTER+LBT that were used to study the nuclear +modification factors for reconstructed jets and high pT +single-charged particles in Ref. [1]. The goal is to explore +the details of the interaction strength at each scale on +the internal structure of the jet. In particular, we exam- +ine the groomed jet observables, which display the effect +of jet-medium interactions at the early high-virtuality +stage, and the jet fragmentation function, which shows +the medium effect on partons throughout a wide range +of scales. In this work, we do not re-tune any parameters +and employ those obtained in our previous work [1]. +The paper is organized as follows. In Sec. II, salient +characteristics of the underlying model are presented. In +Subsec. II A, an overview of the framework and setup is +1 In several other models, e.g., those in Refs [102, 103], coherence +effects are implicitly taken into account, without detailed formu- +lations, by turning off the medium effect at high virtuality. +outlined. Subsection II B is devoted to formulating co- +herence effects. This is followed by an investigation of +the medium modification of jet substructure observables, +focusing on coherence effects, by presenting results from +our model calculations in Sec. III. Here, we also make +predictions for the upcoming measurements of the jet +substructure observables at RHIC. A summary of our re- +sults and concluding remarks are presented in Sec. IV. +The Appendix is dedicated to the presentation of our +predictions of jet RAA at the top RHIC energy for bench- +marking purposes. +II. +MODEL +JETSCAPE is a general-purpose event generator +framework where different sub event generators can be +included in a modular fashion, producing an extensive +end-to-end simulation of a heavy-ion collision. +In this +paper, we will use the results of simulations that were +generated in Ref. [1] to calculate all jet substructure ob- +servables. +This is not just for convenience but rather +to demonstrate how the exact same simulations can si- +multaneously describe both the jet and leading hadron +suppression, as well as several jet substructure observ- +ables. +To that end, only a very brief overview of the compo- +nents of the simulation will be provided in this section. +The reader may refer to Ref. [1] for specific details of the +physics included in a MATTER+LBT simulation within +the JETSCAPE framework. Computational aspects of +the JETSCAPE framework are described in great detail +in Ref. [118], while the basic physics of multi-stage sim- +ulators is described in Ref. [126]. +A. +Overview +To explore the medium modification of jet substruc- +ture, we perform simulations of jet events in high-energy +nucleus-nucleus collisions utilizing the full framework of +JETSCAPE in two separate steps. First, we calculate the +event-by-event space-time profiles of the QGP medium +in nucleus+nucleus (A+A) collisions for the estimation +of the local medium effect on parton shower evolution. +For this part, we perform simulations of (2+1)-D free- +streaming pre-equilibrium evolution [127] and subsequent +viscous hydrodynamic evolution by the (2+1)D VISHNU +code package [128] with the initial condition generated by +TRENTo [129]. Here the MAP parameters obtained by +Bayesian calibration in Ref. [130] are used for the LHC +energy calculations, while hand-tuned parameters were +used for top RHIC energy. +In the second step, the binary collision distribution +from the same TRENTo initial condition as for the +medium is used to sample the transverse position of a +hard scattering. +The hard scattering is produced by +Pythia 8 [131] with initial state radiation (ISR) and + +4 +multiparton interaction (MPI) turned on, and final state +radiation (FSR) turned off. The produced partons in the +hard scattering then undergo the multi-stage in-medium +parton shower evolution within the JETSCAPE frame- +work. In this study, we use a combination of MATTER +and LBT modules as described in Ref. [1]. +The partons produced by hard scattering are first +passed +to +the +MATTER +module, +which +simulates +virtuality-ordered splitting of high-energy partons incor- +porating medium effects [115, 116]. This description by +MATTER is valid for partons with virtuality sufficiently +larger than the accumulated transverse momentum and +virtuality generated by scattering from the medium. +Partons whose virtuality is reduced by showering in +MATTER are then transferred to LBT at a transition +scale. +In LBT, the kinetic theory for on-shell partons +with elastic and inelastic scatterings with medium con- +stituents is applied [90, 92, 132]. +The parton split- +tings under this description are entirely scattering-driven. +In the multi-stage approach of the JETSCAPE frame- +work, virtuality-dependent switching between modules +is done bi-directionally on a per-parton basis using a +switching parameter Q2 +sw. If the virtuality of the par- +ton Q2 = pµpµ − m2 falls below Q2 +sw, it is then sent +from MATTER to LBT. Conversely, the parton is re- +turned to MATTER if its virtuality exceeds Q2 +sw again, +or it goes out of the dense medium. The transition from +medium-like back to vacuum-like emission takes place at +a boundary with a temperature Tc = 0.16 GeV. In this +study, Q2 +sw is set to 4 GeV2. After all the partons are out- +side the QGP medium and have virtuality smaller than +the cut-off scale Q2 +min = 1 GeV2, they are hadronized via +the Colorless Hadronization module, in which the Lund +string model of Pythia 8 is utilized. +In both MATTER and LBT modules, the medium +response effect is taken into account via recoil par- +tons [85, 88, 97, 98, 117, 133, 134]. +In the recoil pre- +scription, the energy-momentum transfer is described by +scatterings between jet partons and medium partons. For +each scattering, a parton is sampled from the thermal +medium. Then, the scattered sampled parton is assumed +to be on-shell, and passed to LBT for its in-medium evo- +lution, assuming weak coupling with the medium. These +recoil partons and further accompanying daughter par- +tons are collectively hadronized with the other jet shower +partons. On the other hand, a deficit of energy and mo- +mentum in the medium is left for each recoil process, +where a parton emanating from the medium is included, +post scattering, as a part of the jet. We treat this deficit +as a freestreaming particle, referred to as a hole parton, +and track it. The hole partons are hadronized separately +from other jet partons, and their energy and momentum +within each positive particle jet cone are subtracted in +the jet clustering routine to ensure energy-momentum +conservation. +In the later stages of evolution, where the energy of +a jet shower parton reaches a comparable scale to the +ambient temperature, the mean free path is no longer +large enough to apply the kinetic theory-based approach +with the recoil prescription. +In principle, such soft +components of jets are supposed to be thermalized and +evolve hydrodynamically as part of the bulk medium +fluid [20, 135–140]. As in Refs. [141–151], implementa- +tion of models based on such a description is proposed, +and there are some studies of the hydrodynamic medium +response to jets using it [72, 74, 95, 112, 117, 152–156]. +However, with such an implementation of the hydrody- +namic medium response, the computational cost for a +systematic and exhaustive study covering various config- +urations, as presented in this paper, is enormously expen- +sive. Thus, in this paper, we mainly discuss the structure +of the hard part of the jet, where the contributions of such +very soft components are relatively small. A further com- +prehensive investigation with more detailed modeling of +the medium response in jet modification is left for future +work. +To investigate the modification of jet substructures by +medium effects in A+A collisions, the calculations of the +same observables for p+p collisions are necessary as ref- +erences. For such calculations, the parton shower evolu- +tion modules are replaced entirely by MATTER with no +in-medium scattering. This setup for p+p collisions of +JETSCAPE, referred to as the JETSCAPE PP19 tune, +is equivalent to the limit of no medium effect in the event +and is detailed in Ref. [119]. +B. +Coherence Effects at High Virtuality +In this study, we focus on coherence effects [114, 125, +157–159] on the interaction of a highly virtual parton +with the medium and explore their manifestation in jet +substructure modification. In Ref. [125], it was demon- +strated that a hard parton with large virtuality resolves +the very short-distance structure of the medium via the +exchange of a gluon whose momentum is much larger +than the medium temperature. +These coherence ef- +fects are formulated with the continuous evolution of the +medium-resolution scale and give a gradual reduction of +jet parton-medium interaction as a function of the virtu- +ality. +For jet quenching calculations, coherence effects can +be effectively implemented by introducing a modulation +factor f(Q2), which diminishes as a function of the parent +parton’s virtuality Q2, in the medium-modified splitting +function: +˜Pa(y, Q2) = P vac +a +(y) +× +� +� +� +� +� +1 + +τ + +form +� +0 +dξ+ˆqa +HTL +ca +ˆqf(Q2) +� +2 − 2 cos +� +ξ+ +τ + +form +�� +y(1 − y)Q2(1 + χa)2 +� +� +� +� +� +. +(1) +In the equation above, P vac +a +(y) is the Altarelli-Parisi +vacuum splitting function [160] for the parent par- +ton species a = (g, q, ¯q) with the forward light-cone + +5 +momentum fraction of the daughter parton y, χa = +(δaq + δa¯q)y2m2 +a/[y(1 − y)Q2 − y2m2 +a] with ma being +the parent parton mass, and ca +ˆq = +� +1 − y +2 (δa,q + δa,¯q) +� +− +χa +� +1 − +� +1 − y +2 +� +χa +� +. The integration in Eq. (1) is taken +over light-cone time ξ+ with the upper bound τ + +form = +2p+/Q2 being the formation time of the radiated par- +ton, where p+ = pµˆnµ/ +√ +2 [with ˆnµ = (1, p/|p|)] is +the forward light-cone momentum of the parent parton. +The formulation of ˜Pa(y, Q2) in Eq. (1) is obtained us- +ing soft collinear effective theory within the higher twist +scheme [161, 162]. +The parameterization of the virtuality-dependent mod- +ulation factor is given as [1] +f(Q2) = +� 1+10 ln2(Q2 +sw)+100 ln4(Q2 +sw) +1+10 ln2(Q2)+100 ln4(Q2) +if Q2 > Q2 +sw +1 +if Q2 ≤ Q2 +sw +. (2) +When this explicit virtuality dependence is eliminated, +the strength of the medium effect is controlled solely by +the conventional transport coefficient for a low virtuality +(near on shell) parton from the hard-thermal-loop (HTL) +calculation [90], +ˆqa +HTL = Ca +42ζ(3) +π +αrun +s +(p0T)αfix +s T 3 ln +�2p0T +m2 +D +� +. +(3) +Here, Ca is the Casimir color factor for the hard parent +parton, ζ(3) ≈ 1.20205 is Ap´ery’s constant, p0 is the en- +ergy of the hard parent parton, T is the temperature at +its location, and m2 +D = 4παfix +s T 2 +3 +� +Nc + Nf +2 +� +is the Debye +screening mass for a QCD plasma with Nc = 3 colors and +Nf = 3 fermion flavors. The coupling strength αrun +s +(p0T) +is evaluated at the scale µ2 = p0T via the running cou- +pling constant, +αrun +s +(µ2) = +� +4π +11−2Nf /3 +1 +ln(µ2/Λ2) +if µ2 > µ2 +0 +αfix +s +if µ2 ≤ µ2 +0 +, +(4) +with Λ being chosen such that αrun +s +(µ2 +0) = αfix +s +at µ2 +0 = +1 GeV2. In this framework, αfix +s +is the free parameter +controlling the overall interaction strength and chosen to +give the best fit to the experimental data of inclusive jet +RAA [1]. +In this paper, we compare results from two different +setups: with and without the virtuality-dependent coher- +ence effects (referred to as Type-3 and Type-2 in Ref. [1], +respectively). For the case with coherence, ˜Pa(y, Q2) in +Eq. (1), with the virtuality-dependent modulation factor +from Eq. (2), is employed in the high virtuality phase +by MATTER, with αfix +s += 0.3.2 +In the setup without +2 This configuration for MATTER+LBT with coherence effects +is referred to as JETSACPEv3.5 AA22 tune, and its results are +provided as defaults for comparisons with experimental and other +data. +coherence effects, the modulation factor is fixed to unity +[f(Q2) = 1] for any Q2 to eliminate the explicit virtual- +ity dependence. The best fit with leading hadron and jet +data is obtained with an αfix +s += 0.25 for this case. We +will present results for jet substructure using events gen- +erated with the above parametrizations, both with and +without coherence effects. +III. +RESULTS +In this section, we present the results for jet sub- +structure observables in Pb+Pb collisions at √sNN = +5.02 TeV based on the multi-stage (MATTER+LBT) jet +quenching model described in the previous section. +A +complementary study of the nuclear modification factor +RAA for reconstructed jets and charged particles using +the same model has been presented in Ref. [1]. Moreover, +this same formalism has been applied to study the heavy- +flavor observables and has been presented in Ref. [124]. +To show the capability of the JETSCAPE framework, +we also provide predictions of the groomed jet observ- +ables, fragmentation function, and jet cone size depen- +dence of inclusive jets and charged jets for the upcom- +ing jet measurements at RHIC. Throughout this work, +the jet reconstruction and Soft Drop grooming are per- +formed using the FastJet package [163, 164] with FastJet +Contrib [165]. +A. +Groomed jet observables +In this subsection, we present the observables ob- +tained via Soft Drop grooming algorithm [166–168]. The +Soft Drop procedure removes the contributions from soft +wide-angle radiation and enables access to the hard par- +ton splittings during the jet evolution. In this algorithm, +first, jets are constructed by a standard jet finding algo- +rithm such as the anti-kt algorithm [169] with a definite +jet cone size R. Then, the constituents of an anti-kt jet +are again reclustered by the Cambridge-Aachen (C/A) +algorithm [170, 171] to form a pairwise clustering tree. +The next step is to trace back the C/A tree. Here, one +declusters the C/A jet by undoing the last step of the +C/A clustering and selecting the resulting two prongs. +The two prongs are checked to see if they satisfy the Soft +Drop condition, given as: +min (pT,1, pT,2) +pT,1 + pT,2 +> zcut +�∆R12 +R +�β +, +(5) +where pT,1 and pT,2 are the transverse momenta of the +prongs, ∆R12 = +� +(η1 − η2)2 + (φ1 − φ2)2 is the radial +distance between the prongs in the rapidity-azimuthal +angle plane, zcut and β are parameters controlling the +grooming procedure. If the condition is failed, the prong +with the larger pT of the pair is further declustered into +a pair of prongs. This process is repeated until one finds + +6 +a pair of prongs satisfying the Soft Drop condition. The +resulting pair of prongs are used to compute the groomed +jet observables. It is worth noting that there may exist +cases in which no prong pair passing the soft-drop condi- +tion is eventually found even if the C/A tree is traversed +back to the end; such cases are referred to as “Soft Drop +fail”. +1. +Jet splitting momentum fraction +Here we study the medium modification of the jet split- +ting momentum fraction zg, which is defined as the left- +hand side of Eq. (5) in the case with the prong pair pass- +ing the Soft Drop condition. +Figure 1 shows zg distributions for charged jets in p+p +collisions at √s = 5.02 TeV defined as +1 +σjet +dσSD,jet +dzg += +1 +Njet +dNSD,jet +dzg +, +(6) +where Njet is the number of inclusive jets, NSD,jet is the +number of jets passing the Soft Drop condition and σjet, +σSD,jet are the corresponding cross sections. +The Soft +Drop parameters are set as zcut = 0.2 and β = 0. The re- +sults from the JETSCAPE PP19 tune for different pch,jet +T +ranges and jet cone sizes are compared with the experi- +mental data from ALICE. Some small discrepancies can +be seen, but they are mostly compatible within uncer- +tainty. +In Fig. 2, the modification of the zg distribution for +charged jets is presented as the ratio of the distribution +in Pb+Pb to p+p collisions at √sNN = 5.02 TeV. Both +results, with and without consideration of coherence ef- +fects, do not exhibit significant modification and are con- +sistent with the experimental data. This indicates that +the medium effects on the functional form for the mo- +mentum fraction y of the splitting function are small in +hard partonic splittings. To be clear, the entire ensemble +of jets in Pb+Pb that are included in this analysis is in- +deed modified by the medium. Looking at these results +and the experimental data, one could imagine two pos- +sibilities: (i) The sample of jets that pass the soft drop +condition is biased towards jets that are unmodified, and +(ii) the jets are modified, but this modification does not +affect the momentum fraction distribution of the prongs +produced in the hardest split. +In the subsequent sub- +section on the angle between the prongs, we will demon- +strate that it is indeed the latter of the two possibilities. +This indicates that most of the modification of the jet +may take place at softer momenta, i.e., the hardest split +is not affected by the medium at all. +Next, for upcoming measurements at RHIC, we present +the prediction of the modification of the zg distribution +for charged jets in 0-10% Au+Au collisions at √sNN = +200 GeV from MATTER+LBT with coherence effects in +Fig. 3. The trend is similar to the results observed at the +LHC collision energy and does not show any significant +nuclear effects for the kinematic configurations consid- +ered. +2. +Jet splitting radius +Next, we study the medium modification of jet split- +ting radius rg, which is defined as the radial distance +∆R12 of the prong pair passing the Soft Drop condition. +In Fig. 4, rg distributions defined as +1 +σjet +dσSD,jet +d (rg/R) = +1 +Njet +dNSD,jet +d (rg/R), +(7) +are shown for charged jets in p+p collisions at √s = +5.02 TeV. The results from the JETSCAPE PP19 tune +show good agreement with the ALICE data, particularly +for the cases with zcut = 0.2. +Figure 5 shows the modification of rg distribution for +charged jets in Pb+Pb collisions at √sNN = 5.02 TeV. +Our full results with coherence effects capture the trend +observed in experimental data: Enhancement at small +rg and suppression at large rg. In particular, the agree- +ments within uncertainties can be seen for the case with +zcut = 0.2. For the 0–10% most central bin, the result +without coherence effects is shown for comparison. +It +gives a slightly smaller slope, but no conclusion can be +drawn within the current uncertainties. Combined with +the results for the zg distribution, we obtain the clear con- +clusion that these jets passing the Soft Drop condition +are indeed modified, but predominantly in their softer +components rather than in the hard partonic splittings. +For jets originally having a larger hard-splitting angle, +the soft component diffusing due to the medium effect is +more likely to leave the jet cone, resulting in more consid- +erable energy loss. Thus, jets with larger hard splitting +angles are less likely to be triggered, and the narrowing +is observed as the yield ratio of jets with smaller splitting +angles increases. +Motivated by the recent analysis by ATLAS [58], we +also calculated the nuclear modification factor RAA for +full jets with different rg. Figures 6 and 7 show the RAA +results as a function of pjet +T +and rg, respectively. Here, +RAA is defined as +RAA = +1 +⟨Ncoll⟩ +d2NSD,jet +drgdpjet +T +��� +AA +d2NSD,jet +drgdpjet +T +��� +pp +, +(8) +for jets passing the Soft Drop condition with a finite value +of rg, and +RAA = +1 +⟨Ncoll⟩ +dN +incl/rg=0 +jet +dpjet +T +���� +AA +dN +incl/rg=0 +jet +dpjet +T +���� +pp +, +(9) +for inclusive jets and jets failing the Soft Drop condition +(rg = 0), where N incl/rg=0 +jet +is the number of triggered jets + +7 +0 +2 +4 +6 +8 +1 +σjet +dσSD,jet +dzg +pp, √s = 5.02 TeV +Charged Jets, anti-kt +Soft Drop zcut = 0.2, β = 0 +R=0.2, |ηch,jet|<0.7 +60 0.2, the prong struc- +ture as the transverse scale of the split exceeds µ⊥ ⪆ +158 GeV × 0.2 ≈ 32 GeV can be completely dominated +by the virtuality acquired by a parent parton at its pro- +duction in the initial hard scattering. This is because, +in this region, the initial virtuality is quite large, and +furthermore, the formation time for the splitting is very +short: τform ⪅ 2·(158 GeV) +(32 GeV)2 ≈ 0.3 GeV−1 ≈ 0.06 fm. Thus, +even without the interaction reduction due to coherence, +no amount of scattering from the medium has much of +an effect on the hard splitting. As a result, the RAA as a +function of pjet +T for the case of 0.2642 < rg < 0.4 shows no +difference between the cases with and without coherence, +as shown in the bottom panel of Fig. 6. This is also the +case for rg ⪆ 0.2 in all the plots of Fig. 7. +We finally address the region with 0.022 < rg < +0.26. Perturbative QCD should be applicable in this re- +gion. +Calculations without coherence effects include a +ˆq · f(Q2) = ˆq that has a large value (growing with the +logarithm of the energy) even in the high virtuality MAT- +TER stage, given by Eq. (3). One notes in Fig. 7, for the +case of the dashed green line (without coherence effects), +that multiple scattering broadens the prong structure. +This creates a depletion at lower rg and an enhancement +around 0.02 ⪅ rg ⪅ 0.06, which eventually begins to dis- +appear at large rg ⪆ 0.1. The broadening can be roughly +estimated using the simple formula that +k2 +⊥ ≊ z(1 − z) +� +2Eˆq ≈ +� +Eˆq/8. +(10) +This yields the simple expression for the peak angle of +the bump of the dashed green line as, +θmax ≊ k⊥ +E ≈ (ˆq/8)1/4 +E3/4 +. +(11) +Using the above equation, one would obtain that if the +energy of the jet were to double, the angle of the bump in +the dashed green line in Fig. 7 would move down in rg by +a factor of 23/4 ≈ 1.6. One notes that this is indeed the +case in the 2nd and 4th panels of Fig. 7. The energy range +between these doubles and the position of the bump in +the green curve shift down in rg by about a factor of 1.5- +2. This different behavior, depending on the presence or +absence of coherence effects, is also evident when shown +as a function of pjet +T from intermediate ranges of rg, as in +Fig. 6. +The bump structure of the jet RAA as a function of rg, +which our results without coherence show, can also be +seen in the prediction results from the JetMed model by +Caucal et al. [102, 114, 173] and semi-analytical calcula- +tion with pT -broadening effect by Ringer et al. [76] for +the ATLAS measurements presented in Ref. [58]. How- +ever, the data from ATLAS exhibit an almost monotoni- +cally decreasing trend with no such clear bump structure +for all pjet +T +intervals, which rather agrees with our MAT- +TER+LBT results with coherence effects. This reveals +that the medium effect is strongly suppressed at high +virtuality, where hard partonic splitting passing the Soft +Drop condition is likely to occur. +Figure 8 presents our prediction for the modification +of rg distribution for charged jets in 0-10% Au+Au col- +lisions at √sNN = 200 GeV from MATTER+LBT with +coherence effects. Similar to the LHC case, one finds en- +hancement at small rg and slight suppression at large rg, +which is more pronounced for jets with larger transverse +momentum. +B. +Jet fragmentation function +We now turn to the last jet substructure observable: +the jet fragmentation function. Jet fragmentation func- +tions are measured as a function of the track-particle +transverse momentum ptrk +T +or longitudinal momentum +fraction relative to the jet, +z = ptrk +T cos(∆r) +pjet +T +, +(12) +where ∆r = +� +(ηtrk − ηjet)2 + (φtrk − φjet)2. The frag- +mentation functions are defined as +D(z) = +1 +Njet +dNtrk +dz +, +(13) +D(ptrk +T ) = +1 +Njet +dNtrk +dptrk +T +, +(14) +where Njet is the number of triggered jets and Ntrk is +the number of charged track particles detected inside the +jet cones, ∆r < R. Our JETSCAPE PP19 results for +the fragmentation functions are compared with the ex- +perimental data by ATLAS in Fig. 9. For all available +pjet +T ranges, the discrepancies from the data are generally +within 20% at most. + +11 +10−2 +10−1 +rg +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +RAA +158 < pjet +T < 1000 GeV +PbPb, 0-10%, √sNN = 5.02 TeV +anti-kt, R = 0.4, |yjet| < 2.1 +10−2 +10−1 +rg +158 < pjet +T < 200 GeV +Soft Drop, zcut = 0.2, β = 0.0 +10−2 +10−1 +rg +200 < pjet +T < 316 GeV +JETSCAPE +[MATTER+LBT (w/ coherence)] +JETSCAPE +[MATTER+LBT (w/o coherence)] +10−2 +10−1 +rg +316 < pjet +T < 501 GeV +ATLAS +[arXiv:2211.11470] +FIG. 7. (Color online) Nuclear modification factor RAA as a function of rg for jets with different pjet +T +in 0-10% Pb+Pb collisions +at √sNN = 5.02 TeV. Jets are reconstructed with R = 0.4 at midrapidity |yjet| < 2.1. The Soft Drop parameters are zcut = 0.2 +and β = 0. The solid and dashed lines with statistical error bars show the results from MATTER+LBT of JETSCAPE with +and without coherence effects, respectively. For comparison, the experimental data from the ATLAS Collaboration [58] are +shown by squares with statistical errors (bars) and systematic uncertainties (bands). The yellow-shaded regions are the bin +areas including the regime where the perturbation approach does not apply (see text for details). +0.00 +0.05 +0.10 +0.15 +rg +0.0 +0.5 +1.0 +1.5 +2.0 +AuAu +pp +� +1 +σjet +dσSD,jet +d(rg/R) +� +AuAu 0-10%, √sNN = 200 GeV +Charged Jets, anti-kt +Soft Drop zcut = 0.2, β = 0 +R=0.2, |ηch,jet|<0.7 +0.0 +0.1 +0.2 +0.3 +0.4 +rg +R=0.4, |ηch,jet|<0.5 +JETSCAPE +MATTER+LBT (w/ coherence) +10 < pch,jet +T +< 30 GeV +30 < pch,jet +T +< 50 GeV +FIG. 8. +(Color online) Ratios of rg distributions for charged jets with R = 0.2 and |ηch,jet| < 0.7 (left), and R = 0.4, +|ηch,jet| < 0.5 (right) between 0-10% Au+Au and p+p collisions at √sNN = 200 GeV, from MATTER+LBT simulations within +JETSCAPE, including coherence effects. The Soft Drop parameters are zcut = 0.2 and β = 0. The solid and dashed lines with +statistical error bars show the results for 10 < pch,jet +T +< 30 GeV and 30 < pch,jet +T +< 50 GeV, respectively. +In Fig. 10, we present the modification of the jet frag- +mentation functions for full jets in 0-10% Pb+Pb col- +lisions at √sNN = 5.02 TeV. Results from the MAT- +TER+LBT simulations, both with and without coher- +ence effects, are compared with the experimental data +from ATLAS. All the simulation results and the data +show qualitatively the same trends. While the track par- +ticles at intermediate z are suppressed by the interactions +with the medium and give the enhancement at small z, +the large-z part is enhanced due to the less affected hard +part of jets. +In jet fragmentation functions, coherence effects are +quantitatively visible as more prominent enhancements +in the large-z region dominated by hadrons from leading +partons of jets. Since the leading parton has the largest +virtuality at the early stage in the jet shower evolution, +the interaction reduction due to coherence affects this +parton the most. As a result, the modification of large-z +jet hadrons is further lessened, and the enhancement be- +comes more substantial than the case without coherence +effects. This is consistent with the weak energy loss of +inclusive charged particles at high pT explained by co- +herence effects presented in Refs. [1, 124]. +In conjunction with the behaviors in the high-z region, +a slight difference can also be seen in the low-z region be- +tween the two settings. Both results with and without +coherence effects show a sizable enhancement at low-z +mainly due to the medium response via recoils but still + +000000 +582000000 +58212 +10−2 +10−1 +100 +101 +102 +103 +D(z) +126< pjet +T <158 GeV +pp, √s = 5.02 TeV +anti-kt, R = 0.4 +|yjet| < 0.3, ptrk +T > 1 GeV +158< pjet +T <200 GeV +JETSCAPE +[MATTER (vacuum)] +200< pjet +T <251 GeV +ATLAS +[PRC 98, no.2, 024908 (2018)] +251< pjet +T <316 GeV +10−2 +10−1 +100 +z +0.5 +1.0 +1.5 +MC/Exp. +10−2 +10−1 +100 +z +10−2 +10−1 +100 +z +10−2 +10−1 +100 +z +10−4 +10−2 +100 +D(ptrk +T ) +126< pjet +T <158 GeV +pp, √s = 5.02 TeV +anti-kt, R = 0.4 +|yjet| < 0.3, ptrk +T > 1 GeV +158< pjet +T <200 GeV +JETSCAPE +[MATTER (vacuum)] +200< pjet +T <251 GeV +ATLAS +[PRC 98, no.2, 024908 (2018)] +251< pjet +T <316 GeV +100 +101 +102 +ptrk +T (GeV) +0.5 +1.0 +1.5 +MC/Exp. +100 +101 +102 +ptrk +T (GeV) +100 +101 +102 +ptrk +T (GeV) +100 +101 +102 +ptrk +T (GeV) +FIG. 9. (Color online) Jet fragmentation functions for jets in p+p collisions at √s = 5.02 TeV and the ratios as a function of +z (top) and ptrk +T +(bottom) for different pjet +T +range. Jets are fully reconstructed including both charged and neutral particles by +anti-kt with R = 0.4 at midrapidity +��yjet�� < 0.3. The solid lines and circles with statistical error bars show the results from +JETSCAPE and the experimental data from the ATLAS Collaboration [50], respectively. The bands indicate the systematic +uncertainties of the experimental data. +underestimate the data. +One possible cause of this is +the visible discrepancy in the suppression at mid-z. Fur- +thermore, for some very soft components of jets giving +contribution in the low-z region, the recoil prescription +may not provide an entirely reasonable description once +their energies become close to the typical scale for the +medium constituents. More comprehensive momentum +structures of jet constituents, including such soft regions +where hydrodynamic medium response needs to be con- +sidered, will be explored in a future effort. +With the current uncertainties, it is not yet possible +to conclude the presence of coherence effects from com- +parisons with only the experimental data on modified +jet fragmentation functions. +However, when taken in +conjunction with the results on the rg distribution, a +stronger case can be made for the existence of coherence +effects at high virtuality. Our results also indicate that +the medium effects over different scales can be discernible +by future measurements with high precision. +In Fig. 11, we present our results of the modifica- +tion of jet fragmentation functions for charged jets in +0-10% Au+Au collisions at √sNN = 200 GeV from MAT- +TER+LBT with coherence effects. Compared to the re- +sults for the top LHC energy, the modifications are quite +small. +IV. +SUMMARY AND OUTLOOK +This paper explored the medium modification of jet +substructure in high-energy heavy-ion collisions, employ- +ing a multi-stage jet evolution model, MATTER+LBT, +with the configuration and parameters established within +the JETSCAPE framework by comparison with leading +hadron and jet data. All parameters were taken from our +previous efforts [1] and were not re-tuned for this study. + +000000 +582000000 +58213 +10−2 +10−1 +100 +z +0.5 +1.0 +1.5 +2.0 +2.5 +PbPb +pp +[D(z)] +126< pjet +T <158 GeV +PbPb 0-10%, √sNN = 5.02 TeV +anti-kt, R = 0.4 +|yjet| < 0.3, ptrk +T > 1 GeV +10−2 +10−1 +100 +z +158< pjet +T <200 GeV +JETSCAPE +[MATTER+LBT (w/ coherence)] +JETSCAPE +[MATTER+LBT (w/o coherence)] +10−2 +10−1 +100 +z +200< pjet +T <251 GeV +ATLAS +[PRC 98, no.2, 024908 (2018)] +10−2 +10−1 +100 +z +251< pjet +T <316 GeV +100 +101 +102 +ptrk +T (GeV) +0.5 +1.0 +1.5 +2.0 +2.5 +PbPb +pp +� +D(ptrk +T ) +� +126< pjet +T <158 GeV +PbPb 0-10%, √sNN = 5.02 TeV +anti-kt, R = 0.4 +|yjet| < 0.3, ptrk +T > 1 GeV +100 +101 +102 +ptrk +T (GeV) +158< pjet +T <200 GeV +JETSCAPE +[MATTER+LBT (w/ coherence)] +JETSCAPE +[MATTER+LBT (w/o coherence)] +100 +101 +102 +ptrk +T (GeV) +200< pjet +T <251 GeV +ATLAS +[PRC 98, no.2, 024908 (2018)] +100 +101 +102 +ptrk +T (GeV) +251< pjet +T <316 GeV +FIG. 10. (Color online) Ratios of jet fragmentation functions for jets between 0-10% Pb+Pb and p+p collisions at √sNN = +5.02 TeV as a function of z (top) and ptrk +T +(bottom) for different pjet +T +range. Jets are fully reconstructed, including both charged +and neutral particles by anti-kt with R = 0.4 at midrapidity +��yjet�� < 0.3. The solid and dashed with statistical error bars +lines show the results from MATTER+LBT of JETSCAPE with and without coherence effects, respectively. For comparison, +the experimental data from the ATLAS Collaboration [50] are shown by squares with statistical errors (bars) and systematic +uncertainties (bands). +10−1 +100 +z +0.5 +1.0 +1.5 +2.0 +2.5 +AuAu +pp +AuAu 0-10% +√sNN = 200 GeV +anti-kt, R = 0.4 +|ηch,jet| < 1, ptrk +T > 1 GeV +D(z) +101 +ptrk +T (GeV) +D(ptrk +T ) +JETSCAPE +MATTER+LBT (w/ coherence) +105 GeV, |ηch,jet|<1−R +R = 0.2 +R = 0.4 +R = 0.6 +FIG. 12. +(Color online) Nuclear modification factor RAA for inclusive full jet with |ηjet| < 1 (left), and charged jet with +|ηch,jet| < 1 − R and leading charged particle pch,lead +T +> 5 GeV (right) in 0 − 10% Au+Au collisions at √sNN = 200 GeV from +MATTER+LBT of JETSCAPE with coherence effects. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Zhao3 (The JETSCAPE Collaboration) 1Akita International University, Yuwa, Akita-city 010-1292, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 2Department of Physics, McGill University, Montr´eal QC H3A 2T8, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 3Department of Physics and Astronomy, Wayne State University, Detroit MI 48201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 4Lawrence Livermore National Laboratory, Livermore CA 94550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 5Research Computing Group, University Technology Solutions, The University of Texas at San Antonio, San Antonio TX 78249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 6Department of Physics, Duke University, Durham, NC 27708, USA 7Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, China 8Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge MA 02139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 9Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 10Department of Physics, Duke University, Durham NC 27708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 11Department of Physics and Astronomy, University of Tennessee, Knoxville TN 37996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 12Physics Division, Oak Ridge National Laboratory, Oak Ridge TN 37830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 13GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 14Institute for Theoretical Physics, Goethe University, 60438 Frankfurt am Main, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 15Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 16Cyclotron Institute, Texas A&M University, College Station TX 77843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 17Department of Physics and Astronomy, Texas A&M University, College Station TX 77843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 18Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 19Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 20Department of Physics, The Ohio State University, Columbus OH 43210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 21Department of Physics, University of California, Berkeley CA 94270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 22Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley CA 94270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 23Department of Statistical Science, Duke University, Durham NC 27708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 24Department of Physics, Brookhaven National Laboratory, Upton NY 11973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 25Department of Physics and Astronomy, Vanderbilt University, Nashville TN 37235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 26Los Alamos National Laboratory, Theoretical Division, Los Alamos, NM 87545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 27Department of Computer Science, Wayne State University, Detroit MI 48202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 28Instituto de F`ısica, Universidade de S˜ao Paulo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 66318, 05315-970 S˜ao Paulo, SP, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 29Institute for Nuclear Theory, University of Washington, Seattle WA, 98195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 30Physics Department, Brookhaven National Laboratory, Upton NY 11973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 31RIKEN BNL Research Center, Brookhaven National Laboratory, Upton NY 11973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 32Department of Physics, Kent State University, Kent, OH 44242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 33Department of Physics, University of Regina, Regina, SK S4S 0A2, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 34Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' We present predictions and postdictions for a wide variety of hard jet-substructure observables using a multi-stage model within the JETSCAPE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The details of the multi-stage model and the various parameter choices are described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A novel feature of this model is the presence of two stages of jet modification: a high virtuality phase (modeled using MATTER), where coherence effects diminish medium-induced radiation, and a lower virtuality phase (modeled using LBT), where parton splits are fully resolved by the medium as they endure multiple scattering in- duced energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Energy loss calculations are carried out on event-by-event viscous fluid dynamic backgrounds constrained by experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The uniformed and consistent descriptions of multi- ple experimental observables demonstrate the essential role of coherence effects and the multi-stage modeling of the jet evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Using the best choice of parameters from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1], and with no further tuning, we present calculations for the medium modified jet fragmentation function, the groomed arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02485v1 [hep-ph] 6 Jan 2023 2 jet momentum fraction zg and angular separation rg distributions, as well as the nuclear modifi- cation factor of groomed jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' These calculations provide accurate descriptions of published and preliminary data from experiments at RHIC and LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Furthermore, we provide predictions from the multi-stage model for future measurements at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' INTRODUCTION In high-energy heavy-ion collisions, the high-transverse momentum (pT ) partons (pT ⪆ 10 GeV) are generated almost at the instant at which the incoming nuclei over- lap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Such high pT partons are generated in parton-parton exchanges with large momentum transfers Q ≫ ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' They are typically produced far from their mass shell and engender multiple collinear emissions produced over a large time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the case of a heavy-ion collision, the propagation and development of these parton show- ers are strongly affected by the produced Quark Gluon Plasma (QGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Studying jet modification in nucleus- nucleus collisions relative to proton-proton collisions, to- gether with constraints from model-to-data comparison provides unique opportunities to probe the properties of the QGP [2–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The experimental attempts started at the Relativis- tic Heavy Ion Collider (RHIC) with the observation of suppression in the yield of single inclusive hadrons [21– 25] and associated hadrons (dihadrons) [26–28] produced with high transverse momentum relative to the yield in proton-proton collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Since 2010, starting at the Large Hadron Collider (LHC) and later at RHIC, the ability of experiments evolved from single hadrons and dihadrons to jets [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Over the last decade, experiments have attained the ability to not just study the energy-momentum and cross section of a jet but also to look at modifications of the internal properties of the jet, often referred to as jet substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Based on current detector improvements and accumulated high statistics data at RHIC and the LHC, it is possible to analyze a vast variety of observ- ables revealing different aspects of the jet-medium in- teraction [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For example, the yield suppression and internal structure of fully reconstructed jets, revealed in observables such as the jet fragmentation function and jet shape (respectively), provide details on the diffusion of jet energy and momentum in momentum or angular space due to the interaction with the medium [29–31, 33– 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Even the structural modification of hard partonic branching is now potentially accessible through groomed jet observables [53–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' On the theory side, many studies have attempted to describe and understand the jet-medium interaction by constructing models that reproduce these various observ- ables or propose predictions and new observables [59–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In particular, to obtain a universal understanding, it is ∗ Corresponding author: ytachibana@aiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='jp † Corresponding author: amit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='kumar3@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='ca essential to simultaneously explain multiple observables, ultimately all observables, with a consistent theoretical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Therefore, Monte Carlo calculations, which can generate experiment-like events by a single model, are a powerful tool for theoretical approaches because they enable one to calculate a wide range of event-by-event defined jet observables [80–113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Jets evolve dynamically, moving through the expand- ing medium, and generating more partons from splits and interactions with the dense medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The original partons start at very high virtuality, and thus, the early splits have a small transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' These splittings from the leading parton and the still highly-virtual daugh- ters are driven by their individual virtualities, with mi- nor medium correction via the scattering, strongly sup- pressed due to their small transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' We refer to these as Vacuum Like Emissions (VLE) [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' To simu- late the VLEs, taking into account the reduction in the effective interaction rate with scale dependence, an event generator such as MATTER [115, 116] can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' With repeated splittings, the virtuality of the partons reduces to the point that splits are widely separated in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' With decreasing virtuality, the transverse size of the parton becomes larger, thereby increasing the rate of interaction with the medium, which in turn triggers more radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Thus, the main mechanism causing par- ton splittings changes dynamically in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The evolution of such partons at lower virtuality but en- ergy still large enough to treat the medium interaction perturbatively can be approximated by kinetic theory- based approaches for on-shell particles, as implemented by generators such as LBT [90, 92, 96, 97], or MAR- TINI [84, 111, 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' As partons transition to energies and virtualities close to those of the QGP, they begin to undergo strong coupling [89] and thermalization with the medium [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Thus, jets interact with the medium over a wide range of scales, which requires incorporating mul- tiple generators at different scales for simulations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' JETSCAPE is a general-purpose framework for Monte Carlo simulations of the complete evolution in high- energy heavy-ion collisions [1, 118–124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The framework is designed to be as general and extensive as possible while modularizing each physics element involved in a collision event, such as the generation of geometric initial conditions, hydrodynamic evolution of the soft sector, jet production by hard scattering, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' so that users can em- ploy a module based on their favorite physical description for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For the in-medium parton shower evolution, the most distinctive feature of the JETSCAPE framework is its support for multi-stage descriptions that, by stitching multiple models together, cover a broader range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Depending on the virtuality or energy of a parton, each model becomes active to handle the parton shower evo- 3 lution interactions with the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Recently, we systematically studied the energy loss of large-transverse momentum particles, jets, and charmed particles using a multi-stage model, combining two mod- ules, MATTER for high-virtual parton shower and LBT for low virtuality, developed within the JETSCAPE framework in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Our simulations indicate that the single high-pT particle spectra are dominated by the large virtuality phase simulated by the MATTER module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' On the other hand, to describe the suppression of reconstructed jets and D mesons, we found that the energy loss of soft daughter partons and heavy quarks is governed by the low-virtuality scattering dominated phase simulated by the LBT module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' One further important insight from our prior work is that the reduction of the interaction with the medium at high virtuality due to coherence effects plays a crucial role in explaining the weak suppression of single charged particles with pT ⪆ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' These coherence effects oc- cur because the partons probing the medium have a small transverse size when the virtuality is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A section of QGP resolved at such a shorter distance scale appears more dilute, resulting in fewer interactions [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='1 Coherence effects implemented in MATTER drasti- cally improve the description of the transverse momen- tum dependence of the nuclear modification factor for inclusive single-charged particles, even at the qualitative functional behavior level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In contrast, for reconstructed jets at the currently available collision energies, coher- ence effects are not visible in the transverse momentum dependence of the nuclear modification factor, which only necessitated a readjustment of the overall medium cou- pling parameter αfix s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Thus, it is essential to search for the role of coherence effects in the evolution of jet show- ering patterns by examining further inner jet structure modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this paper, we systematically analyze the observ- ables characterizing the internal structure of jets using the results of the exact same numerical simulations with MATTER+LBT that were used to study the nuclear modification factors for reconstructed jets and high pT single-charged particles in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The goal is to explore the details of the interaction strength at each scale on the internal structure of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In particular, we exam- ine the groomed jet observables, which display the effect of jet-medium interactions at the early high-virtuality stage, and the jet fragmentation function, which shows the medium effect on partons throughout a wide range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this work, we do not re-tune any parameters and employ those obtained in our previous work [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' II, salient characteristics of the underlying model are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' II A, an overview of the framework and setup is 1 In several other models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=', those in Refs [102, 103], coherence effects are implicitly taken into account, without detailed formu- lations, by turning off the medium effect at high virtuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Subsection II B is devoted to formulating co- herence effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This is followed by an investigation of the medium modification of jet substructure observables, focusing on coherence effects, by presenting results from our model calculations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Here, we also make predictions for the upcoming measurements of the jet substructure observables at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A summary of our re- sults and concluding remarks are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The Appendix is dedicated to the presentation of our predictions of jet RAA at the top RHIC energy for bench- marking purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' MODEL JETSCAPE is a general-purpose event generator framework where different sub event generators can be included in a modular fashion, producing an extensive end-to-end simulation of a heavy-ion collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this paper, we will use the results of simulations that were generated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1] to calculate all jet substructure ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This is not just for convenience but rather to demonstrate how the exact same simulations can si- multaneously describe both the jet and leading hadron suppression, as well as several jet substructure observ- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' To that end, only a very brief overview of the compo- nents of the simulation will be provided in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The reader may refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1] for specific details of the physics included in a MATTER+LBT simulation within the JETSCAPE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Computational aspects of the JETSCAPE framework are described in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [118], while the basic physics of multi-stage sim- ulators is described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Overview To explore the medium modification of jet substruc- ture, we perform simulations of jet events in high-energy nucleus-nucleus collisions utilizing the full framework of JETSCAPE in two separate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' First, we calculate the event-by-event space-time profiles of the QGP medium in nucleus+nucleus (A+A) collisions for the estimation of the local medium effect on parton shower evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For this part, we perform simulations of (2+1)-D free- streaming pre-equilibrium evolution [127] and subsequent viscous hydrodynamic evolution by the (2+1)D VISHNU code package [128] with the initial condition generated by TRENTo [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Here the MAP parameters obtained by Bayesian calibration in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [130] are used for the LHC energy calculations, while hand-tuned parameters were used for top RHIC energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the second step, the binary collision distribution from the same TRENTo initial condition as for the medium is used to sample the transverse position of a hard scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The hard scattering is produced by Pythia 8 [131] with initial state radiation (ISR) and 4 multiparton interaction (MPI) turned on, and final state radiation (FSR) turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The produced partons in the hard scattering then undergo the multi-stage in-medium parton shower evolution within the JETSCAPE frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this study, we use a combination of MATTER and LBT modules as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The partons produced by hard scattering are first passed to the MATTER module, which simulates virtuality-ordered splitting of high-energy partons incor- porating medium effects [115, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This description by MATTER is valid for partons with virtuality sufficiently larger than the accumulated transverse momentum and virtuality generated by scattering from the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Partons whose virtuality is reduced by showering in MATTER are then transferred to LBT at a transition scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In LBT, the kinetic theory for on-shell partons with elastic and inelastic scatterings with medium con- stituents is applied [90, 92, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The parton split- tings under this description are entirely scattering-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the multi-stage approach of the JETSCAPE frame- work, virtuality-dependent switching between modules is done bi-directionally on a per-parton basis using a switching parameter Q2 sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' If the virtuality of the par- ton Q2 = pµpµ − m2 falls below Q2 sw, it is then sent from MATTER to LBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Conversely, the parton is re- turned to MATTER if its virtuality exceeds Q2 sw again, or it goes out of the dense medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The transition from medium-like back to vacuum-like emission takes place at a boundary with a temperature Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='16 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this study, Q2 sw is set to 4 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' After all the partons are out- side the QGP medium and have virtuality smaller than the cut-off scale Q2 min = 1 GeV2, they are hadronized via the Colorless Hadronization module, in which the Lund string model of Pythia 8 is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In both MATTER and LBT modules, the medium response effect is taken into account via recoil par- tons [85, 88, 97, 98, 117, 133, 134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the recoil pre- scription, the energy-momentum transfer is described by scatterings between jet partons and medium partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For each scattering, a parton is sampled from the thermal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Then, the scattered sampled parton is assumed to be on-shell, and passed to LBT for its in-medium evo- lution, assuming weak coupling with the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' These recoil partons and further accompanying daughter par- tons are collectively hadronized with the other jet shower partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' On the other hand, a deficit of energy and mo- mentum in the medium is left for each recoil process, where a parton emanating from the medium is included, post scattering, as a part of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' We treat this deficit as a freestreaming particle, referred to as a hole parton, and track it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The hole partons are hadronized separately from other jet partons, and their energy and momentum within each positive particle jet cone are subtracted in the jet clustering routine to ensure energy-momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the later stages of evolution, where the energy of a jet shower parton reaches a comparable scale to the ambient temperature, the mean free path is no longer large enough to apply the kinetic theory-based approach with the recoil prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In principle, such soft components of jets are supposed to be thermalized and evolve hydrodynamically as part of the bulk medium fluid [20, 135–140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' As in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [141–151], implementa- tion of models based on such a description is proposed, and there are some studies of the hydrodynamic medium response to jets using it [72, 74, 95, 112, 117, 152–156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' However, with such an implementation of the hydrody- namic medium response, the computational cost for a systematic and exhaustive study covering various config- urations, as presented in this paper, is enormously expen- sive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Thus, in this paper, we mainly discuss the structure of the hard part of the jet, where the contributions of such very soft components are relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A further com- prehensive investigation with more detailed modeling of the medium response in jet modification is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' To investigate the modification of jet substructures by medium effects in A+A collisions, the calculations of the same observables for p+p collisions are necessary as ref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For such calculations, the parton shower evolu- tion modules are replaced entirely by MATTER with no in-medium scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This setup for p+p collisions of JETSCAPE, referred to as the JETSCAPE PP19 tune, is equivalent to the limit of no medium effect in the event and is detailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Coherence Effects at High Virtuality In this study, we focus on coherence effects [114, 125, 157–159] on the interaction of a highly virtual parton with the medium and explore their manifestation in jet substructure modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [125], it was demon- strated that a hard parton with large virtuality resolves the very short-distance structure of the medium via the exchange of a gluon whose momentum is much larger than the medium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' These coherence ef- fects are formulated with the continuous evolution of the medium-resolution scale and give a gradual reduction of jet parton-medium interaction as a function of the virtu- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For jet quenching calculations, coherence effects can be effectively implemented by introducing a modulation factor f(Q2), which diminishes as a function of the parent parton’s virtuality Q2, in the medium-modified splitting function: ˜Pa(y, Q2) = P vac a (y) × � � � � � 1 + τ + form � 0 dξ+ˆqa HTL ca ˆqf(Q2) � 2 − 2 cos � ξ+ τ + form �� y(1 − y)Q2(1 + χa)2 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (1) In the equation above, P vac a (y) is the Altarelli-Parisi vacuum splitting function [160] for the parent par- ton species a = (g, q, ¯q) with the forward light-cone 5 momentum fraction of the daughter parton y, χa = (δaq + δa¯q)y2m2 a/[y(1 − y)Q2 − y2m2 a] with ma being the parent parton mass, and ca ˆq = � 1 − y 2 (δa,q + δa,¯q) � − χa � 1 − � 1 − y 2 � χa � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (1) is taken over light-cone time ξ+ with the upper bound τ + form = 2p+/Q2 being the formation time of the radiated par- ton, where p+ = pµˆnµ/ √ 2 [with ˆnµ = (1, p/|p|)] is the forward light-cone momentum of the parent parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The formulation of ˜Pa(y, Q2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (1) is obtained us- ing soft collinear effective theory within the higher twist scheme [161, 162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The parameterization of the virtuality-dependent mod- ulation factor is given as [1] f(Q2) = � 1+10 ln2(Q2 sw)+100 ln4(Q2 sw) 1+10 ln2(Q2)+100 ln4(Q2) if Q2 > Q2 sw 1 if Q2 ≤ Q2 sw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (2) When this explicit virtuality dependence is eliminated, the strength of the medium effect is controlled solely by the conventional transport coefficient for a low virtuality (near on shell) parton from the hard-thermal-loop (HTL) calculation [90], ˆqa HTL = Ca 42ζ(3) π αrun s (p0T)αfix s T 3 ln �2p0T m2 D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (3) Here, Ca is the Casimir color factor for the hard parent parton, ζ(3) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='20205 is Ap´ery’s constant, p0 is the en- ergy of the hard parent parton, T is the temperature at its location, and m2 D = 4παfix s T 2 3 � Nc + Nf 2 � is the Debye screening mass for a QCD plasma with Nc = 3 colors and Nf = 3 fermion flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The coupling strength αrun s (p0T) is evaluated at the scale µ2 = p0T via the running cou- pling constant, αrun s (µ2) = � 4π 11−2Nf /3 1 ln(µ2/Λ2) if µ2 > µ2 0 αfix s if µ2 ≤ µ2 0 , (4) with Λ being chosen such that αrun s (µ2 0) = αfix s at µ2 0 = 1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this framework, αfix s is the free parameter controlling the overall interaction strength and chosen to give the best fit to the experimental data of inclusive jet RAA [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this paper, we compare results from two different setups: with and without the virtuality-dependent coher- ence effects (referred to as Type-3 and Type-2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For the case with coherence, ˜Pa(y, Q2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (1), with the virtuality-dependent modulation factor from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (2), is employed in the high virtuality phase by MATTER, with αfix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2 In the setup without 2 This configuration for MATTER+LBT with coherence effects is referred to as JETSACPEv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='5 AA22 tune, and its results are provided as defaults for comparisons with experimental and other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' coherence effects, the modulation factor is fixed to unity [f(Q2) = 1] for any Q2 to eliminate the explicit virtual- ity dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The best fit with leading hadron and jet data is obtained with an αfix s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='25 for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' We will present results for jet substructure using events gen- erated with the above parametrizations, both with and without coherence effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' RESULTS In this section, we present the results for jet sub- structure observables in Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV based on the multi-stage (MATTER+LBT) jet quenching model described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A complementary study of the nuclear modification factor RAA for reconstructed jets and charged particles using the same model has been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Moreover, this same formalism has been applied to study the heavy- flavor observables and has been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' To show the capability of the JETSCAPE framework, we also provide predictions of the groomed jet observ- ables, fragmentation function, and jet cone size depen- dence of inclusive jets and charged jets for the upcom- ing jet measurements at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Throughout this work, the jet reconstruction and Soft Drop grooming are per- formed using the FastJet package [163, 164] with FastJet Contrib [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Groomed jet observables In this subsection, we present the observables ob- tained via Soft Drop grooming algorithm [166–168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The Soft Drop procedure removes the contributions from soft wide-angle radiation and enables access to the hard par- ton splittings during the jet evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In this algorithm, first, jets are constructed by a standard jet finding algo- rithm such as the anti-kt algorithm [169] with a definite jet cone size R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Then, the constituents of an anti-kt jet are again reclustered by the Cambridge-Aachen (C/A) algorithm [170, 171] to form a pairwise clustering tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The next step is to trace back the C/A tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Here, one declusters the C/A jet by undoing the last step of the C/A clustering and selecting the resulting two prongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The two prongs are checked to see if they satisfy the Soft Drop condition, given as: min (pT,1, pT,2) pT,1 + pT,2 > zcut �∆R12 R �β , (5) where pT,1 and pT,2 are the transverse momenta of the prongs, ∆R12 = � (η1 − η2)2 + (φ1 − φ2)2 is the radial distance between the prongs in the rapidity-azimuthal angle plane, zcut and β are parameters controlling the grooming procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' If the condition is failed, the prong with the larger pT of the pair is further declustered into a pair of prongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This process is repeated until one finds 6 a pair of prongs satisfying the Soft Drop condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The resulting pair of prongs are used to compute the groomed jet observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' It is worth noting that there may exist cases in which no prong pair passing the soft-drop condi- tion is eventually found even if the C/A tree is traversed back to the end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' such cases are referred to as “Soft Drop fail”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Jet splitting momentum fraction Here we study the medium modification of the jet split- ting momentum fraction zg, which is defined as the left- hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' (5) in the case with the prong pair pass- ing the Soft Drop condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Figure 1 shows zg distributions for charged jets in p+p collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV defined as 1 σjet dσSD,jet dzg = 1 Njet dNSD,jet dzg , (6) where Njet is the number of inclusive jets, NSD,jet is the number of jets passing the Soft Drop condition and σjet, σSD,jet are the corresponding cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The Soft Drop parameters are set as zcut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The re- sults from the JETSCAPE PP19 tune for different pch,jet T ranges and jet cone sizes are compared with the experi- mental data from ALICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Some small discrepancies can be seen, but they are mostly compatible within uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 2, the modification of the zg distribution for charged jets is presented as the ratio of the distribution in Pb+Pb to p+p collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Both results, with and without consideration of coherence ef- fects, do not exhibit significant modification and are con- sistent with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This indicates that the medium effects on the functional form for the mo- mentum fraction y of the splitting function are small in hard partonic splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' To be clear, the entire ensemble of jets in Pb+Pb that are included in this analysis is in- deed modified by the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Looking at these results and the experimental data, one could imagine two pos- sibilities: (i) The sample of jets that pass the soft drop condition is biased towards jets that are unmodified, and (ii) the jets are modified, but this modification does not affect the momentum fraction distribution of the prongs produced in the hardest split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In the subsequent sub- section on the angle between the prongs, we will demon- strate that it is indeed the latter of the two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' This indicates that most of the modification of the jet may take place at softer momenta, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=', the hardest split is not affected by the medium at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Next, for upcoming measurements at RHIC, we present the prediction of the modification of the zg distribution for charged jets in 0-10% Au+Au collisions at √sNN = 200 GeV from MATTER+LBT with coherence effects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The trend is similar to the results observed at the LHC collision energy and does not show any significant nuclear effects for the kinematic configurations consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Jet splitting radius Next, we study the medium modification of jet split- ting radius rg, which is defined as the radial distance ∆R12 of the prong pair passing the Soft Drop condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' 4, rg distributions defined as 1 σjet dσSD,jet d (rg/R) = 1 Njet dNSD,jet d (rg/R), (7) are shown for charged jets in p+p collisions at √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' The results from the JETSCAPE PP19 tune show good agreement with the ALICE data, particularly for the cases with zcut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Figure 5 shows the modification of rg distribution for charged jets in Pb+Pb collisions at √sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Our full results with coherence effects capture the trend observed in experimental data: Enhancement at small rg and suppression at large rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' In particular, the agree- ments within uncertainties can be seen for the case with zcut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For the 0–10% most central bin, the result without coherence effects is shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' It gives a slightly smaller slope, but no conclusion can be drawn within the current uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Combined with the results for the zg distribution, we obtain the clear con- clusion that these jets passing the Soft Drop condition are indeed modified, but predominantly in their softer components rather than in the hard partonic splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' For jets originally having a larger hard-splitting angle, the soft component diffusing due to the medium effect is more likely to leave the jet cone, resulting in more consid- erable energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Thus, jets with larger hard splitting angles are less likely to be triggered, and the narrowing is observed as the yield ratio of jets with smaller splitting angles increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Motivated by the recent analysis by ATLAS [58], we also calculated the nuclear modification factor RAA for full jets with different rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Figures 6 and 7 show the RAA results as a function of pjet T and rg, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content=' Here, RAA is defined as RAA = 1 ⟨Ncoll⟩ d2NSD,jet drgdpjet T ��� AA d2NSD,jet drgdpjet T ��� pp , (8) for jets passing the Soft Drop condition with a finite value of rg, and RAA = 1 ⟨Ncoll⟩ dN incl/rg=0 jet dpjet T ���� AA dN incl/rg=0 jet dpjet T ���� pp , (9) for inclusive jets and jets failing the Soft Drop condition (rg = 0), where N incl/rg=0 jet is the number of triggered jets 7 0 2 4 6 8 1 σjet dσSD,jet dzg pp, √s = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='02 TeV Charged Jets, anti-kt Soft Drop zcut = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2, β = 0 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='2, |ηch,jet|<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE0T4oBgHgl3EQflQHp/content/2301.02485v1.pdf'} +page_content='7 60 0) below. Finally, we note that Ψ0(q, xc) +turns out to be not a good approximation to the vibra- +tional polariton ground state under VSC in the herein +discussed asymmetric transfer model, but leads to rich +dynamics from where also H–transfer rates can be deter- +mined. A further analysis of the initial state is given in +the SI, Sec. III. + +3000 +-1.0 +0.10 +-0.5 +0 +00 +0.08 +0.0 +oe/b +7500 +0009 +0.06 +1500 +0.5 +0.04 +1.0 +0.02 +1.5 +0.00 +-400 +-200 +0 +200 +400 +Xc/ymeao3000 +-1.0 +0.12 +-0.5 +0.10 +0.0 +lao +0.08 +7500 +0009 +1500 +4500 +0.5 +0.06 +0.04 +1.0 +0.02 +1.5 +0.00 +-400 +-200 +0 +200 +400 +Xc/ymeao4 +B. +Observables +We describe the time–evolution of the light–matter hy- +brid system by a time–dependent ensemble transfer prob- +ability from the enol form (the more stable configura- +tion of the free molecule or the molecule in the cavity at +η = 0.0) to the enethiol form, which we define as +P ens +SH (t) = +� +Ψ(t) +���� +1 +N +N +� +i=1 +θ(qi) +����Ψ(t) +� +. +(10) +Here, θ(qi), is a Heaviside step function indicating a di- +viding surface located at the transition state of the in- +dividual transfer potentials. Due to the bound nature +of the cPES, the transfer dynamics is subject to recross- +ing events at the dividing–surface, where we characterize +the first recurrence by a recurrence time, τr. The latter +allows us to introduce the notion of short–time dynam- +ics for times, t ≤ τr, and subsequently the extraction +of approximate short–time transfer rates from enol– to +enethiol–configurations as +kens +SH = d +dtP ens +SH (t) +���� +t=tmax +, +(11) +where tmax maximizes kens +SH for t0 < tmax < τr. Further, +time–dependent coordinate expectation values +⟨R⟩ (t) = ⟨Ψ(t)|R|Ψ(t)⟩ +, +R = q, xc +, +(12) +provide a complementary perspective on the dynamics. +In order to address cavity–induced collective quantum ef- +fects, we additionally study entanglement in the strongly +coupled light–matter hybrid system via von Neumann– +entropies +Si(t) = −kB tr{ˆρi(t) ln ˆρi(t)} ≥ 0 +, +(13) +with Boltzmann constant, kB, and reduced density oper- +ators, ˆρi(t), for an individual transfer mode, i = q, or the +cavity mode, i = C. The equality in Eq.(13) holds only +if the reduced system is in a pure state, i.e., when the +reduced subsystem is disentangled from the remaining +degrees of freedom. +Finally, we consider the photon number expectation +value, ⟨ˆnc⟩, and its time evolution, which reads in length– +gauge representation (cf. SI, Sec. IV) +⟨ˆnc⟩ = +1 +ℏωc +� +⟨ ˆHC⟩ + ⟨ ˆHSC⟩ + ⟨ ˆHDSE⟩ +� +− 1 +2 +. +(14) +In the non–interacting limit, Eq.(14) reduces to, ⟨ˆnc⟩ = +1 +ℏωc ⟨ ˆHC⟩ − +1 +2 += +nc, +with +nc +physical +photons, +whereas nc += +0 for the herein studied cavity vac- +uum state. +For non–zero light–matter interaction, +the photon expectation value initially reads, ⟨ˆnc⟩0 = +1 +ℏωc +� +⟨ ˆHC⟩0 + ⟨ ˆHDSE⟩0 +� +− 1 +2 > nc, due to a non–zero +number of virtual photons generated by the strong inter- +action of light and matter.[35] In particular, the number +of virtual photons at t0 is directly determined by the +DSE contribution and therefore ensemble size dependent +(cf. Eq.(6)). Note, the interaction term, ⟨ ˆHSC⟩0, does +initially not contribute to ⟨ˆnc⟩0 but will become relevant +throughout the time–evolution of the hybrid system. +III. +RESULTS AND DISCUSSION +A. +Cavity–induced isomerization: Single molecule +We start our discussion of cavity–induced isomeriza- +tion for the asymmetric hydrogen transfer model in the +single–molecule limit with N = 1 by solving the TDSE +(8) for various coupling strengths, η, always using the +same initial state (9). Tab.I, upper two lines, lists the cor- +responding initial state energies, ⟨ ˆH⟩0, and correspond- +ing photon number expectation values, ⟨ˆnc⟩0, for selected +values of η. We observe an increase for both expectation +values, where ⟨ ˆH⟩0 > ∆Ecl +OH = 1598 cm−1 for relatively +strong couplings of η > 0.05 and ⟨ˆnc⟩0 > 0 correspond to +virtual photons generated by the DSE term. +η +0.00 +0.01 +0.03 +0.05 +0.07 +0.09 +⟨ ˆH⟩0 / cm−1 +1098 +1111 +1218 +1433 +1755 +2184 +⟨ˆnc⟩0 +0.0 +0.16 +1.42 +3.95 +7.74 +12.80 +τr/fs +– +87 +88 +92 +95 +100 +kSH/1011 s−1 +0.00 +0.02 +0.15 +0.46 +0.95 +1.39 +TABLE I. Initial state energies, ⟨ ˆH⟩0 = ⟨ ˆHS⟩0 + ⟨ ˆHC⟩0 + +⟨ ˆHDSE⟩0, photon number expectation values, ⟨ˆnc⟩0, first– +recurrence times, τr, and short–time transfer rates, kSH, +in the single–molecule limit for different light–matter inter- +actions strengths, η. +In all cases, the same initial state, +Ψ0(q, xc) = ψOH(q) ϕ0(xc), was employed. +For η > 0, H–transfer converting the enol (OH) to the +enethiol (SH) form takes place. This can be seen from +Fig.2, where the transfer probability, PSH(t) is shown +(2a), as well as the expectation value of the H–transfer +coordinate, ⟨q⟩ (t) (2b), both as a function of time and +for different values of η. Note, for the transfer probabil- +ity, PSH(t), one initially finds PSH(t0) = 0.11 due to the +weakly delocalized nature of ψOH(q). As time evolves, +PSH(t) increases for η > 0 in an oscillatory fashion, which +indicates formation of the enethiol isomer (Fig.2a). Os- +cillatory signatures in PSH(t) represent recurrences with a +period of 264 fs for η < 0.07, which resembles the cavity– +mode energy, ℏωc. For stronger coupling, the dynamics +turns out to be less regular. +The transfer coordinate +expectation value, ⟨q⟩ (t), closely resembles the transfer +dynamics, with ⟨q⟩ < 0 indicating the enol and ⟨q⟩ > 0 +the enethiol isomer (cf. Fig.2b). +From closer inspection of Fig.2a, we can extract first– +recurrence times, τr, and corresponding short–time trans- +fer rates, kSH, for different values of η. These are given +in Tab.I, lower two rows. In the non–interacting limit + +5 +FIG. 2. Time–evolution of (a) single–molecule (N = 1) trans- +fer probability, PSH(t), and (b) transfer coordinate expecta- +tion value, ⟨q⟩ (t), with black dashed lines indicating the quan- +tum mechanical expectation values, ⟨q⟩OH = ⟨Ψ0|q|Ψ0⟩ and +⟨q⟩SH = ⟨Ψ1|q|Ψ1⟩, respectively, for different light–matter in- +teraction strengths, η. +(η = 0.0), we have kSH = 0.0, i.e., there is no pop- +ulation transfer to the local enethiol minimum without +coupling to the cavity mode. In contrast, for η > 0 we +find transfer rates, kSH ≈ 109 s−1 to 1011 s−1, which in- +crease with η by nearly two orders of magnitude over +the whole VSC regime between η = 0.01 to η = 0.09. +The increase of reaction probability/ transfer rate in a +cavity for this particular system is in contrast to other +systems, where a rate retardation has been found either +experimentally[1] or theoretically[25]. That cavities can +also enhance reactivity is a probably less widespread phe- +nomenon, however, this possibility has been discussed in +recent experimental[13] and theoretical[23] work. +In order to interpret the positive effect of the cav- +ity on the early–time, single–molecule transfer proba- +bility, PSH(t), for TAA, we analyze the properties of +the underlying single–molecule cPES, Vη(q, xc), which +guides the dynamics of the vibro–polaritonic wave packet, +Ψ(q, xc, t). +In Figs.3a and b, we show single–molecule cPES, +Vη(q, xc), +besides +corresponding +vibro–polaritonic +ground state densities for different light–matter in- +teraction strengths, +with initial cavity displacement +coordinate expectation value, ⟨xc⟩0 = 0, indicated by +a red vertical line. For η > 0, the cPES’s minima are +symmetrically shifted to negative values of the cavity +displacement coordinate, such that the cavity contri- +bution of the initial wave packet naturally experiences +an excitation. This is in contrast to a recently studied +class of symmetric double well potentials, e.g., for the +inversion of an NH3 molecule[25] or the ground state +cPES of a cavity Shin–Metiu model[21, 23], which are +asymmetrically distorted at finite light–matter inter- +action due to an antisymmetric, sign–changing dipole +moment. The latter leads to barrier broadening, valley +narrowing and (classical) dynamical caging effects, which +in consequence reduce isomerization probabilities[21, 25]. +The static cPES perspective for TAA translates into a +time–evolution of ⟨xc⟩ (t) as shown in Fig.3c. We find the +vibro–polaritonic wave packet to acquire a significant dy- +namical component along the cavity displacement coordi- +nate as time evolves due to the respective gradient on the +cPES. ⟨xc⟩ (t) reveals coherent oscillations with period +264 fs reflecting ℏωc = ∆ε10 and amplitude increasing +with η, which resembles the enhanced cPES distortion in +terms of altered turning points. Since the dynamics along +cavity displacement and molecular transfer coordinates is +naturally coupled via the interaction term, ˆHSC, we can +interpret the isomerization as cavity–induced excitation +along the transfer coordinate. The corresponding energy +transfer can be related to a virtual photon exchange be- +tween cavity and transfer modes, as will be discussed in +detail below. +Since the dynamics is strictly restricted +to non–zero coupling strengths with η > 0.0, the cavity +can be interpreted as a “catalyst” in this model scenario +– despite the classical barrier height is not affected[34]. +We also note, the studied model system does not ex- +hibit a “reactant resonance effect” as the local OH–/SH– +stretching modes have frequencies, ωOH = 3264 cm−1 and +ωSH = 2737 cm−1, which do not support localized bound +states below the classical activation barriers. +B. +Cavity–induced isomerization: Molecular +ensembles +We now extend our study to an ensemble of N trans- +fer systems coupled to a single cavity mode with initial +state, Ψ0(q, xc), given by Eq.(9). +In what follows, we +set η = 0.05 and concentrate on the influence of varying +ensemble sizes N on the transfer process up to N = 20. +At first, we discuss the time–evolution of ensemble trans- +fer probabilities, P ens +SH (t), for different ensemble sizes N +as shown in Fig.4a. From the short–time dynamics, we +extract an ensemble transfer rate, kens +SH = 3 × 1012 s−1, +which is found to be two orders of magnitude larger than + +6 +FIG. 3. +Single–molecule cavity potential energy surface (cPES), Vη(q, xc), and vibro–polaritonic ground state densities, +|Ψ0(q, xc)|2, for different light–matter interaction strengths, η = 0.0 (a), and η = 0.09 (b), with initial cavity displacement +coordinate expectation value, ⟨xc⟩0 = 0 indicated by red vertical line. (c) Time–evolution of cavity displacement coordinate +expectation value, ⟨xc⟩ (t), for different light–matter interaction strengths, η. +FIG. 4. +Time–evolution of (a) ensemble transfer probabil- +ity, P ens +SH (t), and (b) normalized photon number expectation +value, ⟨ˆnc⟩(t), as function of ensemble size N for light–matter +interaction strength, η = 0.05. +Single molecule properties +(N = 1) as reference indicated by black–dashed graphs. +the single molecule rate (0.46 × 1011 s−1), and nearly in- +dependent of N for ensemble sizes studied here. As time +evolves, we observe an oscillatory evolution of P ens +SH (t), +which can be classified by three different “regimes”: +(i) For N ≤ 6, the dynamics is dominated by a max- +imal probability density transfer at around 500 fs and +P ens +SH (t) is modulated by a series of beats with varying +amplitude and period of 264 fs corresponding to the cav- +ity mode excitation energy of ℏωc = 126.5 cm−1. +(ii) For 10 ≤ N ≤ 20, two prominent maxima occur in +P ens +SH (t) at around 200 fs and 700 fs, with a significantly +increased recurrence time of approximately 472 fs, which +we will again address below in context to entanglement +of the cavity mode. +(iii) Eventually, for an intermediate ensemble size with +N = 8, the probability transfer is significantly reduced +(cf. orange graph in Fig.4a) and no specific recurrence +structure is observed. +In order to provide an explanation for the ensemble +transfer dynamics, we discuss a normalized photon num- +ber expectation value +⟨ˆnc⟩(t) = ⟨ˆnc⟩ (t) +⟨ˆnc⟩ (t0) +, +(15) +with, ⟨ˆnc⟩(t0) = 1, which allows us to address ensem- +ble effects on the virtual photon transfer between cavity +and molecules. We note, due to the different contribu- +tions to ⟨ˆnc⟩ in Eq.(14), a strict assignment of virtual +photons only to the cavity mode is in principle not pos- +sible as both interaction and DSE term also contribute +significantly to ⟨ˆnc⟩ (t). The time–evolution of ⟨ˆnc⟩(t) for +different N is shown in Fig.4b and we find ⟨ˆnc⟩(t) < 1 for +all N (including N = 1) over the studied time–interval, +i.e., virtual photons are transferred to the molecular en- +semble. In particular, we observe ⟨ˆnc⟩(t) to qualitatively +resemble the inverse dynamical trend in P ens +SH (t), i.e., +virtual photon transfer to the molecular ensemble coin- +cides with enhanced population transfer to the enethiol +region (cf. +Fig.4a). +Hence, virtual photons are not +only exchanged with the transfer ensemble but virtually + +4000 +-1.0 +2000 +0.12 +-0.5 +14000 +0.10 +0.0 +oe/b +0.08 +0.5 +0.06 +0.04 +1.0 +0.02 +1.5 +600 +0.00 +-400 +-200 +0 +200 +400 +Xc/ymeao3000 +-1.0 +0.12 +-0.5 +0.10 +0.0 +oel +0.08 +7500 +0009 +a +1500 +4500 +0.5 +0.06 +0.04 +1.0 +0.02 +1.5 +0.00 +-400 +-200 +0 +200 +400 +Xc/Vmeao7 +drive the cavity–induced isomerization. We note, non– +normalized expectation values, ⟨ˆnc⟩, significantly depend +on the interaction regime,i.e., η (cf. SI). +C. +Cavity–induced entanglement in ensembles +In order to gain further insight, we address differences +in entanglement between the single–molecule limit and +the ensemble scenario, which allows us to formulate an +interpretation for ensemble–enhanced isomerization. We +first realize, that in the ensemble scenario, an enhanced +ensemble transfer rate cannot straightforwardly be ex- +plained by the distortion of the (N + 1-dimensional) +cPES, since the distortion in a single molecule–cavity +subspace decreases due to the Dicke–type light-matter +interaction used here, gN, as, gN ∼ +1 +√ +N , for increasing +N (cf. Eq.(7)).[23] However, due to strong coupling be- +tween light and matter constituents, entanglement effects +are in contrast expected to shape the ensemble transfer +dynamics. +We quantify entanglement by transfer and cavity von +Neumann–entropies, Sq(t), and SC(t), as defined in +Eq.(13) with time–evolution depicted in Figs.5a and b for +different ensemble sizes, N. Initially, we have, Sq(t0) = +SC(t0) = 0, independent of ensemble size due to the prod- +uct form of the initial state in Eq.(9), which is by defi- +nition disentangled. From a wave function perspective, +build up of entanglement can therefore be interpreted in +terms of an increase of the multiconfigurational charac- +ter of the full vibro–polaritonic wave function, i.e., as +deviation from a disentangled product state. In general, +we find the time–evolution of both entropies to differ sig- +nificantly due to the different nature of both subsystems +and, SC(t) > Sq(t). +Starting with the reduced transfer system perspective, +we observe Sq(t) to increase faster with time for increas- +ing N. +This indicates a faster build–up of subsystem +entanglement or equivalently multiconfigurational char- +acter in the vibro–polaritonic wave function with respect +to H–transfer systems. Further, for N > 1, we observe +an increase in Sq(t) to accompany the transfer process +to the enethiol configuration as characterized by P ens +SH (t) +(cf. Fig.4a), i.e., H–transfer relates to enhanced entan- +glement. +Further, the transfer von Neumann–entropy +reaches (local) minima, i.e., reduced transfer system en- +tanglement, where extrema are observed in the ensem- +ble transfer probability. This observation is in line with +above discussed dynamics of ⟨ˆnc⟩(t) (cf. +Fig.4b), i.e., +virtual photon transfer drives isomerization, which nat- +urally leads to a stronger entanglement (larger Sq(t)) +between cavity and transfer systems. In contrast, sup- +pressed transfer relates to small values of Sq(t). More- +over, for intermediate ensemble sizes of N = 8, Sq(t) +differs significantly by exhibiting a nearly monotonic but +slow increase with time, which is accompanied by sup- +pressed virtual photon transfer and isomerization proba- +bility (cf. Fig.4), in line with the previous argument. +FIG. 5. Time–evolution of (a) transfer von Neumann–entropy, +Sq(t), (b) cavity von Neumann–entropy, SC(t), and (c) cavity +displacement coordinate expectation value, ⟨xc⟩ (t), as func- +tion of ensemble size N for light–matter interaction strength, +η = 0.05. Single molecule properties (N = 1) as reference +indicated by black–dashed graphs. +Turning to the cavity mode, we find SC(t) to be char- +acterized by an overall increase with time, which be- +comes strongly pronounced for large N, i.e., the cav- +ity mode becomes quickly strongly entangled with the +transfer ensemble. +Further, the cavity von Neumann– + +8 +entropy mimics the vibro–polaritonic wave packet’s dy- +namics along the cavity displacement coordinate as cap- +tured by ⟨xc⟩ (t) (cf. +Fig.5c). +For small ensembles +(N = 2), SC(t) oscillates with a period of 264 fs, re- +covering the cavity mode frequency and is minimized for +⟨xc⟩ (t) reaching the initial position at xc = 0. In con- +trast, for large ensembles with N = 20, the cavity en- +tropy exhibits minima characterized by two different time +scales, which can be related to both maximal and min- +imal displacements of the vibro–polaritonic wave packet +along the displacement coordinate xc. +Eventually, we +note two characteristics, which additionally indicate the +complex nature of the ensemble dynamics: First, for in- +creasing N the amplitude of ⟨xc⟩ (t) is damped as time +evolves since the cavity mode is increasingly immersed in +a bath of strongly coupled and highly anharmonic trans- +fer systems. Second, as ensemble size N increases, the +initial amplitude of ⟨xc⟩ (t) increases too with a maxi- +mum around N = 8. +We attribute this transition to +a change from a small transfer ensemble of multi–mode +Rabi type to a “large” ensemble resembling a system– +bath–type regime with a cavity mode subject to dissipa- +tion on the time scale shown. Naturally, this transition +is equivalently observed in the transfer dynamics. +In summary, we attribute isomerization in the model +studied here to be induced by virtual photon transfer, +which maximizes time–dependent changes in transfer +system entanglement quantified by Sq(t). +In particu- +lar, ensemble–enhanced isomerization rates relate to a +cavity–induced entanglement effect between transfer sys- +tems, which is not explainable by classical cPES distor- +tion arguments valid in the single–molecule limit. Ac- +cordingly, the herein discussed cavity–induced ensemble +transfer process can be interpreted as an inherently col- +lective quantum mechanical effect, which in particular +cannot be captured by scaled single–molecule models. +IV. +SUMMARY AND CONCLUSIONS +In this work, we studied the quantum dynamics of an +entangled molecular ensemble for an asymmetric hydro- +gen transfer model of thioacetylacetone (TAA) interact- +ing with a single cavity mode under vibrational strong +coupling. An N–molecule form of the Pauli–Fierz Hamil- +tonian was used beyond frequently adopted model Hamil- +tonians such as the Tavis–Cummings or Dicke–models. A +N +1–dimensional time–dependent Schrödinger equation +was solved numerically by means of the MCTDH ansatz +to follow the cavity–induced isomerization dynamics from +enol to enethiol isomers of TAA. +At finite light–matter interaction, the cavity acts as a +“catalyst” by inducing population transfer to the enethiol +isomer, which is energetically less favorable than the enol +form of TAA outside the cavity or at vanishing cavity– +molecule coupling. This process is identified to be driven +by virtual photon transfer to the N–molecule subsystem. +We extract approximate short–time transfer rates, which +span in the single–molecule limit two orders of magnitude +for increasing light–matter interaction. In an entangled +ensemble of transfer systems, with a collective Dicke–type +light–matter coupling, a collectively enhanced transfer +rate is observed following from an interplay of virtual +photon–transfer and non–trivial entanglement dynamics +between light and matter components of the hybrid sys- +tem. We furthermore find non–trivial ensemble size de- +pendence of the dynamics as N grows, which we attribute +to a transition from a multi–mode Rabi type scenario to +a system–bath–type regime, where the cavity mode is ef- +fectively immersed in a bath of strongly coupled, anhar- +monic transfer systems. Our study points at the highly +non–trivial role of quantum effects in molecular ensem- +ble models strongly interacting with a quantized cavity +mode beyond scaled single–molecule dynamics. +We close by pointing out several possible extensions of +our model. First, we do not take into account dissipa- +tive effects and decoherence due to leaky cavity modes +or the presence of other molecular degrees of freedom, +which will naturally influence the transfer probability +and allow for a more rigorous definition of transfer rates. +However, if we assume additional degrees of freedom to +be only weakly coupled, which is relevant to ensure the +VSC regime, the main findings of our work should qual- +itatively remain the same for the herein studied time– +interval. +Further, we did not take into account finite +temperature effects, which can be assumed relevant due +to the relatively low energy scale in our model. Finally, +while we went well beyond the Dicke–model by using an +ensemble formulation based on the Pauli–Fierz Hamil- +tonian, we still assumed a Dicke–type coupling in our +work. It might be instructive to also lift the Dicke–type +perspective for the molecule–cavity coupling to carefully +discuss deviations and potentially emerging properties in +vibro–polaritonic chemistry for less restricted coupling +models. Along similar lines, orientational effects (due to +rotation of molecules) on the molecule–cavity coupling in +fluctuating molecular ensembles, their influence on the +related entanglement dynamics as well as the inclusion +of direct intermolecular interactions could be interesting +milestones towards a realistic description of molecular +ensembles in cavities. +ACKNOWLEDGEMENTS +We acknowledge fruitful discussions with Oliver Kühn +(Rostock) and Foudhil Bouakline (Potsdam). This work +was funded by the Deutsche Forschungsgemeinschaft +(DFG, German Research Foundation) under Germany’s +Excellence Strategy – EXC 2008/1-390540038. +E.W. +Fischer acknowledges support by the International Max +Planck Research School for Elementary Processes in +Physical Chemistry. + +9 +DATA AVAILABILITY STATEMENT +The data that support the findings of this study are +available from the corresponding author upon reasonable +request. +CONFLICT OF INTEREST +The authors have no conflicts to disclose. +[1] T. W. Ebbesen, Acc. Chem. Res. 49, 2403, (2016). +[2] R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. +Campos-Gonzalez-Angulo, J. Yuen-Zhou, Chem. Sci. 9, +6325, (2018). +[3] J. Feist, J. Galego, F. J. Garcia-Vidal, ACS Photonics 5, +205, (2018). +[4] A. D. Dunkelberger, B. S. Simpkins, I. Vurgaftman, J. +C. Owrutsky, Ann. Rev. Phys. Chem. 73, 429, (2022). +[5] J. Yuen-Zhou, W. Xiong, T. Shegai, J. Chem. Phys. 156, +030401, (2022). +[6] J. Fregoni, F. J. Garcia-Vidal, J. 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Cummings, textitPhys. Rev. 170, +379, (1968). +[27] R.H. Dicke, Phys. Rev. 93, 99, (1954). +[28] A. Mandal, X. Li, P. Huo, J. Chem. Phys. 156, 014101, +(2022). +[29] D. Wellnitz, G. Pupillo, J. Schachenmayer, Commun. +Phys. 5, 1, (2022). +[30] A. Mandal, M. Taylor, B. Weight, E. Koessler, X. Li, P. +Huo, chemRxiv:2022-g9lr7 (2022). +[31] N. Doslić, K. Sundermann, L. González, O. Mó, J. +Giraud-Girard, O. Kühn, Phys. Chem. Chem. Phys. 1, +1249, (1999). +[32] J. Flick, H. Appel, M. Ruggenthaler, A. Rubio, J. Chem. +Theory Comput. 13, 1616, (2017). +[33] C. Schäfer, M. Ruggenthaler, A. Rubio, Phys. Rev. A 98, +043801, (2018). +[34] E. W. Fischer, P. Saalfrank, J. Chem. Phys. 154, 104311, +(2021). +[35] A. F. Kockum, A. Miranowicz, S. De Liberato, S. +Savasta, F. Nori, Nat. Rev. Phys. 1, 19 (2019). +[36] M. H. Beck, A. Jäckle, G. A. Worth, H.-D. Meyer, Phys. +Rep. 324, 1, (2000). +[37] H.-D. Meyer, WIREs Comput. Mol. Sci. 2, 351, (2012). +[38] H. Wang, M. Thoss, J. Chem. Phys. 119, 1289, (2003). +[39] U. Manthe, J. Chem. Phys. 128, 164116, (2008). +[40] O. Vendrell, H.-D. Meyer, J. Chem. Phys. 134, 044135, +(2011). +[41] G. A. Worth, M. H. Beck, A. Jäckle, and H.-D. Meyer. +The MCTDH Package, Version 8.2, (2000). H.-D. Meyer, +Version 8.3 (2002), Version 8.4 (2007). O. Vendrell and +H.-D. Meyer Version 8.5 (2013). Version 8.5 contains +the ML-MCTDH algorithm. See http://mctdh.uni-hd.de. +Used versions: 8.6.1 (2022). + +arXiv:2301.04074v1 [quant-ph] 10 Jan 2023 +Supplementary Information: +Cavity–Catalyzed Hydrogen Transfer Dynamics in an Entangled Molecular Ensemble +under Vibrational Strong Coupling +Eric W. Fischer∗ +Theoretische Chemie, Institut für Chemie, Universität Potsdam, +Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany +Peter Saalfrank +Theoretische Chemie, Institut für Chemie, Universität Potsdam, +Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany and +Institut für Physik und Astronomie, Universität Potsdam, +Karl-Liebknecht-Straße 24-25, D-14476 Potsdam-Golm, Germany +I. +ONE–DIMENSIONAL HYDROGEN TRANSFER REACTION HAMILTONIAN +A. +Reaction Potential and Minimum Energy Path +We derive the one–dimensional H–transfer Hamiltonian, ˆHS (Eq.(2) with N = 1 in the main text ), from a two– +dimensional asymmetric H–transfer reaction Hamiltonian for thioacetylacetone (TAA) developed by Doslić et al.[1], +which was constructed from ab initio electronic structure calculations and reads +ˆHR = − ℏ2 +2µS +∂2 +∂q2 − ℏ2 +2µB +∂2 +∂Q2 + V (q, Q) +, +(I1) +with a (H–transfer) reaction coordinate, q, a (collective) “heavy” mode coordinate, Q, and corresponding reduced +masses, µS = 1914.028 me and µB = 8622.241 me, respectively.[1] The two–dimensional molecular potential energy +surface (PES), V (q, Q), is given by +V (q, Q) = V (q) + µB ω2 +B +2 +(Q − λS(q))2 +, +(I2) +with “heavy” mode frequency, ωB = 0.0009728 Eh, and nonlinear coupling function, λS(q) = aS q2 +bS q3, determined +by parameters, aS = 0.794 a−1 +0 +and bS = −0.2688 a−2 +0 . +The reaction path potential is described in terms of an +adiabatic potential +V (q) = 1 +2 +� +V+(q) − +� +V 2 +−(q) + 4 K2(q) +� +, +(I3) +where, V±(q) = V1(q)±V2(q), with diabatic harmonic PES, Vi(q), and non–adiabatic coupling function, K(q), defined +as +Vi(q) = µi ω2 +i +2 +(q − qi,0)2 + ∆i +, +K(q) = kc exp +� +−(q − qc)2 +� +. +(I4) +The harmonic potentials resemble the R–OH (V1(q)) and R–SH (V2(q)) configurations in TAA with correspond- +ing harmonic frequencies, ωOH = 0.01487 Eh/ℏ and ωSH = 0.01247 Eh/ℏ, reduced masses, µOH = 1728.46 me and +µSH = 1781.32 me, relative energy shifts, ∆OH = 0.0 Eh and ∆SH = 0.003583 Eh, as well as displacements, qOH,0 = +−0.7181 a0 and qSH,0 = 1.2094 a0. The coupling function, K(q), is determined by an amplitude, kc = 0.15582 Eh, and +a displacement, qc = 0.2872 a0.[1] Further, a molecular dipole function (neglecting the vector character of the dipole +moment) is given in Ref.[1] as +d(q, Q) = d0 + dS(q − q0) + dB(Q − λS(q)) + dSB(q − q0)(Q − λ(q)) +, +(I5) +∗ ericwfischer@posteo.de + +2 +with parameters, d0 = 1.68 ea0, dS = −0.129 ea0/a0, dB = 0.023 ea0/a0, dSB = 0.451 ea0/a2 +0 and q0 = −0.59 a0. +In the present work, where the ensemble character of the isomerizing molecules is in the focus, an effective ap- +proximate one–dimensional Hamiltonian, ˆHS, and corresponding dipole function, d(q), are constructed, which still +resemble the main features of their two–dimensional counterparts. We derive the one–dimensional transfer Hamilto- +nian by minimizing, V (q, Q), with respect to Q as +∂ +∂QV (q, Q) = 0 +⇔ +Q0 = λ(q) +, +(I6) +such that the transfer potential and the dipole function subsequently simplify to one–dimensional functions +V (q, Q0) = V (q) +, +d(q, Q0) = d(q) = d0 + dS(q − q0) +. +(I7) +The latter holds equivalently for an ensemble of N transfer ensembles. In our study, we neglect the “heavy mode”, +Q, which does not couple via a potential–like term to the transfer coordinate, q. In Fig.S1, we show V (q) and d(q), +with the lowest two eigenfunctions, ψ0(q) = ψOH(q) (enol) and ψ1(q) = ψSH(q) (enethiol), indicated. The latter were +obtained by diagonalizing ˆHS in terms of a Colbert–Miller discrete variable representation (DVR)[5] for the transfer +coordinate with Nq = 121 grid points and q ∈ [−1.5, 2.1] a0. The corresponding eigenenergies are ε0 = 988.3 cm−1 +and ε1 = 1092.8 cm−1 as stated in the main text with an energy difference of ∆ε10 = ε1 − ε0 = 126.5cm−1. +FIG. S1. (a) One–dimensional hydrogen–transfer reaction potential, V (q) (in black), with dipole function, d(q), and two lowest +eigenstates, ψ0(q) = ψOH(q) and ψ1(q) = ψSH(q). (b) Ground state, |ψ0(q, Q)|2 = |ψOH(q, Q)|2, and (c) first excited state +densities, |ψ1(q, Q)|2 = |ψSH(q, Q)|2, of two–dimensional reaction Hamiltonian, ˆHR, in Eq.(I1) embedded in two–dimensional +molecular PES, V (q, Q), given in Eq.(I2) with contours in cm−1. +For the two–dimensional Hamiltonian, ˆHR, in Eq.(I1), we numerically obtain energies, ǫ0 = 1037.5 cm−1 and +ǫ1 = 1158.3 cm−1, for the ground and first excited states, respectively, with energy difference of ∆ǫ10 = 120.8 cm−1. +Here, were we again employed a Colbert–Miller DVR with transfer grid parameters equivalent to the one–dimensional +case discussed above and “heavy” mode coordinate Q ∈ [−2.0, 2.0] a0 with NQ = 61 grid points. Eventually, classical +activation energies are by construction equivalent for the one– and two–dimensional PES with ∆Ecl +OH = 1598 cm−1 +and ∆Ecl +SH = 1081 cm−1 as stated in the main text, since V (q) is equivalent to the reaction potential along the +minimum energy path on V (q, Q). +B. +Deviations from a Reaction Path Hamiltonian +We discuss deviations of our approach from a reaction path Hamiltonian, which arises from a two–dimensional +model and additionally involves kinetic energy couplings due to non–zero reaction path curvature. Miller, Handy and +Adams[2] showed that a reaction path Hamiltonian of a two–dimensional system with mass–weighted, cartesian–like +coordinates is given by +ˆH(ˆps, s, ˆPs, Qs) = +ˆp2 +s +2 (1 + Qs κ(s))2 + V0(s) + ˆHvalley(s) +, +(I8) + +6000 +-1.0 +500 +0.05 +-0.5 +1500 +0.04 +lao +0.0 +4500 +0.03 +a +0.5 +4500 +3000 +0.02 +.500 +3000 +1.0 +0.01 +4500 +1.5 +0.00 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Q/ao6000 +-1.0 +500 +0.06 +-0.5 +0.05 +500 +0.04 +lao +0.0 +4500 +a +0.03 +0.5 +4500 +3000 +3000 +1500 +0.02 +1.0 +0.01 +4500 +1.5 +0.00 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Q/ao3 +where the first two terms correspond to kinetic and potential energy contributions along the reaction coordinate, s, +with conjugate momentum, ˆps, whereas the third term provides the “valley” Hamiltonian +ˆHvalley(s) = +ˆP 2 +s +2 + ω(s)2 +2 +Q2 +s +, +(I9) +which accounts for a s–dependent “valley” mode perpendicular to the reaction path with frequency, ω(s), that couples +to the reaction coordinate via the reaction path curvature, κ(s). For the hydrogen transfer system studied here, we +have by construction, V0(s) = V (q, Q0) = V (q). In the following, we discuss deviations from ˆH(ˆps, s, ˆPs, Qs), which +emerge when we approximate the reaction path contribution by the bare transfer Hamiltonian +ˆHS = ˆp2 +q +2 + V (q) = − ℏ2 +2µS +∂2 +∂q2 + V (q) +. +(I10) +This assumption is equivalent to approximately decoupling the “valley” Hamiltonian, ˆHvalley(s), from the reaction +path contribution by assuming the reaction path curvature, κ(s), to be small. In order to access this condition, we +discuss the curvature, κ(s), of the minimum energy or reaction path, s(q), which we introduce as parametric curve in +the mass–weighted q–Q–plane[3] +s(q) = (s1(q), s2(q))T = (√µS q, √µB Q0(q))T +, +(I11) +with components, s1(q) and s2(q). Here, Q0(q) = λS(q), as derived above, which minimizes, V (q, Q), with respect +to variations in the “heavy” mode coordinate. From s(q), which is parameterized in terms of the hydrogen–transfer +coordinate, q, we obtain the corresponding curvature, κs(q), as[4] +κs(q) = det (s′, s′′) +||s′||3 += +|s′ +1s′′ +2 − s′′ +1s′ +2| +[(s′ +1)2 + (s′ +2)2] +3 +2 +, +(I12) +with derivatives, s′ +i = +∂ +∂q si(q) and s′′ +i = +∂2 +∂q2 si(q), respectively. We like to emphasize, that κs(q) depends now on +the hydrogen transfer coordinate, q, which parametrizes the reaction path. +Further, for reaction path elements, +s1(q) = √µS q and s2(q) = √µB λS(q), we find derivatives +s′ +1 = √µS +, +s′′ +1 = 0 +, +s′ +2(q) = √µB +� +2 aS q + 3 bS q2� +, +s′′ +2(q) = √µB (2 aS + 6 bS q) +, +(I13) +which allow us to write the curvature explicitly as +κs(q) = +2√µS µB |aS + 3 bS q| +� +µB q2 (2 aS + 3 bS q)2 + µS +� 3 +2 +. +(I14) +In Fig.S2a, we show the two–dimensional molecular PES, V (q, Q), with reaction path, s(q), in red and in Fig.S2b +the corresponding curvature, κs(q). A kinetic separation of the reaction path from the “valley” coordinate is a good +approximation, if +ˆTs = +ˆp2 +s +2 (1 + Qs κs(q))2 ≈ ˆp2 +q +2 = ˆTq +, +(I15) +which holds for |Qs κs(q)| ≪ 1. By taking into account the maximal curvature, κs(q = 0.0) ≈ 0.078 (√me a0)−1, at +the transition state (q = 0.0), the coupling is solely determined by the “valley” coordinate’s magnitude, which can be +traced back to the excitation of the two–dimensional transfer system along the “heavy” mode coordinate. We shall +estimate the latter by means of the classical turning points +Q± +s = ± +� +ℏ +ωB +(2v + 1) +, +(I16) +of the harmonic “valley” potential with vibrational quantum number, v, and, ω(s) = ωB, at the transition state. +For the first excited state (v = 1), we find, |Q± +s κs(q = 0.0)| ≈ 4.3. Hence, already for the “heavy” mode being +excited to the first excited state, which has to be expected during the transfer process, we observe a coupling to +the reaction coordinate that is assumed to alter the transfer dynamics of the molecular isomerization model system. +However, as the role of the “heavy” mode is not central for the cavity–induced isomerization dynamics, we consider +our approximation to be qualitatively valid and sufficient to discuss entanglement–induced collective effects in the +herein studied reactive vibro–polaritonic model. + +4 +FIG. S2. a) Contour plot of molecular PES, V (q, Q), in mass–weighted coordinates with reaction path, s(q), in red and colorbar +in wave numbers (cm−1) and b) minimum energy path curvature, κs(q), parameterized by mass–weighted transfer coordinate, +√µS q with maximum at the transition state. +II. +NUMERICAL DETAILS FOR QUANTUM DYNAMICS +We solve the TDSE (Eq.(8) in the main text) numerically by means of the multiconfigurational time–dependent +Hartree (MCTDH) method and its multilayer extension (ML–MCTDH) and propagate up to final time tf = 1000 fs. +We employ a Colbert–Miller DVR for transfer reaction coordinates, qi ∈ [−1.5, 2.1]a0, with Nq = 101 grid points and +a harmonic oscillator (HO) DVR for the cavity mode with Nc = 101 grid points and xc ∈ [−561.35, +561.35]√me a0. +We treat ensembles up to N = 4 via the MCTDH method with single particle functions (SPFs), ns = nc = 10. For +ensembles with 4 < N ≤ 20, we employ the ML–MCTDH method with converged trees (max. natural population +≤ 10−4) for all N as displayed in Fig.S3. We employ the same DVR with identical number of primitive basis functions +as above independent of ensemble size, N, but N–dependent numbers of SPFs, as shown next to bonds in ML trees, +due to different entanglement structure in the full vibro–polaritonic wave packet,. +III. +VIBRO–POLARITONIC CHARACTER OF INITIAL STATES +In addition to the information given in the main text, here we analyze the initial states used in there (cf. Eq.(9) +in main text) with respect to their vibro–polaritonic character. +We consider contributions of vibro–polaritonic +states to the initial state by means of infrared spectra, σ(ω), obtained from the autocorrelation function, C(t) = +⟨Ψ⋆(t/2)|Ψ(t/2)⟩, of the vibro–polaritonic wave packet, |Ψ(t)⟩, with initial state, |Ψ(t0)⟩ = |Ψ0⟩, as +σ(ℏω) = A +� 2tf +0 +C(t) ei(E−E0)t/ℏdt = +� +p +| ⟨Ψ0|Ψp⟩ |2 δ(E − (Ep − E0)) +, +(III1) +with constant, A = 1, vibro-polaritonic eigenergies, Ep, and corresponding eigenstates, |Ψp⟩, satisfying the time– +independent Schrödinger equation +� +ˆHS + ˆHC + ˆHSC + ˆHDSE +� +|Ψp⟩ = Ep |Ψp⟩ +. +(III2) +Here, E0 corresponds to the ground state energy, which we obtained by means of imaginary time evolution of the +light–matter hybrid system employing (ML)–MCTDH methods. In Fig.S4, we show σ(ℏω) for different ensemble sizes +N and observe a significant number of states |Ψp⟩ contributing to the initial state. +We attribute the substantial deviation from approximately harmonic vibro–polaritonic models, which commonly +show clear signatures of lower and upper vibro–polaritonic states, to the strong anharmonicity of the hydrogen transfer +potential. A shift of the spectral envelope’s center to higher energies for increasing N results from ⟨ ˆH⟩0 increasing +with N at fixed η due to contributions from both ˆHS and ˆHDSE. Further, the number of states increases with N + +6000 +-40 +500 +16000 +14000 +0 +-20 +e +1500 +12000 +e +w//b sn +-0 +0 +10000 +45 +8000 +20 - +4500 +3000 +6000 +1500 +3000 +40 +4000 +2000 +60 +9000 +4500. +0 +-150 +-100 +-50 +0 +50 +100 +150 +O/yme ao +VUB5 +FIG. S3. Multilayer trees for different ensemble sizes N with S1 to SN transfer systems and cavity mode C. Number of SPFs +are shown next to bonds connecting circular nodes and number of primitive basis functions are shown next to bonds connecting +circular and square nodes. +up to N = 8, which is accompanied by intensity reduction. Around N = 8, the number of vibro–polaritonic states +contributing to the initial state decrease significantly and increase in the following for large ensembles up to N = 20. + +6 +FIG. S4. Infrared spectra, σ(ℏω), for different ensembles sizes N with (a) N = 1 to N = 8 and (b) N = 8 to N = 20. Peak +widths result from finite propagation time of tf = 1000 fs. +Note, energy scales for Fig.S4a and S4b are different due to different ensemble sizes N. +IV. +PHOTON NUMBER OPERATOR IN LENGTH GAUGE REPRESENTATION +In the main text, the expectation value of the photon number operators was used to analyze the cavity–induced +H–transfer dynamics. The common photon number operator, ˆnc = ˆa† +cˆac, can be written in terms of the single–mode +cavity Hamiltonian, ˆHC = ℏωc +� +ˆa† +cˆac + 1 +2 +� +, as +ˆnc = ˆa† +cˆac = +1 +ℏωc +ˆHC − 1 +2 +, +(IV1) +where, ˆa† +c and ˆac are bosonic photon creation and annihilation operators, respectively. However, in length gauge +representation, ˆnc, takes the form[6–9] +S† U† ˆa† +cˆac U S = +1 +ℏωc +� +S† U† ˆHC U S +� +− 1 +2 +, +(IV2) +with unitary operator, U = exp +� +i +ℏ ˆA d(q) +� +, mediating the Power–Zienau–Woolley (PZW) transformation, which is de- +termined by the molecular dipole moment, d(q), and the transverse (single–mode) vector potential, ˆA = +g +ωc +� +ˆa† +c + ˆac +� +. +A second unitary rotation mediated by, S = exp +� +i π +2 ˆa† +cˆac +� +, acts exclusively on the cavity mode subspace and provides +a real light–matter interaction term, ˆHSC. Under U and S, photon creation and annihilation operators transform as +S† U† ˆa† +c U S = −i ˆa† +c − i +ℏ +g +ωc +d(q) +, +S† U† ˆa U S = +i ˆac + i +ℏ +g +ωc +d(q) +. +(IV3) +Employing the latter identities, the transformed number operator turns with +ℏωc +� +S† U† ˆa† +cˆac U S +� += ℏωc +� +−i ˆa† +c − i +ℏ +g +ωc +d(q) +� � ++i ˆac + i +ℏ +g +ωc +d(q) +� +, +(IV4) += ℏωc +� +ˆa† +cˆac + g d(q) +ℏωc +� +ˆa† +c + ˆac +� ++ +g2 +(ℏωc)2 d2(q) +� +, +(IV5) += ℏωc ˆa† +cˆac + g d(q) +� +ˆa† +c + ˆac +� ++ g2 +ℏωc +d2(q) +(IV6) + +7 +as well as identities, +� +ℏ +2ωc +� +ˆa† +c + ˆac +� += xc, and, i +� +ℏωc +2 +� +ˆa† +c − ˆac +� += ˆpc, into +S† U† ˆa† +cˆac U S = +1 +ℏωc + + + + + +ˆp2 +c +2 + ω2 +c +2 x2 +c +� +�� +� += ˆ +HC ++ +� +2ωc +ℏ g d(q) xc +� +�� +� += ˆ +HSC ++ g2 +ℏωc +d2(q) +� +�� +� += ˆ +HDSE + + + + + − 1 +2 +, +(IV7) += +1 +ℏωc +� +ˆHC + ˆHSC + ˆHDSE +� +− 1 +2 +. +(IV8) +This latter expression enters the cavity photon number expectation value, ⟨ˆnc⟩, in Eq.(14) of the main text. The same +arguments generalize to ˆnc for ensembles of N molecules as the unitary operator mediating the PZW transformation, +UN = �N +i +Ui, factorizes due the form of the ensemble dipole function, d(q) = �N +i +d(qi). In Fig.S5, we eventually +FIG. S5. Time-evolution of photon number expectation value, ⟨ˆnc⟩ (t), as function of ensemble size N for light-matter interaction +strength, η = 0.05. +provide the time–evolution of the bare photon number operator expectation value without normalization to the initial +value. +[1] N. Doslić, K. Sundermann, L. González, O. Mó, J. Giraud-Girard, O. Kühn, Phys. Chem. Chem. Phys. 1, 1249, (1999). +[2] W. H. Miller, N. C. Handy, J. E. Adams, J. Chem. Phys. 72, 99, (1979). +[3] E. W. Fischer, J. Anders, P. Saalfrank, J. Chem. Phys. 156, 154305, (2022). +[4] T. Arens, F. Hettlich, C. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel. Mathematik. Springer Spektrum Berlin, +(2018). +[5] D.T. Colbert, W.H. Miller, J. Chem. Phys. 96, 1982, (1992). +[6] V. Rokaj, D. M. Welakuh, M. Ruggenthaler, A. Rubio, J. Phys. B: At., Mol. Opt. Phys. 51, 034005 (2018). +[7] C. Schäfer, M. Ruggenthaler, V. Rokaj, A. Rubio, ACS Photonics 7, 975 (2020). +[8] A. Mandal, T. D. Krauss, P. Huo, J. Phys. Chem. B 124, 6321, (2020). +[9] E. W. Fischer, P. Saalfrank, J. Chem. Phys. 154, 104311 (2021). + diff --git a/VdE2T4oBgHgl3EQftwjJ/content/tmp_files/load_file.txt b/VdE2T4oBgHgl3EQftwjJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ba5b609c44859b27188125e2ab6296965d9148d --- /dev/null +++ b/VdE2T4oBgHgl3EQftwjJ/content/tmp_files/load_file.txt @@ -0,0 +1,1068 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf,len=1067 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04074v1 [quant-ph] 10 Jan 2023 Cavity–Catalyzed Hydrogen Transfer Dynamics in an Entangled Molecular Ensemble under Vibrational Strong Coupling Eric W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fischer∗ Theoretische Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Institut für Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Universität Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Karl-Liebknecht-Strasse 24-25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' D-14476 Potsdam-Golm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Germany Peter Saalfrank Theoretische Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Institut für Chemie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Universität Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Karl-Liebknecht-Strasse 24-25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' D-14476 Potsdam-Golm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Germany and Institut für Physik und Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Universität Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Karl-Liebknecht-Straße 24-25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' D-14476 Potsdam-Golm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Germany Microcavities have been shown to influence the reactivity of molecular ensembles by strong cou- pling of molecular vibrations to quantized cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In quantum mechanical treatments of such scenarios, frequently idealized models with single molecules and scaled, effective molecule–cavity interactions or alternatively ensemble models with simplified model Hamiltonians are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In this work, we go beyond these models by applying an ensemble variant of the Pauli–Fierz Hamiltonian for vibro–polaritonic chemistry and numerically solve the underlying time–dependent Schrödinger equation to study the cavity–induced quantum dynamics in an ensemble of thioacetylacetone (TAA) molecules undergoing hydrogen transfer under vibrational strong coupling (VSC) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Be- ginning with a single molecule coupled to a single cavity mode, we show that the cavity indeed enforces hydrogen transfer from an enol to an enethiol configuration with transfer rates significantly increasing with light–matter interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This positive effect of the cavity on reaction rates is different from several other systems studied so far, where a retarding effect of the cavity on rates was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' It is argued that the cavity “catalyzes” the reaction by transfer of virtual photons to the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The same concept applies to ensembles with up to N = 20 TAA molecules coupled to a single cavity mode, where an additional, significant, ensemble–induced collective isomerization rate enhancement is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter is traced back to complex entanglement dynamics of the ensemble, which we quantify by means of von Neumann–entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A non–trivial dependence of the dynamics on ensemble size is found, clearly beyond scaled single–molecule models, which we interpret as transition from a multi–mode Rabi to a system–bath–type regime as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' INTRODUCTION The interaction of optical modes confined in Fabry– Pérot cavities with optically active molecular degrees of freedom lies at the heart of the emerging field of po- laritonic chemistry[1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In one class of promising ex- periments, molecular vibrations interact strongly with the ground state of infrared active cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In this vibrational strong coupling (VSC) regime, signifi- cantly altered thermal ground state chemistry has been observed[8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' While in early examples of VSC–altered reactivity, rates usually were retarded, meanwhile also experiments exist with accelerated rates[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' VSC ex- periments have generated a significant theoretical effort aiming to understand the complex interplay of light and matter in thermal polariton chemistry[14–25, 28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' As an example, isomerization reactions have been idealized as arising from population transfer from one well of a cavity–distorted double–minimum potential to the other well, with the dynamics being treated classically or quan- tum mechanically[21, 23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' At least in quantum me- chanical treatments, typically single molecules are con- sidered and the transition to ensembles of N molecules ∗ ericwfischer@posteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='de is mimicked via effective single–molecule models with scaled molecule–cavity couplings[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Alternatively, sim- plified N–emitter model Hamiltonians comprising a set of two–level systems and model cavity–emitter couplings are used such as the Tavis–Cummings[26] or Dicke[27]– models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In this contribution, we study hydrogen transfer re- action models for thioacetylacetone (TAA) molecules[31] placed in an infrared cavity both for a single molecule or an ensemble of up to N = 20 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The lat- ter are explicitly treated from a fully quantum mechan- ical perspective and beyond simplified N–emitter model Hamiltonians such as Tavis–Cummings[26], by employing an N–molecule variant of the Pauli–Fierz Hamiltonian and solving a corresponding multi–dimensional time- dependent Schrödinger equation numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For the molecule under investigation, idealized here as an asym- metric double–well, we first demonstrate for the single– molecule scenario a cavity–induced isomerization/ H– transfer from the “enol” configuration, which is the ener- getically favored form outside the cavity, to an “enethiol” form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We extract approximate H–transfer rates from the numerically obtained dynamics, which increase with light–matter coupling and ensemble size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The cavity “catalytic” effect in the single–molecule limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', rate enhancement, is traced back to a symmetric distor- 2 tion/ displacement of the cavity potential energy sur- face (cPES) by a smoothly varying dipole function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In contrast, if the distortion of the cPES is antisymmet- ric (as for the inversion of ammonia, for example, with an antisymmetric dipole function), rates are found to be decelerated by the cavity[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Equivalently, the rate enhancement found here can be understood as being driven by transfer of virtual photons from the cavity to the H–transfer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The concept of virtual pho- ton transfer directly generalizes to the ensemble scenario, where single–molecule cPES distortion arguments no longer hold due to significantly reduced single molecule light–matter coupling in contrast to enhanced ensemble coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [23] In particular, we discuss how virtual pho- ton transfer in the many–molecule scenario leads to a strongly entangled molecular transfer ensemble, which in turn determines the quantum mechanical nature of the transfer dynamics, beyond those arising from scaled single–molecule model Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='II the N– molecule Pauli–Fierz Hamiltonian for an ensemble of TAA molecules is introduced and the computation of ob- servables employed to describe the isomerization dynam- ics is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='III, we demonstrate and analyze the cavity–induced isomerization of single TAA and en- sembles of TAA molecules systematically in dependence of the molecule–cavity coupling strength and the ensem- ble size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='IV summarizes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Most of the numerical details and parameters as well as some further results can be found in the Supplementary Infor- mation (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' THEORY AND MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Hamiltonian and Quantum Dynamics We consider a cavity–altered asymmetric hydrogen transfer reaction by extending a well studied reaction model Hamiltonian for TAA[31] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The light– matter hybrid system is described by an effective Pauli– Fierz Hamiltonian in length–gauge representation, cavity Born–Oppenheimer type and long–wavelength approxi- mations, which reads[32–34] ˆH = ˆHS + ˆHC + ˆHSC + ˆHDSE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (1) The first term resembles N non–interacting H–transfer systems, idealized here as a sum of one–dimensional Hamiltonians along a transfer coordinate, qi, for the i–th molecule with ˆHS = N � i=1 � − ℏ2 2µS ∂2 ∂q2 i + V (qi) � , (2) and corresponding reduced mass, µS, close to the hy- drogen mass (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' As shown in the SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='I, this Hamiltonian is a one–dimensional approximation ob- tained from a two–dimensional Hamiltonian developed originally in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The single–molecule potential, V (q) (where we suppress the index i here, as all poten- tials are identical), constitutes an asymmetric double– well potential with a global minimum at q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='572 a0, which relates to the enol (OH) configuration, and a local minimum at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='947 a0 corresponding to the enethiol (SH) configuration of TAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Both minima are charac- terized by classical over–the–barrier activation energies, ∆Ecl OH = 1598 cm−1 and ∆Ecl SH = 1081 cm−1, with re- spect to the transition state located at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', the classical energy difference between the two isomers is 517 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' By diagonalizing ˆHS for the single–molecule case with N = 1, the two energetically lowest lying eigenstates are found to correspond to the ground state enol configuration, ψ0(q) = ψOH(q), and the first excited state, enethiol configuration, ψ1(q) = ψSH(q), with en- ergies, ε0 = 966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3 cm−1 and ε0 = 1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='8 cm−1, respec- tively, giving a corresponding quantum mechanical en- ergy difference of ∆ε10 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Details on the transfer potential, V (q), with all numerical parameters are provided in the SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (1) describes the cavity Hamil- tonian, which we restrict here to a single mode, given in coordinate representation as ˆHC = −ℏ2 2 ∂2 ∂x2c + ω2 c 2 x2 c , (3) with cavity displacement coordinate, xc (of dimension mass1/2× length), and harmonic cavity frequency, ωc, chosen to be resonant with the lowest vibrational tran- sition of the transfer system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', ℏωc = ∆ε10 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, the light–matter interaction term, ˆHSC, is given by ˆHSC = � 2ωc ℏ gN d(q) xc , (4) with collective transfer coordinate, q = (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' While going well beyond Dicke–Hamiltonians, the collective light–matter interaction constant, gN, is still chosen to be of Dicke form[27], gN = g √ N , where g is a single–molecule coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The collective dipole moment is given by, d(q) = �N i d(qi), with ground state dipole function, d(qi), for the i–th H–transfer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter is a lin- ear approximation to the dipole function given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [31] and reads d(qi) = d0 + dS (qi − q0) , (5) with parameters d0, dS and q0 given in the SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We note that the dipole moment takes positive values at both the enol minimum (dOH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='678 ea0) and the enethiol minimum (dSH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='482 ea0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', d(qi) changes smoothly without sign change along the (classical) reac- tion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Note further that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (4), we assume the dipole moment of each molecule to be aligned with both the respective H–transfer coordinate and the polarization axis of the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The single–molecule coupling 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (a) Schematic setup for thioacetylacetone (TAA) isomerization in a single–mode cavity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Indicated are the enol (with OH group, O in red) and enethiol configurations (with SH group, S in yellow), other molecules in the ensemble, and the cavity mode between two parallel mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' b) Ground state density, |Ψ0(q, xc)|2, corresponding mainly to the enol configuration and c) first excited state density, |Ψ1(q, xc)|2, corresponding mainly to the enethiol configuration, both embedded in a two–dimensional cPES, V (q, xc) (contours in cm−1), which arises from a cavity–hydrogen–transfer model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The single–molecule cPES is a function of the H–transfer coordinate, q, and a cavity displacement coordinate, xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The uncoupled case with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 is shown with lowest–state energies, E0 = ε0 + ℏωc 2 , and, E1 = ε1 + ℏωc 2 , where ε0 = 966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3 cm−1 and ε1 = 1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='8 cm−1 are the lowest two eigenenergies of the one–dimensional H–transfer Hamiltonian, ˆHS, with potential, V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We set the cavity frequency to be resonant with the lowest molecular vibrational transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', ℏωc = ∆ε10 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' constant, g, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (4) has dimension of an electric field strength and is modeled here as, g = ℏωc d10 η[34], where, d10 = ⟨ψ0|d(q)|ψ1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='042 ea0, is the fundamental tran- sition dipole moment of the H–transfer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, η is a dimensionless function of the effective cavity vol- ume and the dielectric constant within the cavity [32–34], but treated here as a variable parameter chosen between η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 (no molecule–cavity coupling) and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The vibrational strong coupling (VSC) regime is determined by 0 < η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [35] Note, we do not refer to alterna- tive definitions of VSC related to dissipation strengths here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [30] Finally, the dipole self–energy (DSE) reads ˆHDSE = g2 N ℏωc N � i=1 d2(qi) + g2 N ℏωc N � i̸=j d(qi) d(qj) , (6) containing both diagonal and off–diagonal contributions, where the latter couples all N H–transfer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For the N–molecule plus one–cavity mode system, an N +1– dimensional cavity potential energy surface (cPES) can be defined as Vη(q, xc) = V (q) + ω2 c 2 � xc + � ℏ 2ω3c gN d(q) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (7) For the uncoupled, single–molecule case (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0, N = 1), the cPES, V (q, xc), is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1b and 1c su- perimposed with probability densities of the two low- est eigenfunctions, |Ψ0(q, xc)|2 and |Ψ1(q, xc)|2 obtained from diagonalizing the corresponding total Hamiltonian ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' These states are simple product states for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 and therefore also correspond to the enol and enethiol forms, weakly delocalized with small contributions at the other minimum as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For N molecules, in our fully quantum mechanical ap- proach, the time–evolution of the light-matter hybrid sys- tem is governed by an N +1–dimensional time-dependent Schrödinger equation (TDSE) iℏ ∂ ∂tΨ(q, xc, t) = ˆH Ψ(q, xc, t) , (8) which we solve numerically by means of the mul- ticonfigurational time–dependent Hartree (MCTDH) approach[36, 37] and its multilayer (ML–MCTDH) extension[38–40] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' II for details) as imple- mented in the Heidelberg MCTDH package[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' More- over, we consider as initial state Ψ0(q, xc) = � N � i=1 ψOH(qi) � � �� � =ψN(q) ϕ0(xc) , (9) with ground state enol configuration, ψOH(qi), for the i– th H–transfer system and cavity vacuum state, ϕ0(xc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter corresponds to the ground state of the bare cavity with zero physical photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We will discuss differences to photon number expectation values for a molecule–cavity system at finite light–matter interaction strength (η > 0) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Finally, we note that Ψ0(q, xc) turns out to be not a good approximation to the vibra- tional polariton ground state under VSC in the herein discussed asymmetric transfer model, but leads to rich dynamics from where also H–transfer rates can be deter- mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A further analysis of the initial state is given in the SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 3000 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 lao 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='08 7500 0009 1500 4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 400 200 0 200 400 Xc/ymeao4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Observables We describe the time–evolution of the light–matter hy- brid system by a time–dependent ensemble transfer prob- ability from the enol form (the more stable configura- tion of the free molecule or the molecule in the cavity at η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0) to the enethiol form, which we define as P ens SH (t) = � Ψ(t) ���� 1 N N � i=1 θ(qi) ����Ψ(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (10) Here, θ(qi), is a Heaviside step function indicating a di- viding surface located at the transition state of the in- dividual transfer potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Due to the bound nature of the cPES, the transfer dynamics is subject to recross- ing events at the dividing–surface, where we characterize the first recurrence by a recurrence time, τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter allows us to introduce the notion of short–time dynam- ics for times, t ≤ τr, and subsequently the extraction of approximate short–time transfer rates from enol– to enethiol–configurations as kens SH = d dtP ens SH (t) ���� t=tmax , (11) where tmax maximizes kens SH for t0 < tmax < τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, time–dependent coordinate expectation values ⟨R⟩ (t) = ⟨Ψ(t)|R|Ψ(t)⟩ , R = q, xc , (12) provide a complementary perspective on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In order to address cavity–induced collective quantum ef- fects, we additionally study entanglement in the strongly coupled light–matter hybrid system via von Neumann– entropies Si(t) = −kB tr{ˆρi(t) ln ˆρi(t)} ≥ 0 , (13) with Boltzmann constant, kB, and reduced density oper- ators, ˆρi(t), for an individual transfer mode, i = q, or the cavity mode, i = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (13) holds only if the reduced system is in a pure state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', when the reduced subsystem is disentangled from the remaining degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Finally, we consider the photon number expectation value, ⟨ˆnc⟩, and its time evolution, which reads in length– gauge representation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' IV) ⟨ˆnc⟩ = 1 ℏωc � ⟨ ˆHC⟩ + ⟨ ˆHSC⟩ + ⟨ ˆHDSE⟩ � − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (14) In the non–interacting limit, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (14) reduces to, ⟨ˆnc⟩ = 1 ℏωc ⟨ ˆHC⟩ − 1 2 = nc, with nc physical photons, whereas nc = 0 for the herein studied cavity vac- uum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For non–zero light–matter interaction, the photon expectation value initially reads, ⟨ˆnc⟩0 = 1 ℏωc � ⟨ ˆHC⟩0 + ⟨ ˆHDSE⟩0 � − 1 2 > nc, due to a non–zero number of virtual photons generated by the strong inter- action of light and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [35] In particular, the number of virtual photons at t0 is directly determined by the DSE contribution and therefore ensemble size dependent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='(6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Note, the interaction term, ⟨ ˆHSC⟩0, does initially not contribute to ⟨ˆnc⟩0 but will become relevant throughout the time–evolution of the hybrid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Cavity–induced isomerization: Single molecule We start our discussion of cavity–induced isomeriza- tion for the asymmetric hydrogen transfer model in the single–molecule limit with N = 1 by solving the TDSE (8) for various coupling strengths, η, always using the same initial state (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='I, upper two lines, lists the cor- responding initial state energies, ⟨ ˆH⟩0, and correspond- ing photon number expectation values, ⟨ˆnc⟩0, for selected values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We observe an increase for both expectation values, where ⟨ ˆH⟩0 > ∆Ecl OH = 1598 cm−1 for relatively strong couplings of η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05 and ⟨ˆnc⟩0 > 0 correspond to virtual photons generated by the DSE term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='09 ⟨ ˆH⟩0 / cm−1 1098 1111 1218 1433 1755 2184 ⟨ˆnc⟩0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='74 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='80 τr/fs – 87 88 92 95 100 kSH/1011 s−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='39 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Initial state energies, ⟨ ˆH⟩0 = ⟨ ˆHS⟩0 + ⟨ ˆHC⟩0 + ⟨ ˆHDSE⟩0, photon number expectation values, ⟨ˆnc⟩0, first– recurrence times, τr, and short–time transfer rates, kSH, in the single–molecule limit for different light–matter inter- actions strengths, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In all cases, the same initial state, Ψ0(q, xc) = ψOH(q) ϕ0(xc), was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For η > 0, H–transfer converting the enol (OH) to the enethiol (SH) form takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2, where the transfer probability, PSH(t) is shown (2a), as well as the expectation value of the H–transfer coordinate, ⟨q⟩ (t) (2b), both as a function of time and for different values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Note, for the transfer probabil- ity, PSH(t), one initially finds PSH(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='11 due to the weakly delocalized nature of ψOH(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' As time evolves, PSH(t) increases for η > 0 in an oscillatory fashion, which indicates formation of the enethiol isomer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Os- cillatory signatures in PSH(t) represent recurrences with a period of 264 fs for η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='07, which resembles the cavity– mode energy, ℏωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For stronger coupling, the dynamics turns out to be less regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The transfer coordinate expectation value, ⟨q⟩ (t), closely resembles the transfer dynamics, with ⟨q⟩ < 0 indicating the enol and ⟨q⟩ > 0 the enethiol isomer (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' From closer inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2a, we can extract first– recurrence times, τr, and corresponding short–time trans- fer rates, kSH, for different values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' These are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='I, lower two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In the non–interacting limit 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Time–evolution of (a) single–molecule (N = 1) trans- fer probability, PSH(t), and (b) transfer coordinate expecta- tion value, ⟨q⟩ (t), with black dashed lines indicating the quan- tum mechanical expectation values, ⟨q⟩OH = ⟨Ψ0|q|Ψ0⟩ and ⟨q⟩SH = ⟨Ψ1|q|Ψ1⟩, respectively, for different light–matter in- teraction strengths, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0), we have kSH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', there is no pop- ulation transfer to the local enethiol minimum without coupling to the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In contrast, for η > 0 we find transfer rates, kSH ≈ 109 s−1 to 1011 s−1, which in- crease with η by nearly two orders of magnitude over the whole VSC regime between η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='01 to η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The increase of reaction probability/ transfer rate in a cavity for this particular system is in contrast to other systems, where a rate retardation has been found either experimentally[1] or theoretically[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' That cavities can also enhance reactivity is a probably less widespread phe- nomenon, however, this possibility has been discussed in recent experimental[13] and theoretical[23] work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In order to interpret the positive effect of the cav- ity on the early–time, single–molecule transfer proba- bility, PSH(t), for TAA, we analyze the properties of the underlying single–molecule cPES, Vη(q, xc), which guides the dynamics of the vibro–polaritonic wave packet, Ψ(q, xc, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3a and b, we show single–molecule cPES, Vη(q, xc), besides corresponding vibro–polaritonic ground state densities for different light–matter in- teraction strengths, with initial cavity displacement coordinate expectation value, ⟨xc⟩0 = 0, indicated by a red vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For η > 0, the cPES’s minima are symmetrically shifted to negative values of the cavity displacement coordinate, such that the cavity contri- bution of the initial wave packet naturally experiences an excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This is in contrast to a recently studied class of symmetric double well potentials, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', for the inversion of an NH3 molecule[25] or the ground state cPES of a cavity Shin–Metiu model[21, 23], which are asymmetrically distorted at finite light–matter inter- action due to an antisymmetric, sign–changing dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter leads to barrier broadening, valley narrowing and (classical) dynamical caging effects, which in consequence reduce isomerization probabilities[21, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The static cPES perspective for TAA translates into a time–evolution of ⟨xc⟩ (t) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We find the vibro–polaritonic wave packet to acquire a significant dy- namical component along the cavity displacement coordi- nate as time evolves due to the respective gradient on the cPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' ⟨xc⟩ (t) reveals coherent oscillations with period 264 fs reflecting ℏωc = ∆ε10 and amplitude increasing with η, which resembles the enhanced cPES distortion in terms of altered turning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Since the dynamics along cavity displacement and molecular transfer coordinates is naturally coupled via the interaction term, ˆHSC, we can interpret the isomerization as cavity–induced excitation along the transfer coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The corresponding energy transfer can be related to a virtual photon exchange be- tween cavity and transfer modes, as will be discussed in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Since the dynamics is strictly restricted to non–zero coupling strengths with η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0, the cavity can be interpreted as a “catalyst” in this model scenario – despite the classical barrier height is not affected[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We also note, the studied model system does not ex- hibit a “reactant resonance effect” as the local OH–/SH– stretching modes have frequencies, ωOH = 3264 cm−1 and ωSH = 2737 cm−1, which do not support localized bound states below the classical activation barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Cavity–induced isomerization: Molecular ensembles We now extend our study to an ensemble of N trans- fer systems coupled to a single cavity mode with initial state, Ψ0(q, xc), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In what follows, we set η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05 and concentrate on the influence of varying ensemble sizes N on the transfer process up to N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' At first, we discuss the time–evolution of ensemble trans- fer probabilities, P ens SH (t), for different ensemble sizes N as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' From the short–time dynamics, we extract an ensemble transfer rate, kens SH = 3 × 1012 s−1, which is found to be two orders of magnitude larger than 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Single–molecule cavity potential energy surface (cPES), Vη(q, xc), and vibro–polaritonic ground state densities, |Ψ0(q, xc)|2, for different light–matter interaction strengths, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 (a), and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='09 (b), with initial cavity displacement coordinate expectation value, ⟨xc⟩0 = 0 indicated by red vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (c) Time–evolution of cavity displacement coordinate expectation value, ⟨xc⟩ (t), for different light–matter interaction strengths, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Time–evolution of (a) ensemble transfer probabil- ity, P ens SH (t), and (b) normalized photon number expectation value, ⟨ˆnc⟩(t), as function of ensemble size N for light–matter interaction strength, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Single molecule properties (N = 1) as reference indicated by black–dashed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' the single molecule rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='46 × 1011 s−1), and nearly in- dependent of N for ensemble sizes studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' As time evolves, we observe an oscillatory evolution of P ens SH (t), which can be classified by three different “regimes”: (i) For N ≤ 6, the dynamics is dominated by a max- imal probability density transfer at around 500 fs and P ens SH (t) is modulated by a series of beats with varying amplitude and period of 264 fs corresponding to the cav- ity mode excitation energy of ℏωc = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (ii) For 10 ≤ N ≤ 20, two prominent maxima occur in P ens SH (t) at around 200 fs and 700 fs, with a significantly increased recurrence time of approximately 472 fs, which we will again address below in context to entanglement of the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (iii) Eventually, for an intermediate ensemble size with N = 8, the probability transfer is significantly reduced (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' orange graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4a) and no specific recurrence structure is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In order to provide an explanation for the ensemble transfer dynamics, we discuss a normalized photon num- ber expectation value ⟨ˆnc⟩(t) = ⟨ˆnc⟩ (t) ⟨ˆnc⟩ (t0) , (15) with, ⟨ˆnc⟩(t0) = 1, which allows us to address ensem- ble effects on the virtual photon transfer between cavity and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We note, due to the different contribu- tions to ⟨ˆnc⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (14), a strict assignment of virtual photons only to the cavity mode is in principle not pos- sible as both interaction and DSE term also contribute significantly to ⟨ˆnc⟩ (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The time–evolution of ⟨ˆnc⟩(t) for different N is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4b and we find ⟨ˆnc⟩(t) < 1 for all N (including N = 1) over the studied time–interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', virtual photons are transferred to the molecular en- semble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In particular, we observe ⟨ˆnc⟩(t) to qualitatively resemble the inverse dynamical trend in P ens SH (t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', virtual photon transfer to the molecular ensemble coin- cides with enhanced population transfer to the enethiol region (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Hence, virtual photons are not only exchanged with the transfer ensemble but virtually 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 14000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 oe/b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 400 200 0 200 400 Xc/ymeao3000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 oel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='08 7500 0009 a 1500 4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 400 200 0 200 400 Xc/Vmeao7 drive the cavity–induced isomerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We note, non– normalized expectation values, ⟨ˆnc⟩, significantly depend on the interaction regime,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', η (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Cavity–induced entanglement in ensembles In order to gain further insight, we address differences in entanglement between the single–molecule limit and the ensemble scenario, which allows us to formulate an interpretation for ensemble–enhanced isomerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We first realize, that in the ensemble scenario, an enhanced ensemble transfer rate cannot straightforwardly be ex- plained by the distortion of the (N + 1-dimensional) cPES, since the distortion in a single molecule–cavity subspace decreases due to the Dicke–type light-matter interaction used here, gN, as, gN ∼ 1 √ N , for increasing N (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='(7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [23] However, due to strong coupling be- tween light and matter constituents, entanglement effects are in contrast expected to shape the ensemble transfer dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We quantify entanglement by transfer and cavity von Neumann–entropies, Sq(t), and SC(t), as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (13) with time–evolution depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5a and b for different ensemble sizes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Initially, we have, Sq(t0) = SC(t0) = 0, independent of ensemble size due to the prod- uct form of the initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (9), which is by defi- nition disentangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' From a wave function perspective, build up of entanglement can therefore be interpreted in terms of an increase of the multiconfigurational charac- ter of the full vibro–polaritonic wave function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', as deviation from a disentangled product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In general, we find the time–evolution of both entropies to differ sig- nificantly due to the different nature of both subsystems and, SC(t) > Sq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Starting with the reduced transfer system perspective, we observe Sq(t) to increase faster with time for increas- ing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This indicates a faster build–up of subsystem entanglement or equivalently multiconfigurational char- acter in the vibro–polaritonic wave function with respect to H–transfer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, for N > 1, we observe an increase in Sq(t) to accompany the transfer process to the enethiol configuration as characterized by P ens SH (t) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', H–transfer relates to enhanced entan- glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, the transfer von Neumann–entropy reaches (local) minima, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', reduced transfer system en- tanglement, where extrema are observed in the ensem- ble transfer probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This observation is in line with above discussed dynamics of ⟨ˆnc⟩(t) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', virtual photon transfer drives isomerization, which nat- urally leads to a stronger entanglement (larger Sq(t)) between cavity and transfer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In contrast, sup- pressed transfer relates to small values of Sq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' More- over, for intermediate ensemble sizes of N = 8, Sq(t) differs significantly by exhibiting a nearly monotonic but slow increase with time, which is accompanied by sup- pressed virtual photon transfer and isomerization proba- bility (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4), in line with the previous argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Time–evolution of (a) transfer von Neumann–entropy, Sq(t), (b) cavity von Neumann–entropy, SC(t), and (c) cavity displacement coordinate expectation value, ⟨xc⟩ (t), as func- tion of ensemble size N for light–matter interaction strength, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Single molecule properties (N = 1) as reference indicated by black–dashed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Turning to the cavity mode, we find SC(t) to be char- acterized by an overall increase with time, which be- comes strongly pronounced for large N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=', the cav- ity mode becomes quickly strongly entangled with the transfer ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, the cavity von Neumann– 8 entropy mimics the vibro–polaritonic wave packet’s dy- namics along the cavity displacement coordinate as cap- tured by ⟨xc⟩ (t) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For small ensembles (N = 2), SC(t) oscillates with a period of 264 fs, re- covering the cavity mode frequency and is minimized for ⟨xc⟩ (t) reaching the initial position at xc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In con- trast, for large ensembles with N = 20, the cavity en- tropy exhibits minima characterized by two different time scales, which can be related to both maximal and min- imal displacements of the vibro–polaritonic wave packet along the displacement coordinate xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Eventually, we note two characteristics, which additionally indicate the complex nature of the ensemble dynamics: First, for in- creasing N the amplitude of ⟨xc⟩ (t) is damped as time evolves since the cavity mode is increasingly immersed in a bath of strongly coupled and highly anharmonic trans- fer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Second, as ensemble size N increases, the initial amplitude of ⟨xc⟩ (t) increases too with a maxi- mum around N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We attribute this transition to a change from a small transfer ensemble of multi–mode Rabi type to a “large” ensemble resembling a system– bath–type regime with a cavity mode subject to dissipa- tion on the time scale shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Naturally, this transition is equivalently observed in the transfer dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In summary, we attribute isomerization in the model studied here to be induced by virtual photon transfer, which maximizes time–dependent changes in transfer system entanglement quantified by Sq(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In particu- lar, ensemble–enhanced isomerization rates relate to a cavity–induced entanglement effect between transfer sys- tems, which is not explainable by classical cPES distor- tion arguments valid in the single–molecule limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ac- cordingly, the herein discussed cavity–induced ensemble transfer process can be interpreted as an inherently col- lective quantum mechanical effect, which in particular cannot be captured by scaled single–molecule models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this work, we studied the quantum dynamics of an entangled molecular ensemble for an asymmetric hydro- gen transfer model of thioacetylacetone (TAA) interact- ing with a single cavity mode under vibrational strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' An N–molecule form of the Pauli–Fierz Hamil- tonian was used beyond frequently adopted model Hamil- tonians such as the Tavis–Cummings or Dicke–models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A N +1–dimensional time–dependent Schrödinger equation was solved numerically by means of the MCTDH ansatz to follow the cavity–induced isomerization dynamics from enol to enethiol isomers of TAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' At finite light–matter interaction, the cavity acts as a “catalyst” by inducing population transfer to the enethiol isomer, which is energetically less favorable than the enol form of TAA outside the cavity or at vanishing cavity– molecule coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This process is identified to be driven by virtual photon transfer to the N–molecule subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We extract approximate short–time transfer rates, which span in the single–molecule limit two orders of magnitude for increasing light–matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In an entangled ensemble of transfer systems, with a collective Dicke–type light–matter coupling, a collectively enhanced transfer rate is observed following from an interplay of virtual photon–transfer and non–trivial entanglement dynamics between light and matter components of the hybrid sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We furthermore find non–trivial ensemble size de- pendence of the dynamics as N grows, which we attribute to a transition from a multi–mode Rabi type scenario to a system–bath–type regime, where the cavity mode is ef- fectively immersed in a bath of strongly coupled, anhar- monic transfer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Our study points at the highly non–trivial role of quantum effects in molecular ensem- ble models strongly interacting with a quantized cavity mode beyond scaled single–molecule dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We close by pointing out several possible extensions of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' First, we do not take into account dissipa- tive effects and decoherence due to leaky cavity modes or the presence of other molecular degrees of freedom, which will naturally influence the transfer probability and allow for a more rigorous definition of transfer rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' However, if we assume additional degrees of freedom to be only weakly coupled, which is relevant to ensure the VSC regime, the main findings of our work should qual- itatively remain the same for the herein studied time– interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, we did not take into account finite temperature effects, which can be assumed relevant due to the relatively low energy scale in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Finally, while we went well beyond the Dicke–model by using an ensemble formulation based on the Pauli–Fierz Hamil- tonian, we still assumed a Dicke–type coupling in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' It might be instructive to also lift the Dicke–type perspective for the molecule–cavity coupling to carefully discuss deviations and potentially emerging properties in vibro–polaritonic chemistry for less restricted coupling models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Along similar lines, orientational effects (due to rotation of molecules) on the molecule–cavity coupling in fluctuating molecular ensembles, their influence on the related entanglement dynamics as well as the inclusion of direct intermolecular interactions could be interesting milestones towards a realistic description of molecular ensembles in cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge fruitful discussions with Oliver Kühn (Rostock) and Foudhil Bouakline (Potsdam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2008/1-390540038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fischer acknowledges support by the International Max Planck Research School for Elementary Processes in Physical Chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 9 DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' CONFLICT OF INTEREST The authors 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Yuen-Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Xiong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Shegai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 156, 030401, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fregoni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Garcia-Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Feist, ACS Photonics 9, 1096, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [7] T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 73, 43, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' George, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Shalabney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Hutchison, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Genet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ebbesen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 6, 1027, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Thomas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' George, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Shalabney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Dryzhakov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Varma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Moran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chervy, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Zhong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' De- vaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Genet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Hutchison, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ebbesen, Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 55, 11462, (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Thomas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lethuillier-Karl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Nagarajan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} 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10635, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Thomas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Jayachandran, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lethuillier-Karl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Vergauwe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Nagarajan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Devaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Genet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Moran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ebbesen, Nanophotonics 9, 249, (2020).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 143, 16877 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Galego, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Climent, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Garcia-Vidal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Feist, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' X 9, 021057, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Campos-Gonzalez-Angulo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ribeiro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Yuen- Zhou, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 10, 4685, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Campos-Gonzalez-Angulo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Yuen-Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 152, 161101, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 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(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Vurgaftman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Simpkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Dunkelberger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 154, 191103, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [21] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Mandal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Huo, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 12, 1315, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 12, 9531, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Sun, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Vendrell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 13, 4441, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Lindoy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Mandal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Schachenmayer, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 5, 1, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Mandal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Taylor, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Weight, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Koessler, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Huo, chemRxiv:2022-g9lr7 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Doslić, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Sundermann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' González, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Mó, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Giraud-Girard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Kühn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 1, 1249, (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Flick, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Appel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ruggenthaler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Rubio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 154, 104311, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Kockum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Miranowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Meyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 134, 044135, (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [41] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Worth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Beck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Jäckle, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Meyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The MCTDH Package, Version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2, (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Meyer, Version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3 (2002), Version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='4 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Vendrell and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Meyer Version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 contains the ML-MCTDH algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' See http://mctdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='uni-hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Used versions: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04074v1 [quant-ph] 10 Jan 2023 Supplementary Information: Cavity–Catalyzed Hydrogen Transfer Dynamics in an Entangled Molecular Ensemble under Vibrational Strong Coupling Eric W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fischer∗ Theoretische Chemie, Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany Peter Saalfrank Theoretische Chemie, Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany and Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam-Golm, Germany I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' ONE–DIMENSIONAL HYDROGEN TRANSFER REACTION HAMILTONIAN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Reaction Potential and Minimum Energy Path We derive the one–dimensional H–transfer Hamiltonian, ˆHS (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (2) with N = 1 in the main text ), from a two– dimensional asymmetric H–transfer reaction Hamiltonian for thioacetylacetone (TAA) developed by Doslić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [1], which was constructed from ab initio electronic structure calculations and reads ˆHR = − ℏ2 2µS ∂2 ∂q2 − ℏ2 2µB ∂2 ∂Q2 + V (q, Q) , (I1) with a (H–transfer) reaction coordinate, q, a (collective) “heavy” mode coordinate, Q, and corresponding reduced masses, µS = 1914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='028 me and µB = 8622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='241 me, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [1] The two–dimensional molecular potential energy surface (PES), V (q, Q), is given by V (q, Q) = V (q) + µB ω2 B 2 (Q − λS(q))2 , (I2) with “heavy” mode frequency, ωB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0009728 Eh, and nonlinear coupling function, λS(q) = aS q2 +bS q3, determined by parameters, aS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='794 a−1 0 and bS = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2688 a−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The reaction path potential is described in terms of an adiabatic potential V (q) = 1 2 � V+(q) − � V 2 −(q) + 4 K2(q) � , (I3) where, V±(q) = V1(q)±V2(q), with diabatic harmonic PES, Vi(q), and non–adiabatic coupling function, K(q), defined as Vi(q) = µi ω2 i 2 (q − qi,0)2 + ∆i , K(q) = kc exp � −(q − qc)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I4) The harmonic potentials resemble the R–OH (V1(q)) and R–SH (V2(q)) configurations in TAA with correspond- ing harmonic frequencies, ωOH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='01487 Eh/ℏ and ωSH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='01247 Eh/ℏ, reduced masses, µOH = 1728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='46 me and µSH = 1781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='32 me, relative energy shifts, ∆OH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 Eh and ∆SH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='003583 Eh, as well as displacements, qOH,0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='7181 a0 and qSH,0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2094 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The coupling function, K(q), is determined by an amplitude, kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='15582 Eh, and a displacement, qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='2872 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [1] Further, a molecular dipole function (neglecting the vector character of the dipole moment) is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [1] as d(q, Q) = d0 + dS(q − q0) + dB(Q − λS(q)) + dSB(q − q0)(Q − λ(q)) , (I5) ∗ ericwfischer@posteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='de 2 with parameters, d0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='68 ea0, dS = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='129 ea0/a0, dB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='023 ea0/a0, dSB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='451 ea0/a2 0 and q0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='59 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In the present work, where the ensemble character of the isomerizing molecules is in the focus, an effective ap- proximate one–dimensional Hamiltonian, ˆHS, and corresponding dipole function, d(q), are constructed, which still resemble the main features of their two–dimensional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We derive the one–dimensional transfer Hamilto- nian by minimizing, V (q, Q), with respect to Q as ∂ ∂QV (q, Q) = 0 ⇔ Q0 = λ(q) , (I6) such that the transfer potential and the dipole function subsequently simplify to one–dimensional functions V (q, Q0) = V (q) , d(q, Q0) = d(q) = d0 + dS(q − q0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I7) The latter holds equivalently for an ensemble of N transfer ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In our study, we neglect the “heavy mode”, Q, which does not couple via a potential–like term to the transfer coordinate, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S1, we show V (q) and d(q), with the lowest two eigenfunctions, ψ0(q) = ψOH(q) (enol) and ψ1(q) = ψSH(q) (enethiol), indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The latter were obtained by diagonalizing ˆHS in terms of a Colbert–Miller discrete variable representation (DVR)[5] for the transfer coordinate with Nq = 121 grid points and q ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1] a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The corresponding eigenenergies are ε0 = 988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3 cm−1 and ε1 = 1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='8 cm−1 as stated in the main text with an energy difference of ∆ε10 = ε1 − ε0 = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (a) One–dimensional hydrogen–transfer reaction potential, V (q) (in black), with dipole function, d(q), and two lowest eigenstates, ψ0(q) = ψOH(q) and ψ1(q) = ψSH(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (b) Ground state, |ψ0(q, Q)|2 = |ψOH(q, Q)|2, and (c) first excited state densities, |ψ1(q, Q)|2 = |ψSH(q, Q)|2, of two–dimensional reaction Hamiltonian, ˆHR, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I1) embedded in two–dimensional molecular PES, V (q, Q), given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I2) with contours in cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For the two–dimensional Hamiltonian, ˆHR, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I1), we numerically obtain energies, ǫ0 = 1037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 cm−1 and ǫ1 = 1158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3 cm−1, for the ground and first excited states, respectively, with energy difference of ∆ǫ10 = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='8 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Here, were we again employed a Colbert–Miller DVR with transfer grid parameters equivalent to the one–dimensional case discussed above and “heavy” mode coordinate Q ∈ [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0] a0 with NQ = 61 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Eventually, classical activation energies are by construction equivalent for the one– and two–dimensional PES with ∆Ecl OH = 1598 cm−1 and ∆Ecl SH = 1081 cm−1 as stated in the main text, since V (q) is equivalent to the reaction potential along the minimum energy path on V (q, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Deviations from a Reaction Path Hamiltonian We discuss deviations of our approach from a reaction path Hamiltonian, which arises from a two–dimensional model and additionally involves kinetic energy couplings due to non–zero reaction path curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Miller, Handy and Adams[2] showed that a reaction path Hamiltonian of a two–dimensional system with mass–weighted, cartesian–like coordinates is given by ˆH(ˆps, s, ˆPs, Qs) = ˆp2 s 2 (1 + Qs κ(s))2 + V0(s) + ˆHvalley(s) , (I8) 6000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='04 lao 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='03 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 4500 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='500 3000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 Q/ao6000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='06 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='01 4500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0 Q/ao3 where the first two terms correspond to kinetic and potential energy contributions along the reaction coordinate, s, with conjugate momentum, ˆps, whereas the third term provides the “valley” Hamiltonian ˆHvalley(s) = ˆP 2 s 2 + ω(s)2 2 Q2 s , (I9) which accounts for a s–dependent “valley” mode perpendicular to the reaction path with frequency, ω(s), that couples to the reaction coordinate via the reaction path curvature, κ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For the hydrogen transfer system studied here, we have by construction, V0(s) = V (q, Q0) = V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In the following, we discuss deviations from ˆH(ˆps, s, ˆPs, Qs), which emerge when we approximate the reaction path contribution by the bare transfer Hamiltonian ˆHS = ˆp2 q 2 + V (q) = − ℏ2 2µS ∂2 ∂q2 + V (q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I10) This assumption is equivalent to approximately decoupling the “valley” Hamiltonian, ˆHvalley(s), from the reaction path contribution by assuming the reaction path curvature, κ(s), to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In order to access this condition, we discuss the curvature, κ(s), of the minimum energy or reaction path, s(q), which we introduce as parametric curve in the mass–weighted q–Q–plane[3] s(q) = (s1(q), s2(q))T = (√µS q, √µB Q0(q))T , (I11) with components, s1(q) and s2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Here, Q0(q) = λS(q), as derived above, which minimizes, V (q, Q), with respect to variations in the “heavy” mode coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' From s(q), which is parameterized in terms of the hydrogen–transfer coordinate, q, we obtain the corresponding curvature, κs(q), as[4] κs(q) = det (s′, s′′) ||s′||3 = |s′ 1s′′ 2 − s′′ 1s′ 2| [(s′ 1)2 + (s′ 2)2] 3 2 , (I12) with derivatives, s′ i = ∂ ∂q si(q) and s′′ i = ∂2 ∂q2 si(q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We like to emphasize, that κs(q) depends now on the hydrogen transfer coordinate, q, which parametrizes the reaction path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, for reaction path elements, s1(q) = √µS q and s2(q) = √µB λS(q), we find derivatives s′ 1 = √µS , s′′ 1 = 0 , s′ 2(q) = √µB � 2 aS q + 3 bS q2� , s′′ 2(q) = √µB (2 aS + 6 bS q) , (I13) which allow us to write the curvature explicitly as κs(q) = 2√µS µB |aS + 3 bS q| � µB q2 (2 aS + 3 bS q)2 + µS � 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (I14) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S2a, we show the two–dimensional molecular PES, V (q, Q), with reaction path, s(q), in red and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S2b the corresponding curvature, κs(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A kinetic separation of the reaction path from the “valley” coordinate is a good approximation, if ˆTs = ˆp2 s 2 (1 + Qs κs(q))2 ≈ ˆp2 q 2 = ˆTq , (I15) which holds for |Qs κs(q)| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' By taking into account the maximal curvature, κs(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='078 (√me a0)−1, at the transition state (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0), the coupling is solely determined by the “valley” coordinate’s magnitude, which can be traced back to the excitation of the two–dimensional transfer system along the “heavy” mode coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We shall estimate the latter by means of the classical turning points Q± s = ± � ℏ ωB (2v + 1) , (I16) of the harmonic “valley” potential with vibrational quantum number, v, and, ω(s) = ωB, at the transition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For the first excited state (v = 1), we find, |Q± s κs(q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='0)| ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Hence, already for the “heavy” mode being excited to the first excited state, which has to be expected during the transfer process, we observe a coupling to the reaction coordinate that is assumed to alter the transfer dynamics of the molecular isomerization model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' However, as the role of the “heavy” mode is not central for the cavity–induced isomerization dynamics, we consider our approximation to be qualitatively valid and sufficient to discuss entanglement–induced collective effects in the herein studied reactive vibro–polaritonic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' a) Contour plot of molecular PES, V (q, Q), in mass–weighted coordinates with reaction path, s(q), in red and colorbar in wave numbers (cm−1) and b) minimum energy path curvature, κs(q), parameterized by mass–weighted transfer coordinate, √µS q with maximum at the transition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' NUMERICAL DETAILS FOR QUANTUM DYNAMICS We solve the TDSE (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (8) in the main text) numerically by means of the multiconfigurational time–dependent Hartree (MCTDH) method and its multilayer extension (ML–MCTDH) and propagate up to final time tf = 1000 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We employ a Colbert–Miller DVR for transfer reaction coordinates, qi ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='1]a0, with Nq = 101 grid points and a harmonic oscillator (HO) DVR for the cavity mode with Nc = 101 grid points and xc ∈ [−561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='35, +561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='35]√me a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We treat ensembles up to N = 4 via the MCTDH method with single particle functions (SPFs), ns = nc = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' For ensembles with 4 < N ≤ 20, we employ the ML–MCTDH method with converged trees (max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' natural population ≤ 10−4) for all N as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We employ the same DVR with identical number of primitive basis functions as above independent of ensemble size, N, but N–dependent numbers of SPFs, as shown next to bonds in ML trees, due to different entanglement structure in the full vibro–polaritonic wave packet,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' VIBRO–POLARITONIC CHARACTER OF INITIAL STATES In addition to the information given in the main text, here we analyze the initial states used in there (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (9) in main text) with respect to their vibro–polaritonic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We consider contributions of vibro–polaritonic states to the initial state by means of infrared spectra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' σ(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' obtained from the autocorrelation function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' C(t) = ⟨Ψ⋆(t/2)|Ψ(t/2)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' of the vibro–polaritonic wave packet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' |Ψ(t)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' with initial state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' |Ψ(t0)⟩ = |Ψ0⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' as σ(ℏω) = A � 2tf 0 C(t) ei(E−E0)t/ℏdt = � p | ⟨Ψ0|Ψp⟩ |2 δ(E − (Ep − E0)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (III1) with constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' vibro-polaritonic eigenergies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' and corresponding eigenstates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' |Ψp⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' satisfying the time– independent Schrödinger equation � ˆHS + ˆHC + ˆHSC + ˆHDSE � |Ψp⟩ = Ep |Ψp⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (III2) Here, E0 corresponds to the ground state energy, which we obtained by means of imaginary time evolution of the light–matter hybrid system employing (ML)–MCTDH methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S4, we show σ(ℏω) for different ensemble sizes N and observe a significant number of states |Ψp⟩ contributing to the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' We attribute the substantial deviation from approximately harmonic vibro–polaritonic models, which commonly show clear signatures of lower and upper vibro–polaritonic states, to the strong anharmonicity of the hydrogen transfer potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A shift of the spectral envelope’s center to higher energies for increasing N results from ⟨ ˆH⟩0 increasing with N at fixed η due to contributions from both ˆHS and ˆHDSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Further, the number of states increases with N 6000 40 500 16000 14000 0 20 e 1500 12000 e w//b sn 0 0 10000 45 8000 20 - 4500 3000 6000 1500 3000 40 4000 2000 60 9000 4500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 0 150 100 50 0 50 100 150 O/yme ao VUB5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Multilayer trees for different ensemble sizes N with S1 to SN transfer systems and cavity mode C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Number of SPFs are shown next to bonds connecting circular nodes and number of primitive basis functions are shown next to bonds connecting circular and square nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' up to N = 8, which is accompanied by intensity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Around N = 8, the number of vibro–polaritonic states contributing to the initial state decrease significantly and increase in the following for large ensembles up to N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Infrared spectra, σ(ℏω), for different ensembles sizes N with (a) N = 1 to N = 8 and (b) N = 8 to N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Peak widths result from finite propagation time of tf = 1000 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Note, energy scales for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S4a and S4b are different due to different ensemble sizes N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' PHOTON NUMBER OPERATOR IN LENGTH GAUGE REPRESENTATION In the main text, the expectation value of the photon number operators was used to analyze the cavity–induced H–transfer dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The common photon number operator, ˆnc = ˆa† cˆac, can be written in terms of the single–mode cavity Hamiltonian, ˆHC = ℏωc � ˆa† cˆac + 1 2 � , as ˆnc = ˆa† cˆac = 1 ℏωc ˆHC − 1 2 , (IV1) where, ˆa† c and ˆac are bosonic photon creation and annihilation operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' However, in length gauge representation, ˆnc, takes the form[6–9] S† U† ˆa† cˆac U S = 1 ℏωc � S† U† ˆHC U S � − 1 2 , (IV2) with unitary operator, U = exp � i ℏ ˆA d(q) � , mediating the Power–Zienau–Woolley (PZW) transformation, which is de- termined by the molecular dipole moment, d(q), and the transverse (single–mode) vector potential, ˆA = g ωc � ˆa† c + ˆac � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' A second unitary rotation mediated by, S = exp � i π 2 ˆa† cˆac � , acts exclusively on the cavity mode subspace and provides a real light–matter interaction term, ˆHSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Under U and S, photon creation and annihilation operators transform as S† U† ˆa† c U S = −i ˆa† c − i ℏ g ωc d(q) , S† U† ˆa U S = +i ˆac + i ℏ g ωc d(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (IV3) Employing the latter identities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' the transformed number operator turns with ℏωc � S† U† ˆa† cˆac U S � = ℏωc � −i ˆa† c − i ℏ g ωc d(q) � � +i ˆac + i ℏ g ωc d(q) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (IV4) = ℏωc � ˆa† cˆac + g d(q) ℏωc � ˆa† c + ˆac � + g2 (ℏωc)2 d2(q) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (IV5) = ℏωc ˆa† cˆac + g d(q) � ˆa† c + ˆac � + g2 ℏωc d2(q) (IV6) 7 as well as identities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' � ℏ 2ωc � ˆa† c + ˆac � = xc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' i � ℏωc 2 � ˆa† c − ˆac � = ˆpc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' into S† U† ˆa† cˆac U S = 1 ℏωc \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed ˆp2 c 2 + ω2 c 2 x2 c � �� � = ˆ HC + � 2ωc ℏ g d(q) xc � �� � = ˆ HSC + g2 ℏωc d2(q) � �� � = ˆ HDSE \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (IV7) = 1 ℏωc � ˆHC + ˆHSC + ˆHDSE � − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (IV8) This latter expression enters the cavity photon number expectation value, ⟨ˆnc⟩, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' (14) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' The same arguments generalize to ˆnc for ensembles of N molecules as the unitary operator mediating the PZW transformation, UN = �N i Ui, factorizes due the form of the ensemble dipole function, d(q) = �N i d(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='S5, we eventually FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Time-evolution of photon number expectation value, ⟨ˆnc⟩ (t), as function of ensemble size N for light-matter interaction strength, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' provide the time–evolution of the bare photon number operator expectation value without normalization to the initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Doslić, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Sundermann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' González, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Mó, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Giraud-Girard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Kühn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Adams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 72, 99, (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Fischer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Anders, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Saalfrank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 156, 154305, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} +page_content=' 154, 104311 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf'} diff --git a/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/2301.02500v1.pdf.txt b/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/2301.02500v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7139720854c45c3dbee6303ec138705f235380b1 --- /dev/null +++ b/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/2301.02500v1.pdf.txt @@ -0,0 +1,1047 @@ +arXiv:2301.02500v1 [quant-ph] 6 Jan 2023 +Violation of Diagonal Non-Invasiveness: A Hallmark of Quantum Memory Effects +Adri´an A. Budini +Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), +Centro At´omico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, +Argentina, and Universidad Tecnol´ogica Nacional (UTN-FRBA), +Fanny Newbery 111, (8400) Bariloche, Argentina +(Dated: January 9, 2023) +An operational (measurement based) scheme that connects in a univocal way measurement in- +vasivity and the presence of memory effects is defined. Its underlying theoretical basis relies on a +non-invasive measurability of (memoryless) Markovian dynamics when the corresponding observ- +able is diagonal in the same basis as the system density matrix. In contrast, (operational defined) +quantum memory effects always lead to violation of diagonal non-invasiveness. Related conditions +for violation of Leggett-Garg inequality due to non-Markovian memory effects are also established. +Several well understood physical principles allow dis- +tinguishing classical from quantum realms. +For exam- +ple, the principle of locality, valid in a classical regime, +is violated in presence of quantum entanglement. This +property is captured by Bell inequality (BI), which in- +troduces a severe constraint on the (measurement) cor- +relations that can be established between two spatially- +separated systems [1, 2]. Non-invasive measurability is +another principle that distinguish classical and quantum +systems. In fact, a quantum measurement process leads +in general to an unavoidable modification of the system +state. Added to macroscopic realism, this feature is the +basis of Leggett-Garg inequality (LGI) [3, 4]. It defines +a constraint on the correlations that can be established +between measurements performed over a single system +at two different times. Similarly to BI, a broad class of +theoretical an experimental related results were analyzed +and proposed in the literature [5–24], being from different +perspectives a topic of current interest [25–30]. +Both BI and LGI assume a similar structure after map- +ping the distance between the measured systems in the +former case with the time-interval between measurements +in the last case. For this reason LGIs are also termed as +“temporal Bell Inequalities” [4, 5]. Nevertheless, there +is an intrinsic unavoidable difference. Temporal correla- +tions can only be studied after introducing a non-trivial +(non-vanishing) system dynamics. +In fact, discussion +about the influence of the (open) system time-evolution +can be found in the original LGI presentation [3]. +From a simplified point of view, one can affirm that +both closed and open quantum systems are always al- +tered by a measurement process, leading to LGI viola- +tion. This kind of simplification contrast with the strong +advancements achieved in the last years in the classifica- +tion and understanding of open quantum dynamics [31– +33]. In particular, the usual association between memory +effects and time-convoluted contributions in the density +matrix evolution was abandoned. Instead, memory ef- +fects are defined from two complementary perspectives. +In non-operational approaches [34–42], memory effects +are related to departures of the system time-evolution +from a (memoryless) Markovian Lindblad master equa- +tion [31, 32]. +Alternatively, in operational approaches +the system is subjected to a set of measurement pro- +cesses [43–53]. Thus, non-Markovianity is defined in a +standard probabilistic way [54] from the corresponding +outcome statistics. +Given the previous advancements in the characteriza- +tion of open system dynamics, it is compelling to find out +if there exist any general relation between measurement +invasivity and the presence of memory effects. +Stud- +ies along these lines were performed previously [55–58]. +Nevertheless, not any clear boundary in the properties +of measurement invasivity seems to be defined by the +presence or absence of memory effects in the system dy- +namics, independently of which definition is taken. In +fact, the possible relations turn up to be strongly model- +dependent. Hence, a criteria that allow relating measure- +ment invasivity with the properties of the system dynam- +ics, Markovian or non-Markovian, is still lacking. +The aim of this work is to establishing a clear and rig- +orous general relation between measurement invasivity +and the presence of memory effects. It provides a fun- +damental step forward in the understanding and char- +acterization of measurement processes in open quantum +system dynamics. The main theoretical ingredient relies +on a diagonal non-invasiveness (DNI) of Markovian dy- +namics, which applies when the measurement observable +commutates with the (pre-measurement) system density +matrix. Hence, the observable and system state are diag- +onal in the same basis of states. In contrast, memory ef- +fects, as defined in operational approaches [43, 44], break +the previous condition. An operational scheme, based on +performing three consecutive system measurement pro- +cesses, allow to witnessing these properties. In addition, +these results enable us to establish under which condi- +tions violations of LGI can be related univocally to the +presence of memory effects. +Measurement Invasivity.—In operational approaches, +memory effects can be witnessed with (a minimum of) +three consecutive system measurement processes. Conse- +quently, measurement invasivity is defined from the same + +2 +(operational) basis. +The measurements are performed +at times 0, t, and t + τ, delivering correspondingly the +outcomes {x}, {y}, and {z}. Their joint probability is +denoted as P3(z, y, x), where the subindex indicates the +number of performed measurements. Margination over +the intermediate measurement outcomes lead to +P3(z, x) ≡ +� +y P3(z, y, x). +(1) +Alternatively, one can perform only two measurements +at times 0 and t + τ, which defines the joint probability +P2(z, x). For quantum systems, measurement invasivity +implies that P2(z, x) ̸= P3(z, x). In order to quantify this +disagreement, we use a Kolmogorov (trace) distance [59] +I ≡ +� +zx |P3(z, x) − P2(z, x)|. +(2) +For classical systems I = 0, while I > 0 is a direct wit- +ness of measurement invasiveness. This property is valid +independently of the system dynamics, closed or open, +Markovian or non-Markovian. In the following analysis, +the quantifier I is studied by assuming different underly- +ing system-environment (s-e) dynamics. +Markovian dynamics.—Similarly to closed systems, a +Markovian dynamics is defined by a density matrix prop- +agator Λt,t′ that is completely independent of the system +or environment initial states [43, 44]. Hence, in the case +of two measurement processes it is simple to obtain +P2(z, x) +P1(x) += Trs(EzΛt+τ,0[ρx]). +(3) +Similarly, when performing three measurements it follows +P3(z, y, x) +P1(x) += Trs(EzΛt+τ,t[ρy]) Trs(EyΛt,0[ρx]). +(4) +In both cases, P1(x) = Trs(Exρ0), where ρ0 is the ini- +tial system state. Tr(•) is the trace operation. {Em} +and {ρm} (m = x, y, z) are the (positive) measurement +operators and system post-measurement states respec- +tively [32, 59]. For Hermitian observables, both {Em} +and {ρm} are the projectors associated to each observable +spectral representation. Notice that the Markov property +P3(z, y, x) = P3(z|y)P2(y|x)P1(x) is fulfilled. P(b|a) de- +notes the conditional probability of b given a. +From +Eq. +(3) +and +(4) +it +follows +that +in +gen- +eral I +̸= 0. In fact, measurement invasivity applies +when the system dynamics is Markovian. +Neverthe- +less, given that the measurements are arbitrary ones, +the intermediate y-measurement can be chosen such +that [ρt|x, Ey] += +0, where ρt|x +≡ +Λt,0[ρx]. Thus, +{Trs(EyΛt,0[ρx])} can be read as the eigenvalues of ρt|x +implying � +y ρyTrs(EyΛt,0[ρx]) = ρt|x. Consequently, +ID +M= 0. +(5) +This property defines the DNI of Markovian dynamics: +a measurement process at a given time is non-invasive if +the corresponding observable commutates with the pre- +measurement system density matrix. In the three mea- +surement scheme DNI at (any) time t is valid for arbi- +trary x- and z-measurements. These are central results +for the developed scheme. They are violated in presence +of memory effects. +Stochastic Hamiltonian models.—Here the open sys- +tem is driven by random fluctuations such that Λt,t′ = +Λst +t,t′, where Λst +t,t′ is the stochastic propagator for each +noise realization while the overline denotes the corre- +sponding average. In similarity with Eq. (3) it follows +P2(z, x) +P1(x) += Trs(EzΛst +t+τ,0[ρx]), +(6) +while in similarity with Eq. (4), +P3(z, y, x) +P1(x) += Trs(EzΛst +t+τ,t[ρy]) Trs(EyΛst +t,0[ρx]). +(7) +Hence, DNI is not valid in general. It is recovered when +the average defining P3(z, y, x) split in two terms as in +Eq. (4). This property is only valid for white noise fluctu- +ations [54], that is, when the dynamics is Markovian [44]. +Completely positive system-environment dynamics.— +We consider an open dynamics where the system (s) and +its environment (e) obey a completely positive dynamics. +Taking separable initial conditions, given the (bipartite) +propagator Gt,t′, it follows +P2(z, x) +P1(x) += Trse(EzGt+τ,0[ρx ⊗ σ0]). +(8) +where σ0 is the environment initial state. Furthermore, +P3(z, y, x) +P1(x) += Trse(EzGt+τ,t[ρy ⊗ Trs(EyGt,0[ρx ⊗ σ0])]). +(9) +From these expressions it follows that DNI is hardly sat- +isfied. Additionally, P3(z, y, x) ̸= P3(z|y)P2(y|x)P1(x). +For unitary s-e interactions, when a Born-Markov ap- +proximation applies [32], Gt,0[ρ0 ⊗ σ0] ≈ ρt ⊗ σ0, DNI +and Markovianity are simultaneously recovered. Com- +plementarily, memory effects lead to violation of DNI. +Alternatively, one can also consider non-unitary s-e (dis- +sipative) Lindblad dynamics [31, 32]. The same relation +between violation of DNI and non-Markovianity remains +valid. +From Eqs. (8) and (9), it is simple to check that even +in presence of memory effects (for example for environ- +ments of finite dimension) DNI is accidentally valid if the +bipartite (pre-measurement) state ρse +t|x ≡ Gt,0[ρx⊗σ0] has +(∀t) a null discord [60–62], ρse +t|x = � +c |ct⟩⟨ct|⊗σ(c) +t , where +{|ct⟩} is a (in general time-dependent) complete orthog- +onal basis of system states while {σ(c) +t } are bath states. +The intermediate y-measurement must be defined by the +projectors {|ct⟩⟨ct|}. This case is consistent with the re- +sults of Ref. [58], which relate classicality with a bipartite + +3 +state with vanishing discord. Nevertheless, only for very +specific interactions and particular s-e initial conditions +bipartite evolutions (unitary or non-unitary) lead to a +state with vanishing discord ∀t [63–65]. +Thus, for the +proposed continuous-in-time open system dynamics, the +relation between violation of DNI and memory effects is +always recovered by considering the arbitrariness of the +x- and z-measurements. +In fact, an underlying local- +in-time completely positive bipartite quantum dynamics +cannot lead to a vanishing discord state ∀t when consid- +ering arbitrary system initial conditions [63]. +Leggett-Garg inequality—By denoting with {Om} the +observable values associated to each outcome, m = +x, y, z, LGI reads [4] +− 3 ≤ ⟨OyOx⟩ + ⟨OzOy⟩ − ⟨OzOx⟩ ≤ 1. +(10) +This result is valid for dichotomic observables, Om = ±1. +The correlators, which are obtained by performing only +two measurements, are ⟨OjOk⟩ ≡ � +jk OjOkP2(j, k). +Eq. (10) can be derived from P2(y, x) = � +z P3(z, y, x), +and by assuming that P2(z, y) +LGI += +� +x P3(z, y, x), and +P2(z, x) +LGI += +� +y P3(z, y, x) [4]. +Due to measurement +invasiveness and independently of the system dynamics +(Markovian or non-Markovian), these equalities are not +valid in general. Nevertheless, from the present analysis +[DNI, Eq. (5)] we conclude that Markovian dynamics al- +ways obey LGI if at each pair of measurement processes +the observables commutate with the pre-measurement sys- +tem density matrix. On the other hand, even under this +election of measurement processes, non-Markovian dy- +namics may or may not obey LGI. In this sense, the dis- +tance I [Eq. (2)] provides a deeper characterization of +measurement invasivity. +Examples.—The previous results establish a univocal +relation between the presence of memory effects and vi- +olation of DNI. Here we study different s-e interactions +that lead to a dephasing non-Markovian dynamics. The +system is a qubit (two-level system) with basis of states +|±⟩. Its density matrix reads +ρt = +� +⟨+|ρ0|+⟩ +d(t)⟨+|ρ0|−⟩ +d(t)∗⟨−|ρ0|+⟩ +⟨−|ρ0|−⟩ +� +. +(11) +The populations remain constant while the coherences +are characterized by the decay function d(t). +In correspondence with the analyzed dynamics, the +first evolution is set by a stochastic Hamiltonian [66, 67] +dρst +t +dt += −iξ(t)[σz, ρst +t ]. +(12) +Here, σz is the z-Pauli matrix. The noise has a null aver- +age ξ(t) = 0 and correlation ξ(t)ξ(t′) = (γ/2τc) exp[−(t− +t′)/τc]. From the system density matrix ρt = ρst +t , the co- +herence decay reads d(t) = exp +� +−2γ(t − τc(1 − e−t/τc) +� +. +In the limit of a vanishing correlation time, τc/γ → 0, an +exponential (Markovian) decay is recovered. +A unitary s-e model is defined by a spin bath [68, 69], +dρse +t +dt += −ig +� +σz ⊗ +�n +j=1 σ(j) +z , ρse +t +� +. +(13) +Here, g is a coupling parameter while σ(j) +z +is the z- +Pauli matrix of the j environment spin. +The system +density matrix ρt = Tre[ρse +t ] follows by tracing the bi- +partite s-e state. Assuming that each bath spin begins +in an identity state, the system coherence decay reads +d(t) = [cos(2gt)]n. +A dissipative s-e model is set by a non-diagonal mul- +tipartite Lindblad equation [70, 71] +dρse +t +dt += +n +� +j,k=1 +Γjk(σ(j) +z ρse +t σ(k) +z +− 1 +2{σ(k) +z σ(j) +z , ρse +t }+), (14) +where Γjk = (γ − χ)δjk + χ(i)j−1(−i)k−1. When χ = 0, +each qubit obey a Markovian dephasing evolution with +rate γ. When χ ̸= 0 all subsystems are coupled to +each other, leading to the development of memory ef- +fects. Taking the first qubit as the system of interest, +the coherence decay reads d(t) = e−2γt[cos(2χt)]¯n, where +¯n = Int(n/2) is the integer part of n/2. This result re- +lies on assuming that all environment qubit subsystems +begin in a completely mixed state [71]. +For the three previous models it is possible to calcu- +late P3(z, y, x) in an exact analytical way, where z = ±1, +y = ±1, and x = ±1. We assume that three measurement +processes are performed successively in the Bloch direc- +tions ˆx − ˆn − ˆx, where the vector ˆn = ˆn(θ, φ) is defined +by polar angles (θ, φ). Using that d(t) = d(t)∗, we get +P3(z, y, x) +P1(x) += 1 +4[1 + yxf(t) + zyf(τ) + zxf(t, τ)], (15) +where f(t) = sin(θ) cos(φ)d(t), while +f(t, τ) = 1 +2 sin2(θ)[d(t + τ) + cos(2φ)d(t, τ)]. +(16) +The function d(t) is the coherence decay of each model, +while d(t, τ) differ in each case [72]. In all cases P3(z, y, x) +does not fulfill (in general) a Markovian property. +From Eqs. (1) and (15) it is easy to obtain P3(z, x) = +[1 + zxf(t, τ)]P1(x)/2. Given that the last measurement +is performed in the ˆx-Bloch direction, P2(z, x) can also +be obtained from Eq. (15) under the steps � +z, the re- +placements y → z, t → t + τ, and taking θ = π/2, φ = 0, +which deliver P2(z, x) = [1 + zxd(t + τ)]P1(x)/2. The +invasivity distance Eq. (2) therefore is +I = I(t, τ) = |f(t, τ) − d(t + τ)|. +(17) +This expression for I is valid for an arbitrary intermedi- +ate measurement defined by the angles (θ, φ). The DNI of +Markovian dynamics [Eq. (5)] is valid when this measure- +ment is performed in the same basis where ρt|x is diago- +nal. Given that the first measurement is performed in the + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.1 +0.2 +0.3 +t + +=0.3 + +=0.35 + +=0.4 + +=0.5 +I(t,t) + + +c +/ +=1/2 +t +I(t,t) + + +(d) +(c) +(b) +n=3 + +=0.3 + +=0.35 + +=0.4 + +=0.5 +gt/ +I(t,t) + +(a) +c +/ +=10 +-3 +/ +=0.3 +n=100 + +=0.3 + +=0.35 + +=0.4 + +=0.5 +t +I(t,t) + +FIG. 1: +Invasivity measure I(t, τ) at equal time-intervals +(τ += t) for different underlying open system dynamics, +where the measurements are performed in Bloch directions +ˆx− ˆn(θ, φ)− ˆx with φ = 0. (a) and (b) Stochastic Hamiltonian +model [Eq. (12)]. (c) Hamiltonian s-e model [Eq. (13)]. (d) +Dissipative model [Eq. (14)]. The parameters are indicated +in each plot. The DNI Bloch-direction is θ = π/2, φ = 0. +ˆx-Bloch direction, Eq. (11) implies ⟨±|ρt|x|±⟩ = 1/2 and +⟨±|ρt|x|∓⟩ = xd(t)/2. Defining Mˆı ≡ Trs[σiρt|x] where +σi are the Pauli matrixes, we get Mˆx = xd(t), while +Mˆy = Mˆz = 0. Hence, ρt|x is diagonal (∀t) in the ˆx-Bloch +direction [73]. Consequently, the intermediate observable +commutates with ρt|x when θ = π/2 and φ = 0. +In Fig. 1 we plot I(t, τ) for the different open dynamics. +The intermediate measurement is in the ˆz-ˆx Bloch plane: +φ = 0. Fig. 1(a) and (b) correspond to the stochastic +Hamiltonian evolution [Eq. (12)]. In (a) the parameters +approach a Markovian white noise limit, τc/γ ≃ 0 ⇒ +d(t) ≃ exp(−2γt) ⇒ P3(z, y, x) ≃ P3(z|y)P2(y|x)P1(x) +[Eq. (15)]. Invasiveness is clearly observed, I(t, τ) ̸= 0. +Nevertheless, when θ → π/2, the DNI of Markovian dy- +namics is corroborated I(t, τ) → 0. In contrast, in (b) +when τc/γ > 0, even when the intermediate observable +commutates with the system density matrix (θ = π/2 +and φ = 0), consistently with our results, due to the +presence of memory effects I(t, τ) does not vanish. +In Fig. 1(c) we plot I(t, τ) for the spin environment +model [Eq. (13)]. Given that the number of spins is fi- +nite all behaviors are periodic in time. Again, even when +the measurement and system state commutate (θ = π/2 +and φ = 0), DNI is violated, I(t, τ) > 0. The same re- +sult is valid for the dissipative model [Eq. (14)] when +χ/γ ̸= 0, Fig. 1(d). When χ/γ → 0, a Markovian regime +is approached. The behavior of I(t, τ) becomes indistin- +guishable from that shown in Fig. 1(a), corroborating the +DNI of Markovian dynamics in this alternative model. +Assuming that ρ0 is diagonal in the ˆx-Bloch direction, +the density matrix [Eq. (11)] remains diagonal in that +base. +Hence, violation of LGI due to memory effects +0 +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +-3 +-2 +-1 +0 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t + +c +/ +=1 + +c +/ +=0.5 + +c +/ +=0.15 + +c +/ +=10 +-3 +K(t,t) + + + n=3 + n=4 +gt/ +K(t,t) + + +(d) +(c) +(b) + +/ +=0.17 + +/ +=0.3 + +/ +=0.5 + +/ +=1 +t +K(t,t) + +(a) +n=100 + +/ +=0.17 + +/ +=0.3 + +/ +=0.5 + +/ +=1 +t +d(t) + +FIG. 2: +LGI parameter K(t, τ) ≡ d(t) + d(τ) − d(t + τ) +[Eq. (18)] for equal time-intervals. (a) Stochastic Hamilto- +nian model [Eq. (12)]. (b) Hamiltonian s-e model [Eq. (13)]. +(c) Dissipative model [Eq. (14)], while (d) shows the associ- +ated system coherence decay. The dotted line corresponds to +the Markovian limit d(t) = exp[−2γt]. +can be checked by choosing the three observables {Om} +(m = x, y, z) as diagonal in the same ˆx-direction. Given +that only two measurements are explicitly performed, for +the three studied s-e models, Eq. (10) (Om = m) can be +expressed in terms of the corresponding coherence decay, +− 3 ≤ d(t) + d(τ) − d(t + τ) ≤ 1. +(18) +In Fig. 2 we study the validity of this inequality. For +the stochastic Hamiltonian model (a), LGI is only valid +when a Markovian white noise limit is attained, that is +τc/γ = 0. For the unitary s-e model (b), given the ab- +sence of a Markovian limit (in probability), LGI is vio- +lated independently of the number of environment spins. +The dissipative model (c) presents a memory induced +transition. In fact, for χ/γ ≲ 0.17 LGI is valid, which +includes the Markovian case χ/γ = 0. Nevertheless, LGI +is violated for χ/γ ≳ 0.17. In (d) we show the corre- +sponding system coherence decay d(t). All of them are +quasi-monotonic. Thus, the memory induced transition, +as defined in the present operational approach [Eq. (15)], +does not relies on any revival in the coherence decay. +Conclusions.—A deep relation between measurement +invasivity and the presence of memory effects in open +quantum systems has been established. Based on an op- +erational (measurement based) approach, it was found +that non-Markovian dynamics are intrinsically modified +by a measurement process even when the corresponding +observable commutates with the system state. This vio- +lation of DNI disappears when the dynamic approaches +a memoryless Markovian regime. +A measure of the previous relation was introduced. +It relies on performing three system measurement pro- +cesses, where the observable of the intermediate one must + +5 +to commutate with the (pre-measurement) system den- +sity matrix. By taking into account the arbitrariness of +the first and last measurements, the invasiveness indica- +tor only vanishes in a Markovian regime. In stretched +relation, we found that LGI is always obeyed by Marko- +vian dynamics if the pairs of involved measurements com- +mutate with the system state. The studied models sup- +port the main conclusions. All of them can be imple- +mented in different experimental platforms. +The DNI +measurement-basis can be determine from standard to- +mographic techniques. The examples also allowed us to +demonstrate that pure memory effects can drive a tran- +sition in the validity of LGI. +Acknowledgments.—The author thanks M´onica M. +Guraya for a critical reading of the manuscript. +This +paper was supported by Consejo Nacional de Investiga- +ciones Cient´ıficas y T´ecnicas (CONICET), Argentina. +[1] J. S. 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Budini, Memory effects in multipartite systems +coupled by non-diagonal dephasing mechanisms, arXiv: +2209.00400 (2022). +[72] For the model Eq. (12), d(t, τ) = d(t + τ) exp[4γτc(1 − +e−t/τc)(1 − e−τ/τc)], for the model Eq. (13), d(t, τ) = +d(t−τ), and for the model Eq. (14), d(t, τ) = exp[−2γ(t+ +τ)] cos[2χ(t − τ)]¯n. +[73] K. Blum, +Density Matrix Theory and Applications, +(Plenum Press, New York, 1996). + diff --git a/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/load_file.txt b/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a6fe91e5ee77ee9b5d776cf1075e45138143bf0 --- /dev/null +++ b/X9E0T4oBgHgl3EQfmgGg/content/tmp_files/load_file.txt @@ -0,0 +1,936 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf,len=935 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='02500v1 [quant-ph] 6 Jan 2023 Violation of Diagonal Non-Invasiveness: A Hallmark of Quantum Memory Effects Adri´an A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Budini Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Centro At´omico Bariloche, Avenida E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Bustillo Km 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5, (8400) Bariloche, Argentina, and Universidad Tecnol´ogica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina (Dated: January 9, 2023) An operational (measurement based) scheme that connects in a univocal way measurement in- vasivity and the presence of memory effects is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Its underlying theoretical basis relies on a non-invasive measurability of (memoryless) Markovian dynamics when the corresponding observ- able is diagonal in the same basis as the system density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In contrast, (operational defined) quantum memory effects always lead to violation of diagonal non-invasiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Related conditions for violation of Leggett-Garg inequality due to non-Markovian memory effects are also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Several well understood physical principles allow dis- tinguishing classical from quantum realms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For exam- ple, the principle of locality, valid in a classical regime, is violated in presence of quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This property is captured by Bell inequality (BI), which in- troduces a severe constraint on the (measurement) cor- relations that can be established between two spatially- separated systems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Non-invasive measurability is another principle that distinguish classical and quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, a quantum measurement process leads in general to an unavoidable modification of the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Added to macroscopic realism, this feature is the basis of Leggett-Garg inequality (LGI) [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' It defines a constraint on the correlations that can be established between measurements performed over a single system at two different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Similarly to BI, a broad class of theoretical an experimental related results were analyzed and proposed in the literature [5–24], being from different perspectives a topic of current interest [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Both BI and LGI assume a similar structure after map- ping the distance between the measured systems in the former case with the time-interval between measurements in the last case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For this reason LGIs are also termed as “temporal Bell Inequalities” [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, there is an intrinsic unavoidable difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Temporal correla- tions can only be studied after introducing a non-trivial (non-vanishing) system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, discussion about the influence of the (open) system time-evolution can be found in the original LGI presentation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' From a simplified point of view, one can affirm that both closed and open quantum systems are always al- tered by a measurement process, leading to LGI viola- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This kind of simplification contrast with the strong advancements achieved in the last years in the classifica- tion and understanding of open quantum dynamics [31– 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In particular, the usual association between memory effects and time-convoluted contributions in the density matrix evolution was abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Instead, memory ef- fects are defined from two complementary perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In non-operational approaches [34–42], memory effects are related to departures of the system time-evolution from a (memoryless) Markovian Lindblad master equa- tion [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Alternatively, in operational approaches the system is subjected to a set of measurement pro- cesses [43–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Thus, non-Markovianity is defined in a standard probabilistic way [54] from the corresponding outcome statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Given the previous advancements in the characteriza- tion of open system dynamics, it is compelling to find out if there exist any general relation between measurement invasivity and the presence of memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Stud- ies along these lines were performed previously [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, not any clear boundary in the properties of measurement invasivity seems to be defined by the presence or absence of memory effects in the system dy- namics, independently of which definition is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, the possible relations turn up to be strongly model- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Hence, a criteria that allow relating measure- ment invasivity with the properties of the system dynam- ics, Markovian or non-Markovian, is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The aim of this work is to establishing a clear and rig- orous general relation between measurement invasivity and the presence of memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' It provides a fun- damental step forward in the understanding and char- acterization of measurement processes in open quantum system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The main theoretical ingredient relies on a diagonal non-invasiveness (DNI) of Markovian dy- namics, which applies when the measurement observable commutates with the (pre-measurement) system density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Hence, the observable and system state are diag- onal in the same basis of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In contrast, memory ef- fects, as defined in operational approaches [43, 44], break the previous condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' An operational scheme, based on performing three consecutive system measurement pro- cesses, allow to witnessing these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In addition, these results enable us to establish under which condi- tions violations of LGI can be related univocally to the presence of memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Measurement Invasivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—In operational approaches, memory effects can be witnessed with (a minimum of) three consecutive system measurement processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Conse- quently, measurement invasivity is defined from the same 2 (operational) basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The measurements are performed at times 0, t, and t + τ, delivering correspondingly the outcomes {x}, {y}, and {z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Their joint probability is denoted as P3(z, y, x), where the subindex indicates the number of performed measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Margination over the intermediate measurement outcomes lead to P3(z, x) ≡ � y P3(z, y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (1) Alternatively, one can perform only two measurements at times 0 and t + τ, which defines the joint probability P2(z, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For quantum systems, measurement invasivity implies that P2(z, x) ̸= P3(z, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In order to quantify this disagreement, we use a Kolmogorov (trace) distance [59] I ≡ � zx |P3(z, x) − P2(z, x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (2) For classical systems I = 0, while I > 0 is a direct wit- ness of measurement invasiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This property is valid independently of the system dynamics, closed or open, Markovian or non-Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In the following analysis, the quantifier I is studied by assuming different underly- ing system-environment (s-e) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—Similarly to closed systems, a Markovian dynamics is defined by a density matrix prop- agator Λt,t′ that is completely independent of the system or environment initial states [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Hence, in the case of two measurement processes it is simple to obtain P2(z, x) P1(x) = Trs(EzΛt+τ,0[ρx]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (3) Similarly, when performing three measurements it follows P3(z, y, x) P1(x) = Trs(EzΛt+τ,t[ρy]) Trs(EyΛt,0[ρx]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (4) In both cases, P1(x) = Trs(Exρ0), where ρ0 is the ini- tial system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Tr(•) is the trace operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' {Em} and {ρm} (m = x, y, z) are the (positive) measurement operators and system post-measurement states respec- tively [32, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For Hermitian observables, both {Em} and {ρm} are the projectors associated to each observable spectral representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Notice that the Markov property P3(z, y, x) = P3(z|y)P2(y|x)P1(x) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' P(b|a) de- notes the conditional probability of b given a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (3) and (4) it follows that in gen- eral I ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, measurement invasivity applies when the system dynamics is Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Neverthe- less, given that the measurements are arbitrary ones, the intermediate y-measurement can be chosen such that [ρt|x, Ey] = 0, where ρt|x ≡ Λt,0[ρx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Thus, {Trs(EyΛt,0[ρx])} can be read as the eigenvalues of ρt|x implying � y ρyTrs(EyΛt,0[ρx]) = ρt|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Consequently, ID M= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (5) This property defines the DNI of Markovian dynamics: a measurement process at a given time is non-invasive if the corresponding observable commutates with the pre- measurement system density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In the three mea- surement scheme DNI at (any) time t is valid for arbi- trary x- and z-measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' These are central results for the developed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' They are violated in presence of memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Stochastic Hamiltonian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—Here the open sys- tem is driven by random fluctuations such that Λt,t′ = Λst t,t′, where Λst t,t′ is the stochastic propagator for each noise realization while the overline denotes the corre- sponding average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In similarity with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (3) it follows P2(z, x) P1(x) = Trs(EzΛst t+τ,0[ρx]), (6) while in similarity with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (4), P3(z, y, x) P1(x) = Trs(EzΛst t+τ,t[ρy]) Trs(EyΛst t,0[ρx]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (7) Hence, DNI is not valid in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' It is recovered when the average defining P3(z, y, x) split in two terms as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This property is only valid for white noise fluctu- ations [54], that is, when the dynamics is Markovian [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Completely positive system-environment dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='— We consider an open dynamics where the system (s) and its environment (e) obey a completely positive dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Taking separable initial conditions, given the (bipartite) propagator Gt,t′, it follows P2(z, x) P1(x) = Trse(EzGt+τ,0[ρx ⊗ σ0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (8) where σ0 is the environment initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Furthermore, P3(z, y, x) P1(x) = Trse(EzGt+τ,t[ρy ⊗ Trs(EyGt,0[ρx ⊗ σ0])]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (9) From these expressions it follows that DNI is hardly sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Additionally, P3(z, y, x) ̸= P3(z|y)P2(y|x)P1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For unitary s-e interactions, when a Born-Markov ap- proximation applies [32], Gt,0[ρ0 ⊗ σ0] ≈ ρt ⊗ σ0, DNI and Markovianity are simultaneously recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Com- plementarily, memory effects lead to violation of DNI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Alternatively, one can also consider non-unitary s-e (dis- sipative) Lindblad dynamics [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The same relation between violation of DNI and non-Markovianity remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (8) and (9), it is simple to check that even in presence of memory effects (for example for environ- ments of finite dimension) DNI is accidentally valid if the bipartite (pre-measurement) state ρse t|x ≡ Gt,0[ρx⊗σ0] has (∀t) a null discord [60–62], ρse t|x = � c |ct⟩⟨ct|⊗σ(c) t , where {|ct⟩} is a (in general time-dependent) complete orthog- onal basis of system states while {σ(c) t } are bath states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The intermediate y-measurement must be defined by the projectors {|ct⟩⟨ct|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This case is consistent with the re- sults of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' [58], which relate classicality with a bipartite 3 state with vanishing discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, only for very specific interactions and particular s-e initial conditions bipartite evolutions (unitary or non-unitary) lead to a state with vanishing discord ∀t [63–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Thus, for the proposed continuous-in-time open system dynamics, the relation between violation of DNI and memory effects is always recovered by considering the arbitrariness of the x- and z-measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, an underlying local- in-time completely positive bipartite quantum dynamics cannot lead to a vanishing discord state ∀t when consid- ering arbitrary system initial conditions [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Leggett-Garg inequality—By denoting with {Om} the observable values associated to each outcome, m = x, y, z, LGI reads [4] − 3 ≤ ⟨OyOx⟩ + ⟨OzOy⟩ − ⟨OzOx⟩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (10) This result is valid for dichotomic observables, Om = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The correlators, which are obtained by performing only two measurements, are ⟨OjOk⟩ ≡ � jk OjOkP2(j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (10) can be derived from P2(y, x) = � z P3(z, y, x), and by assuming that P2(z, y) LGI = � x P3(z, y, x), and P2(z, x) LGI = � y P3(z, y, x) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Due to measurement invasiveness and independently of the system dynamics (Markovian or non-Markovian), these equalities are not valid in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, from the present analysis [DNI, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (5)] we conclude that Markovian dynamics al- ways obey LGI if at each pair of measurement processes the observables commutate with the pre-measurement sys- tem density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' On the other hand, even under this election of measurement processes, non-Markovian dy- namics may or may not obey LGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In this sense, the dis- tance I [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (2)] provides a deeper characterization of measurement invasivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—The previous results establish a univocal relation between the presence of memory effects and vi- olation of DNI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Here we study different s-e interactions that lead to a dephasing non-Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The system is a qubit (two-level system) with basis of states |±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Its density matrix reads ρt = � ⟨+|ρ0|+⟩ d(t)⟨+|ρ0|−⟩ d(t)∗⟨−|ρ0|+⟩ ⟨−|ρ0|−⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (11) The populations remain constant while the coherences are characterized by the decay function d(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In correspondence with the analyzed dynamics, the first evolution is set by a stochastic Hamiltonian [66, 67] dρst t dt = −iξ(t)[σz, ρst t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (12) Here, σz is the z-Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The noise has a null aver- age ξ(t) = 0 and correlation ξ(t)ξ(t′) = (γ/2τc) exp[−(t− t′)/τc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' From the system density matrix ρt = ρst t , the co- herence decay reads d(t) = exp � −2γ(t − τc(1 − e−t/τc) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In the limit of a vanishing correlation time, τc/γ → 0, an exponential (Markovian) decay is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' A unitary s-e model is defined by a spin bath [68, 69], dρse t dt = −ig � σz ⊗ �n j=1 σ(j) z , ρse t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (13) Here, g is a coupling parameter while σ(j) z is the z- Pauli matrix of the j environment spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The system density matrix ρt = Tre[ρse t ] follows by tracing the bi- partite s-e state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Assuming that each bath spin begins in an identity state, the system coherence decay reads d(t) = [cos(2gt)]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' A dissipative s-e model is set by a non-diagonal mul- tipartite Lindblad equation [70, 71] dρse t dt = n � j,k=1 Γjk(σ(j) z ρse t σ(k) z − 1 2{σ(k) z σ(j) z , ρse t }+), (14) where Γjk = (γ − χ)δjk + χ(i)j−1(−i)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' When χ = 0, each qubit obey a Markovian dephasing evolution with rate γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' When χ ̸= 0 all subsystems are coupled to each other, leading to the development of memory ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Taking the first qubit as the system of interest, the coherence decay reads d(t) = e−2γt[cos(2χt)]¯n, where ¯n = Int(n/2) is the integer part of n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This result re- lies on assuming that all environment qubit subsystems begin in a completely mixed state [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For the three previous models it is possible to calcu- late P3(z, y, x) in an exact analytical way, where z = ±1, y = ±1, and x = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' We assume that three measurement processes are performed successively in the Bloch direc- tions ˆx − ˆn − ˆx, where the vector ˆn = ˆn(θ, φ) is defined by polar angles (θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Using that d(t) = d(t)∗, we get P3(z, y, x) P1(x) = 1 4[1 + yxf(t) + zyf(τ) + zxf(t, τ)], (15) where f(t) = sin(θ) cos(φ)d(t), while f(t, τ) = 1 2 sin2(θ)[d(t + τ) + cos(2φ)d(t, τ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (16) The function d(t) is the coherence decay of each model, while d(t, τ) differ in each case [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In all cases P3(z, y, x) does not fulfill (in general) a Markovian property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (1) and (15) it is easy to obtain P3(z, x) = [1 + zxf(t, τ)]P1(x)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Given that the last measurement is performed in the ˆx-Bloch direction, P2(z, x) can also be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (15) under the steps � z, the re- placements y → z, t → t + τ, and taking θ = π/2, φ = 0, which deliver P2(z, x) = [1 + zxd(t + τ)]P1(x)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The invasivity distance Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (2) therefore is I = I(t, τ) = |f(t, τ) − d(t + τ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (17) This expression for I is valid for an arbitrary intermedi- ate measurement defined by the angles (θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The DNI of Markovian dynamics [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (5)] is valid when this measure- ment is performed in the same basis where ρt|x is diago- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Given that the first measurement is performed in the 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 t =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='35 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 I(t,t) c / =1/2 t I(t,t) (d) (c) (b) n=3 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='35 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 gt/ I(t,t) (a) c / =10 3 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 n=100 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='35 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 t I(t,t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1: Invasivity measure I(t, τ) at equal time-intervals (τ = t) for different underlying open system dynamics, where the measurements are performed in Bloch directions ˆx− ˆn(θ, φ)− ˆx with φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (a) and (b) Stochastic Hamiltonian model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (12)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (c) Hamiltonian s-e model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (d) Dissipative model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (14)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The parameters are indicated in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The DNI Bloch-direction is θ = π/2, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' ˆx-Bloch direction, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (11) implies ⟨±|ρt|x|±⟩ = 1/2 and ⟨±|ρt|x|∓⟩ = xd(t)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Defining Mˆı ≡ Trs[σiρt|x] where σi are the Pauli matrixes, we get Mˆx = xd(t), while Mˆy = Mˆz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Hence, ρt|x is diagonal (∀t) in the ˆx-Bloch direction [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Consequently, the intermediate observable commutates with ρt|x when θ = π/2 and φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1 we plot I(t, τ) for the different open dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The intermediate measurement is in the ˆz-ˆx Bloch plane: φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1(a) and (b) correspond to the stochastic Hamiltonian evolution [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (12)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In (a) the parameters approach a Markovian white noise limit, τc/γ ≃ 0 ⇒ d(t) ≃ exp(−2γt) ⇒ P3(z, y, x) ≃ P3(z|y)P2(y|x)P1(x) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (15)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Invasiveness is clearly observed, I(t, τ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, when θ → π/2, the DNI of Markovian dy- namics is corroborated I(t, τ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In contrast, in (b) when τc/γ > 0, even when the intermediate observable commutates with the system density matrix (θ = π/2 and φ = 0), consistently with our results, due to the presence of memory effects I(t, τ) does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1(c) we plot I(t, τ) for the spin environment model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Given that the number of spins is fi- nite all behaviors are periodic in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Again, even when the measurement and system state commutate (θ = π/2 and φ = 0), DNI is violated, I(t, τ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The same re- sult is valid for the dissipative model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (14)] when χ/γ ̸= 0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' When χ/γ → 0, a Markovian regime is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The behavior of I(t, τ) becomes indistin- guishable from that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 1(a), corroborating the DNI of Markovian dynamics in this alternative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Assuming that ρ0 is diagonal in the ˆx-Bloch direction, the density matrix [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (11)] remains diagonal in that base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Hence, violation of LGI due to memory effects 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='0 t c / =1 c / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 c / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='15 c / =10 3 K(t,t) n=3 n=4 gt/ K(t,t) (d) (c) (b) / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='17 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 / =1 t K(t,t) (a) n=100 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='17 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='3 / =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='5 / =1 t d(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 2: LGI parameter K(t, τ) ≡ d(t) + d(τ) − d(t + τ) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (18)] for equal time-intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (a) Stochastic Hamilto- nian model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (12)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (b) Hamiltonian s-e model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (c) Dissipative model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (14)], while (d) shows the associ- ated system coherence decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The dotted line corresponds to the Markovian limit d(t) = exp[−2γt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' can be checked by choosing the three observables {Om} (m = x, y, z) as diagonal in the same ˆx-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Given that only two measurements are explicitly performed, for the three studied s-e models, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (10) (Om = m) can be expressed in terms of the corresponding coherence decay, − 3 ≤ d(t) + d(τ) − d(t + τ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (18) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' 2 we study the validity of this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For the stochastic Hamiltonian model (a), LGI is only valid when a Markovian white noise limit is attained, that is τc/γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' For the unitary s-e model (b), given the ab- sence of a Markovian limit (in probability), LGI is vio- lated independently of the number of environment spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The dissipative model (c) presents a memory induced transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In fact, for χ/γ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='17 LGI is valid, which includes the Markovian case χ/γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Nevertheless, LGI is violated for χ/γ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In (d) we show the corre- sponding system coherence decay d(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' All of them are quasi-monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Thus, the memory induced transition, as defined in the present operational approach [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' (15)], does not relies on any revival in the coherence decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—A deep relation between measurement invasivity and the presence of memory effects in open quantum systems has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Based on an op- erational (measurement based) approach, it was found that non-Markovian dynamics are intrinsically modified by a measurement process even when the corresponding observable commutates with the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' This vio- lation of DNI disappears when the dynamic approaches a memoryless Markovian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' A measure of the previous relation was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' It relies on performing three system measurement pro- cesses, where the observable of the intermediate one must 5 to commutate with the (pre-measurement) system den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' By taking into account the arbitrariness of the first and last measurements, the invasiveness indica- tor only vanishes in a Markovian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' In stretched relation, we found that LGI is always obeyed by Marko- vian dynamics if the pairs of involved measurements com- mutate with the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The studied models sup- port the main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' All of them can be imple- mented in different experimental platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The DNI measurement-basis can be determine from standard to- mographic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' The examples also allowed us to demonstrate that pure memory effects can drive a tran- sition in the validity of LGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} +page_content='—The author thanks M´onica M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E0T4oBgHgl3EQfmgGg/content/2301.02500v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..281612ad978fda04237c6c9bc1d1b8732fb3a173 --- /dev/null +++ b/XdE1T4oBgHgl3EQfJQP0/content/tmp_files/2301.02951v1.pdf.txt @@ -0,0 +1,2133 @@ +arXiv:2301.02951v1 [math.NT] 8 Jan 2023 +Class Number of the Imaginary Quadratic +Field and Quadratic Residues Identities +Jorge Garcia +January 10, 2023 +Abstract +A formula for the sum of quadratic residues modulus a prime +p = 4n − 1 is studied. We relate some terms on this formula with +roots of quadratics and provide an exhaustive analysis of new con- +cepts based on these roots. A number of formulas for the sum of the +quadratic residues are obtained. We finalize the paper by obtaining +several identities involving h(−p) the class number of the imaginary +quadratic field Q(√−p). +1 +Introduction +Consider a prime p = 4n − 1 and 1 ≤ k ≤ p − 1. By rp (k2) , we denote +the remainder of k2 when we divide by p. We call this number rp (k2) the +quadratic residue of k2 modulus p. When we add all these residues we obtain +the sum of quadratic residues relative to the prime number p. There is a +complicated formula for such sum, +p−1 +� +k=1 +rp +� +k2� += +�p +2 +� +− p · h(−p), +(1.1) +where h(−p) is the class number of the imaginary quadratic field Q(√−p). +Important formulas for the class number when the prime is of the form p = +4n − 1 > 3 include Dirichlet class number formula +1 + +h(−p) += +√p +2π +∞ +� +r=1 +χ(r) +r +, +(1.2) +where χ is the Dirichlet character. +Formula 1.2 can be found in [2]. Another formula that involves the Kro- +necker symbol +� +p +r +� +can be found in [5] (Corollary 1 p. 428). Finally a formula +developed by Cohen [1] [Corollary 5.3.16] provides the class number too. +The main purpose of this paper is to obtain some identities for this num- +ber h(−p) by computing the quadratic residues in a different manner. +Here we provide a summary of the main formulas obtained in this paper. +Here Qk = +� +1 + √4kp + 12n − 7 +� +/2, M = ⌊n2/p⌋ and Jn is the number of +jumps (see Section 2). +M +� +k=1 +�� +kp +� +− +M−1 +� +k=0 +⌊Qk⌋ += +Jn − M − 1. +n +� +k=1 +�k2 − k + 2 − 3n +p +� ++ +M−1 +� +k=0 +⌊Qk⌋ += +(M − 1)n. +p−1 +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p−1 +� +k=1 +rp +� +k2� ++ 3n − 2. +2n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +2n +� +k=1 +rp +� +k2� ++ n. +n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +n +� +k=1 +rp +� +k2� ++ n(n + 1) +2 +− p(Jn − 1 − M). +p−1 +� +k=1 +�k2 − k + 2 − 3n +p +� += +p(p − 5) + 6 − n +3 +− 1 +p +p−1 +� +k=1 +rp +� +k2� +. +2n +� +k=1 +�k2 − k + 2 − 3n +p +� += +n(2n − 1)(4n − 7) +3p +− 1 +p +2n +� +k=1 +rp +� +k2� +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +(n + 1)(2n2 − 23n + 6) +3p +− 1 +p +n +� +k=1 +rp +� +k2� ++ Jn − M. +2 + +This is a summary of the identities involving the class number h = h(−p) +found on this paper. +Jn +2 + M(n − 1) += +h +4 + +M +� +k=1 +�� +kp +� ++ n2 − 5n + 9 +12 +. +−Jn +2 + Mn += +h +4 + +M−1 +� +k=0 +⌊Qk⌋ + n2 − 5n − 3 +12 +. +Jn +2 + +n−1 +� +k=1 +�k2 +p +� += +h +4 + n2 − 5n + 9 +12 +. +p−1 +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p · (2n − h) − n − 1. +2n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p · (2n − h) + 1 +2 +. +n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p +4(3n + 2 − 2Jn − h) − (n + 1)(n − 1) +4 +. +n +� +k=1 +rp +� +k2� += +p +4(2Jn + 2n − 3 − 4M − h) + n(n + 1) +4 +. +p−1 +� +k=1 +�k2 − k + 2 − 3n +p +� += +h + 16n2 − 35n + 15 +3 +. +2n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +2 + 4n2 − 14n + 9 +6 +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +4 + Jn +2 + n2 − 17n − 3 +12 +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +2 + M(1 − n) + n2 − 11n + 3 +6 ++ +M +� +k=1 +�� +kp +� +. +3 + +In Section 2 we provide the main definitions and the notation used in the +whole paper. Here, we develop the concepts of jump and total residue. We +also identify when these jumps occur. In Section 3 we organize the jumps on +six different sets and we state the main lemmas that will be used to count +the jumps which is done in Section 4. It is here where we define bijective +functions among different pairs of sets to compute their cardinalities. +In +Section 5 we obtain the sums of quadratic residues of terms of the form +k2 − k + 2 − 3n where k ranges on the different intervals [1, 4n − 2], [1, 2n] +or [1, n]. In this section we also count to total amount of jumps. Finally in +Section 6 we establish several identities involving the class number h(−p) +of the imaginary quadratic field Q(√−p). These identities are based on the +sums found in previous sections and in some of these identities the jumps +quantity appears. +Whereas in [4] we computed several sums of quadratic residues when +p = 4n + 1, in this paper, we perform a similar but different analysis when +p = 4n − 1. It is on this paper where the class number is involved. +On +some occasions we present both formulas (p = 4n − 1 and p = 4n + 1) for +comparison purposes. +2 +Sum of Quadratic Residues and Jumps +Consider p = 4n − 1 a prime number. It will be understood that when we +write n, we mean a natural number n ≥ 1. +Definition 2.1 Let q be a positive integer and x ∈ Z. By rq(x) we denote +the remainder of x when we divide by q. Hence rq(x) ∈ {0, 1, 2, ..., q − 1} +satisfies +x = m · q + rq(x), +for some m ∈ Z. Clearly, m = ⌊x/q⌋ . +The following notation found in [3] will be useful during the whole paper. +4 + +Notation 2.2 For m ∈ Z, p = 4n − 1 prime and m ≥ 0 we denote +Qm = 1 +2 + 1 +2 +� +1 + 4 [(m + 1)p − n − 1] = 1 +2 + 1 +2 +� +4mp + 3p − 4 , +Rm = √mp , +M = +�n2 +p +� +. +In [3], we obtained a theorem that involves the sum of quadratic residues +when p = 4n − 1. For reference purposes, we write here such theorem. +Theorem 2.3 Let p = 4n − 1 be prime. Using Notation 2.2 we have +1 +2 +p−1 +� +k=1 +rp +� +k2� += p +� M +� +m=1 +⌊Rm⌋ + +M−1 +� +m=0 +⌊Qm⌋ +� +− Mp(2n − 1) + p · (n2 + n) +6 +. +A concept arises naturally when we study the term Qm. This term is the +positive root of the quadratic polynomial x2 − x + 2 − 3n − mp which is the +same as (x − 1)2 + x + 1 − 3n − mp. +Definition 2.4 Let p = 4n − 1 be a prime and 0 ≤ k ≤ p. The total residue +of k is defined and denoted by +Γ(k) = rp +� +(k − 1)2� ++ rp (k + 1 − 3n) . +We also say that k is a jump if its total residue is p or more, i.e. if +Γ(k) ≥ p. +What is the importance of the jumps? Firstly, in the proof of Theorem 2.3 +(see [3]), a key technique is adding rp (k2) and rp ((2n − k)2) . It happens that +when k ∈ (Rm, Qm] this sum is constant, but as soon as k exceeds Qm, we +need to subtract p. Therefore knowing when we need to subtract p is key +to comprehend better the formula in Theorem 2.3. Secondly, the amount of +jumps in the interval [2, n+2] will allow us to establish a formula to compute +the terms �M−1 +m=0 ⌊Qm⌋ , �M +m=1 ⌊Rm⌋ as well as �n +k=0 rp (k2) as a function of +n, h(−p) and the number of jumps. This is achieved in Corollaries 6.2 and 6.4. +The following three lemmas allow us to identify some jumps and when +they occur. For notation purposes we define Z0 = {0, 1, 2, ...}. +5 + +Lemma 2.5 Let p = 4n − 1 be a prime and m ∈ Z0. Then +⌊Qm⌋ < Qm. +Proof. Assume there is j ∈ Z such that j = +� +1 + +� +4mp + 12n − 7 +� +/2. +Then j2 − j + 2 − 3n = mp. Taking x0 = 2n − j gives +x2 +0 = 4n2 − n + mp + j − 4nj + n + 3n − 2 = (n + m − j + 1)p − 1, +hence x2 +0 ≡ −1 (mod p) which contradicts Fermat Little Theorem as p = +4n − 1. +□ +Remark 2.6 By using the same argument, there is no integer k with k2 − +3k + 4 − 3n = mp, else taking k − 1 = j we obtain j2 − j + 2 − 3n = mp +which leads to a contradiction. +Lemma 2.7 Let p = 4n − 1 be a prime with n ≥ 3. Let m ∈ Z0 with +0 ≤ m ≤ +�n2 − 4n + 5 +p +� +and km = 1 + +�1 + √4mp + 12n − 7 +2 +� +. +Then +(i) +3 ≤ km ≤ n + 2. +(ii) +1 + √4mp + 12n − 7 < 2km < 3 + √4mp + 12n − 7. +(iii) km is a jump and +(iv) +3n − km − 1 < rp +� +(km − 1)2� += (km − 1)2 − mp ≤ 3n + km − 4. +Proof. +(i) Notice that 4mp + 12n − 7 ≤ 4n2 + 4n + 1, hence km ≤ n + 2. Since +2 ≤ n, 9 ≤ 4mp + 12n − 7. Therefore 3 ≤ km. +(ii) Notice that km is strictly above the positive root of x2−x+2−3n−mp, +hence +k2 +m − km + 2 − 3n > mp. +(2.3) +6 + +Now km ≤ n + 2 ≤ 3n and 2 ≤ n imply +(km−1)2−mp = k2 +m−km+2−3n−mp+3n−km−1 ≥ 3n−km. (2.4) +From Lemma 2.5, km < +� +3 + √4mp + 12n − 7 +� +/2. The other inequal- +ity is obvious. +(iii) From (ii), km is less than the positive root of x2 − 3x + 4 − mp − 3n, +hence k2 +m − 3km + 4 − mp − 3n ≤ −1. Therefore +(km − 1)2 − mp ≤ 3n + km − 4 ≤ 4n. +(2.5) +It is impossible that (km−1)2−mp = 4n, otherwise (km−1)2−1 = (m+ +1)p, hence p would divide (km − 2)km which forces km = 0 or km = 2, +which contradicts km ≥ 3. Hence (km − 1)2 − mp ≤ 4n − 1, however, +if (km − 1)2 − mp = 4n − 1, then km = 1 which is again, impossible. +Then (km − 1)2 − mp < p and hence rp ((km − 1)2) = (km − 1)2 − mp. +Now Γ(km) = rp ((km − 1)2) + rp (km + 1 − 3n) = (km − 1)2 − mp + +km + 1 − 3n − p as km ≤ n + 2 ≤ 3n − 2. Now Inequality 2.3 implies +Γ(km) = k2 +m − km + 2 − 3n − mp + p ≥ p, hence km is a jump. +(iv) To finish the proof, we observe that from Inequalities 2.4 and 2.5 we +obtain +3n − km ≤ rp +� +(km − 1)2� += (km − 1)2 − mp ≤ 3n + km − 4. +□ +The following lemma allows us to find more jumps based on the ones +found in Lemma 2.7. +Lemma 2.8 Consider km be the jump in Lemma 2.7 and k ≤ n + 2. If +km < k ≤ 1 + +�√mp + p − 1 +� +then k is a jump and +3n + k − 3 < rp +� +(k − 1)2� += (k − 1)2 − mp. +Proof. Clearly km < k implies mp ≤ (km − 1)2 < (k − 1)2. Since k ≤ +�√mp + p − 1 +� +, (k −1)2 ≤ mp+p−1. Hence rp ((k − 1)2) = (k −1)2 −mp. +7 + +Since k ≤ n+2, rp (k + 1 − 3n) = k+1−3n−p. We know that k is strictly +above the positive root of x2 −x+2−3n−mp, hence k2 −k +2−3n > mp. +This implies that +Γ(k) =rp +� +(k − 1)2� ++ rp (k + 1 − 3n) = (k − 1)2 − mp + k + 1 − 3n + p +=k2 − k + 2 − 3n − mp + p ≥ p. +and then k is a jump. +From Lemma 2.7, +� +1 + √4mp + 12n − 7 +� +/2 < km ≤ k−1, hence +� +3 + √4mp + 12n − 7 +� +/2 < +k. Therefore 0 < k2 − 3k + 4 − mp − 3n, then +3n + k − 3 < (k − 1)2 − mp = rp +� +(k − 1)2� +. +□ +Remark 2.9 Note that by Lemma 2.7, the jumps km satisty 3n − km − 1 < +rp ((k − 1)2) ≤ 3n + km − 4 and by Lemma 2.8, the jumps k > km satisfy +3n + k − 3 < rp ((k − 1)2) . Now if k is not a jump and 2 ≤ k ≤ n + 2, then +necessarily rp ((k − 1)2) < 3n − k − 1 as the following lemma shows. +The following two lemmas are about the total residues and will help us +to count the total amount of jumps in the interval [1, 4n]. We will only prove +the fist one as the proof of the second one is similar. +Lemma 2.10 Let n ≥ 2 and 2 ≤ k ≤ n + 2 and p = 4n − 1 prime. +(a) If rp ((k − 1)2) < 3n − k − 1 then +Γ(k) < p +and Γ(p + 2 − k) < p. +(b) If 3n−k −1 ≤ rp ((k − 1)2) < 3n+ k −3 then Γ(p + 2 −k) < p ≤ Γ(k). +(c) If 3n + k − 3 ≤ rp ((k − 1)2) then +p ≤ Γ(p + 2 − k) and +p ≤ Γ(k). +Proof. +Note that −(4n − 1) ≤ k + 1 − 3n ≤ −1, therefore rp (k + 1 − 3n) = +k + 1 − 3n + p. Similarly, k ≤ n + 2 implies 0 ≤ p + 3 − k − 3n ≤ p − 1 and +hence rp (k2) p + 3 − k − 3n = p + 3 − k − 3n. +8 + +(a) Clearly +Γ(k) =rp +� +(k − 1)2� ++ rp (k + 1 − 3n) , +=rp +� +(k − 1)2� ++ p + k + 1 − 3n < p. +Γ(p + 2 − k) =rp +� +(p − (k − 1))2� ++ rp (p + 3 − k − 3n) , +=rp +� +(k − 1)2� ++ p + 3 − k − 3n < p. +(b) Here Γ(k) = rp ((k − 1)2) + p + k + 1 − 3n ≥ p and Γ(p + 2 − k) = +rp ((k − 1)2) + p + 3 − k − 3n < p. +(c) Finally Γ(k) = rp ((k − 1)2) + p + k + 1 − 3n ≥ p and Γ(p + 2 − k) = +rp ((k − 1)2) + p + 3 − k − 3n ≥ p. +Note that 2 ≤ k ≤ n + 2 implies 0 ≤ p + 2 − k ≤ p, hence Γ(p + 2 − k) is +defined. Also when k = 0 or k = 1, +rp (k2) < 3n − k − 1 and Γ(k) < p. +Then Γ(p + 2 − k) is not defined as p + 2 − k > p. +□ +Lemma 2.11 Let n ≥ 2, p = 4n − 1 prime and n + 3 ≤ k ≤ 2n. +(a) If rp ((k − 1)2) < k − n − 2 then +Γ(k) < p +and Γ(p + 2 − k) < p. +(b) If k − n − 2 ≤ rp ((k − 1)2) < 3n − k − 1 then Γ(k) < p ≤ Γ(p + 2 − k). +(c) If 3n − k − 1 ≤ rp ((k − 1)2) then +p ≤ Γ(k) and +Γ(p + 2 − k). +Proof. (very similar to the one of Lemma 2.10.) +□ +3 +Splitting the Jumps +For future reference we define the following sets. +9 + +Notation 3.1 For p = 4n − 1 prime, denote +C< = +� +k ∈ {2, ..., n + 2} : rp +� +(k − 1)2� +< 3n − k − 1 +� +, +C[−−) = +� +k ∈ {2, ..., n + 2} : rp +� +(k − 1)2� +∈ [3n − k − 1, 3n + k − 3) +� +, +C≥ = +� +k ∈ {2, ..., n + 2} : rp +� +(k − 1)2� +≥ 3n + k − 3 +� +, +D< = +� +k ∈ {n + 3, ..., 2n} : rp +� +(k − 1)2� +< k − n − 2 +� +, +D[−−) = +� +k ∈ {n + 3, ..., 2n} : rp +� +(k − 1)2� +∈ [k − n − 2, 3n − k − 1) +� +, +D≥ = +� +k ∈ {n + 3, ..., 2n} : rp +� +(k − 1)2� +≥ 3n − k − 1 +� +. +Lemma 3.2 Let ℓ ∈ Z and p = 4n − 1 prime. +(a) Let ℓ ≥ 6. If n = 4ℓ + 2 or n = 4ℓ + 3 then n − 3, n − 1, n + 1 ∈ C[−−) +and n − 2, n, n + 2 ∈ C<. +(b) Let ℓ ≥ 4. If n = 4ℓ + 1 or n = 4ℓ + 4 then n − 3, n − 1, n + 1 ∈ C< and +n − 2, n, n + 2 ∈ C[−−). +Proof. +Table 1 summarizes the residues of (k − 1)2 for k = n − 3, n − +2, n − 1, n, n + 1 and n + 2 given the four different cases for n which are +4ℓ + 1, 4ℓ + 2, 4ℓ + 3 and 4ℓ + 4, +ℓ ≥ 4. +We only verify the first column of Table 1, that is, we will compute the +residues of (k − 1)2 when k = n − 3. +(4ℓ − 3)2 += +(16ℓ + 3)(ℓ − 2) + 5ℓ + 15, +0 ≤ 5ℓ + 15 ≤ 16ℓ + 2, +(4ℓ − 2)2 += +(16ℓ + 7)(ℓ − 2) + 9ℓ + 18, +0 ≤ 9ℓ + 18 ≤ 16ℓ + 6, +(4ℓ − 1)2 += +(16ℓ + 11)(ℓ − 2) + 13ℓ + 23, +0 ≤ 13ℓ + 23 ≤ 16ℓ + 10, +(4ℓ)2 += +(16ℓ + 15)(ℓ − 1) + ℓ + 15, +0 ≤ ℓ + 15 ≤ 16ℓ + 14. +For a given n, k, denote Ik +n = [3n−k −1, 3n+k −3) and ∆k +n = rp ((k − 1)2) . +(a) Consider n = 4ℓ+2 and k = n−3 then Ik +n = [8ℓ+6, 16ℓ+2). According +to Table 1, ∆k +n = 9ℓ + 18. We observe that if ℓ ≥ 3, ∆k +n ∈ Ik +n. Similarly, +if k = n − 1, Ik +n = [8ℓ + 4, 16ℓ + 4) and ∆k +n = 9ℓ + 7 ∈ Ik +n for ℓ ≥ 1. If +k = n + 1, Ik +n = [8ℓ + 2, 16ℓ + 6) and ∆k +n = 9ℓ + 4 ∈ Ik +n for ℓ ≥ 0. Hence +for ℓ ≥ 6, +n − 3, n − 1, n + 1 ∈ C[−−). +Notice that if k = n − 2, ∆k +n = ℓ + 8 /∈ [8ℓ + 5, 16ℓ + 3) = Ik +n when ℓ ≥ 1. +Likewise, if k = n, ∆k +n = ℓ + 1 /∈ [8ℓ + 3, 16ℓ + 5) = Ik +n when ℓ ≥ 0. +10 + +k +n − 3 +n − 2 +n − 1 +n +n + 1 +n + 2 +n = 4ℓ + 1 +p = 16ℓ + 3 +5ℓ + 15 +13ℓ + 10 +5ℓ + 4 +13ℓ + 3 +5ℓ + 1 +13ℓ + 4 +n = 4ℓ + 2 +p = 16ℓ + 7 +9ℓ + 18 +ℓ + 8 +9ℓ + 7 +ℓ + 1 +9ℓ + 4 +ℓ + 2 +n = 4ℓ + 3 +p = 16ℓ + 11 +13ℓ + 23 +5ℓ + 11 +13ℓ + 12 +5ℓ + 4 +13ℓ + 9 +5ℓ + 5 +n = 4ℓ + 4 +p = 16ℓ + 15 +ℓ + 15 +9ℓ + 16 +ℓ + 4 +9ℓ + 9 +ℓ + 1 +9ℓ + 10 +Table 1: Residues of (k − 1)2 when k = n − 3, n − 2, ..., n + 2 for the different +cases of n. +Finally, if k = n + 2, ∆k +n = ℓ + 2 /∈ [8ℓ + 1, 16ℓ + 7) = Ik +n when ℓ ≥ 1. +Therefore n − 2, n, n + 2 /∈ C[−−) when ℓ ≥ 1, in fact, n − 2, n, n + 2 ∈ C< +for ℓ ≥ 1. The case n = 4ℓ + 3 is analogous. +(b) This case is done analogously. +□ +Remark 3.3 If n > 3 either Γ(n + 2) < p ≤ Γ(n + 1) or Γ(n + 1) < p ≤ +Γ(n + 2). +Proof. From Lemma 3.2, if n = 4ℓ + 2 or n = 4ℓ + 3 and ℓ ≥ 6, then +n + 1 ∈ C[−−) and n + 2 ∈ C<. By Lemma 2.10, Γ(n + 2) < p ≤ Γ(n + 1). +From Lemma 3.2, if n = 4ℓ + 1 or n = 4ℓ + 4 and ℓ ≥ 4, then n + 1 ∈ C< +and n + 2 ∈ C[−−). By Lemma 2.10, Γ(n + 1) < p ≤ Γ(n + 2). +We only need to verify the cases n = 5, 6, 8, 11, 12, 15 and 18 as the cases +n = 4, 7, 9, 10, 13, 16 and 19 do not give prime numbers. +(i) If n = 5 then n + 2 ∈ C[−−) = {5, 7} and n + 1 ∈ C< = {2, 3, 4, 6}. +(ii) If n = 6 then n + 1 ∈ C[−−) = {5, 7} and n + 2 ∈ C< = {2, 3, 4, 6, 8}. +11 + +(iii) If n = 8 then n+2 ∈ C[−−) = {6, 8, 10} and n+1 ∈ C< = {2, 3, 4, 5, 7, 9}. +(iv) If n = 11 then n + 1 ∈ C[−−) = {7, 10, 12} and n + 2 ∈ C<. +(v) If n = 12 then n + 2 ∈ C[−−) = {7, 10, 12, 14} and n + 1 ∈ C<. +(vi) If n = 15 then n + 1 ∈ C[−−) = {8, 11, 14, 16} and n + 2 ∈ C<. +(vii) If n = 18 then n + 1 ∈ C[−−) = {8, 12, 15, 17, 19} and n + 2 ∈ C<. +An application of Lemma 2.10 gives us the result. +Observe that when n = 3, C[−−) = {4, 5}, C< = {2, 3}, in this case both +Γ(n + 1) = 16 and Γ(n + 2) = 13 are greater than p = 11. +□ +The following two lemmas allow us to compute specifically the cardinality +of C≥. +Lemma 3.4 Let n > 3, p = 4n − 1 prime and M0 = ⌊(n2 − 4n + 5)/p⌋. +Define ℓm = 1 + +�√mp + p − 1 +� +and consider km defined as in Lemma 2.7. +Let 2 ≤ k ≤ n + 2 and 0 ≤ m ≤ M0 − 1. +If k < k0, k > ℓM0 or ℓm < k ≤ km+1 then k /∈ C≥. Also if k < k0 or +ℓm < k < km+1 then k is not a jump. +Proof. Consider k < k0 = 1+ +� +(1 + √12n − 7)/2 +� +. Then (2k−1)2 < 12n−7 +from which (k − 1)2 < 3n − k − 1 ≤ 4n − 1. Hence rp ((k − 1)2) = (k − 1)2 < +3n − k − 1. Therefore by Lemma 2.10, k ∈ C< and k is not a jump. +Consider k > ℓM0. Notice that ℓM0 = 1 + +�√Mp +� +≥ n − 1. By Lemma 3.2, +either k ∈ C[−−) or k ∈ C<. If k = km+1, by Lemma 2.7 (iv), k ∈ C[−−). +Finally consider ℓm < k < km+1. Then +1 + +� +(m + 1)p < k < 1 + +� +4(m + 1)p + 12n − 7 +2 +. +Hence +(m + 1)p < (k − 1)2 < (m + 1)p + 3n − k − 1. +Therefore rp ((k − 1)2) = (k−1)2−(m+1)p < 3n−k−1. By Lemmas 2.10 +and 3.2, k is not a jump and k ∈ C<. +□ +12 + +Lemma 3.5 Consider n, p and M0 as in Lemma 2.7. Then k ∈ C≥ if and +only if there is m ∈ {0, 1, ..., M0} such that km < k ≤ 1 + +�√mp + p − 1 +� +. +Proof. +From Lemma 2.8, if km < k ≤ ℓm then k is a jump and rp ((k − 1)2) > +3n + k − 3. Note that 2 ≤ k0 < k. Since m ≤ M0 we have ℓm ≤ n + 2, hence +2 ≤ k ≤ n + 2. Therefore k ∈ C≥. +Conversely, let k ∈ C≥. Then 2 ≤ k ≤ n+2 and rp ((k − 1)2) ≥ 3n+k−3. +Observe that +[2, n + 2] = [2, k0] ∪ (k0, ℓ0] ∪ (ℓ0, k1] ∪ (k1, ℓ1] ∪ · · · ∪ (kM0, ℓM0] ∪ (ℓM0, n + 2]. +If k < k0, k > ℓM0 or ℓm < k ≤ km+1, by Lemma 3.4, k /∈ C≥. +This forces k to be in (km, ℓm] for some m ∈ {0, ..., M0}, which is what +we wanted. +□ +4 +Counting the Jumps +In this section we will relate the jumps in the different sets C<, C[−−), C≥, D<, D[−−) +and D≥. We will compute different cardinalities when possible. In this section +we consider n > 3 and p = 4n − 1 a prime number. +Theorem 4.1 The function f : C≥ −→ D< defined by +f(k) = 2n + 2 − k, +is well-defined and bijective. +Proof. +Consider k ∈ C≥. By Lemmas 2.8, 3.5 and Observation 2.9, there is +m ∈ {0, 1, ..., M0} with M0 = ⌊(n2 − 4n + 5)/p⌋ such that km < k ≤ 1 + +�√mp + p − 1 +� +and 3n + k − 3 ≤ (k − 1)2 − mp = rp ((k − 1)2) , where km is +the jump given in Lemma 2.10. Hence +0 ≤ k2 − 3k − 3n + 4 − mp, +(4.6) +k2 − 2k + 1 − mp ≤ 4n − 2. +(4.7) +13 + +Let kf = f(k) = 2n + 2 − k. We will first prove that kf ∈ D<. Now +(kf − 1)2 = (n − k + m + 2)p + (k − 1)2 − mp + n − k + 1 − p. +Let w = (k − 1)2 − mp + n − k + 1 − p. From Inequality 4.6, w ≥ −1. From +Remark 2.6, k2 − 3k + 4 − 3n − mp ̸= 0, hence w ≥ 0. +Note that in the case of equality in Inequality 4.7, we would have (k−1)2 = +mp + p − 1, hence the congruency x2 ≡ −1 (mod p) would have a solution, +which is impossible. Therefore +k2 − 2k + 1 − mp < 4n − 2. +(4.8) +From Inequality 4.7, +w = k2 − 2k + 1 − mp + 2 − k − 3n < n − k. +(4.9) +Clearly n − k ≤ 4n − 2, hence w < p − 1. Therefore rp ((kf − 1)2) = w. From +Inequality 4.9, 0 ≤ w < n−k, this forces k ≤ n−1. Also from Inequality 4.9, +since kf − n − 2 = n − k, we conclude that rp ((kf − 1)2) = w < kf − n − 2. +Finally, 2 ≤ k ≤ n − 1 implies n + 3 ≤ kf ≤ 2n, hence f is well defined +as kf ∈ D<. +Clearly f is injective. Take now �k ∈ D<. Then n + 3 ≤ �k ≤ 2n and +rp +� +(�k − 1)2� +< �k − n − 2. +(4.10) +Consider k = 2n + 2 − �k and �m ∈ Z with +�m ≤ (�k − 1)2 < �m · p + p. +(4.11) +Then 2 ≤ k ≤ n − 1 and +(k − 1)2 = (n − �k + �m) · p + (�k − 1)2 − �mp + n − �k + 1 − p. +Consider m = n − �k + �m and u = (�k − 1)2 − �mp + n − �k + 1 − p. +Since �k ≤ 5n, Inequality 4.11 implies +0 ≤ n − �k + 1 − p ≤ (k − 1)2 − �mp + n − �k + 1 − p. +(4.12) +Also from Inequality 4.10, we have (k − 1)2 − �mp + n − �k + 1 − p < p − 1, +therefore 0 ≤ u < p − 1. Hence rp ((k − 1)2) = (k − 1)2 − mp = u. +Finally Inequality 4.12 implies u ≥ 5n − �k ≥ 5n − 1 − �k = 3n + k − 3. +Therefore rp ((k − 1)2) ≥ 3n + k − 3, i.e. k ∈ C≥. Clearly f(k) = �k, then f +is bijective. +□ +14 + +Lemma 4.2 Let y, z be the last two elements in C[−−). If k ∈ C[−−) − {y, z} +then +mp + p ≤ (n − 2)2 + 1, +(4.13) +where m = m(k) = ⌊(k − 1)2/p⌋ . +Proof. +(a) Case n = 4ℓ + 2 or n = 4ℓ + 3, ℓ ≥ 6. By Lemma 3.2, k ∈ C[−−) − {y, z} +implies k ≤ n − 3. Hence m ≤ ℓ − 2 = ⌊(n − 4)2/p⌋ . If n = 4ℓ + 2 then +mp + p ≤ (ℓ − 1)(16ℓ + 7) ≤ 16ℓ2 + 1 = (n − 2)2 + 1. +If n = 4ℓ+3 then mp+p ≤ (ℓ−1)(16ℓ+11) ≤ 16ℓ2+8ℓ+2 = (n−2)2+1. +(b) Case n = 4ℓ + 1 or n = 4ℓ + 4, ℓ ≥ 4. By Lemma 3.2, k ∈ C[−−) − {y, z} +implies k ≤ n − 2. If n = 4ℓ + 1 then m ≤ ⌊(n − 3)2/p⌋ = ℓ − 2. Hence +mp + p ≤ (ℓ − 1)(16ℓ + 3) ≤ 16ℓ2 − 8ℓ + 2 = (n − 2)2 + 1. +If n = 4ℓ+4 then m ≤ ⌊(n − 3)2/p⌋ = ℓ−1. Hence mp+p ≤ ℓ·(16ℓ+15) ≤ +16ℓ2 + 16ℓ + 5 = (n − 2)2 + 1. +(c) The only left cases are n = 5, 6, 8, 11, 12, 15 and 18 as the choices n = +4, 7, 9, 10, 13, 14, 16, 19, 22, 23 do not provide prime numbers. +(i) If n = 5 or 6 then C[−−) has only two elements. Hence Inequality 4.13 +is trivial. +(ii) If n = 8 then C[−−) = {6, 8, 10} and k = 6. Therefore m(k) = 0 = +⌊(k − 1)2/p⌋ = ⌊25/31⌋ clearly satisfies Inequality 4.13. +(iii) If n = 11 then C[−−) = {7, 10, 12} and k = 7. Therefore m(k) = 0 +clearly satisfies Inequality 4.13. +(iv) If n = 12 then C[−−) = {7, 10, 12, 14}. If k = 10, then m(k) = 1 = +⌊(k − 1)2/p⌋ = ⌊81/47⌋ clearly satisfies Inequality 4.13 as 94 ≤ 101. +Clearly m(7) also does. +(v) If n = 15 then C[−−) = {8, 11, 14, 16}. If k = 11, then m = 1 = +⌊(k − 1)2/p⌋ = ⌊100/59⌋ clearly satisfies Inequality 4.13 as 118 ≤ +169. Clearly m(8) also does. +(vi) If n = 18 then n + 1 ∈ C[−−) = {8, 12, 15, 17, 19}. If k = 15, then +m(k) = 2 = ⌊(k − 1)2/p⌋ = ⌊196/71⌋ clearly satisfies Inequal- +ity 4.13 as 213 ≤ 256. Clearly m(8), m(12) also satisfy Inequal- +ity 4.13. +15 + +□ +Lemma 4.3 k ∈ D[−−) if and only if there is an integer m with 1 ≤ m ≤ +⌊(n2 − 4n + 5)/p⌋ and +k = 2n − +�� +mp − 1 +� +. +Proof. +⇒) Let k ∈ D[−−) and define α = 2n + 1 − k. Hence 1 ≤ α ≤ n − 2 and +(k − 1)2 = 4n2 + α2 − 4nα = (n − α + m)p + α2 + n − α − mp, +where m satisfies mp ≤ α2+n−α < (m+1)p. Therefore rp ((k − 1)2) = +α2+n−α−mp. Since k−n−2 ≤ rp ((k − 1)2) < 3n−k−1, mp−1 ≤ α2 +and (α − 1)2 < mp − 1. +Hence √mp − 1 ≤ α < √mp − 1 + 1. Since x2 ≡ −1 (mod p) has no +solution, √mp − 1 is not an integer. Therefore α = +�√mp − 1 +� ++ 1. +Then k = 2n − +�√mp − 1 +� +. +Since 0 ≤ α2 + n − α, m ≥ 0. Now (α − 1)2 < mp − 1 implies 1 ≤ m. +Clearly mp ≤ α2 + 1 ≤ (n − 2)2 + 1. Therefore m ≤ (n2 − 4n + 5)/p. +⇐) Consider an integer m with 1 ≤ m ≤ (n2 − 4n + 5)/p and k = 2n − +�√mp − 1 +� +. +Since mp − 1 ≤ (n − 2)2, +n + 2 ≤ k. Clearly n + 2 = k leads us to an +integer solution of x2 ≡ −1 (mod p), therefore n + 3 ≤ k. Also 1 ≤ m +implies k ≤ 2n. +Consider α = +�√mp − 1 +� ++ 1. Then √mp − 1 ≤ α < √mp − 1 + 1. +Clearly α ̸= √mp − 1 + 1 (otherwise x2 ≡ −1 (mod p) has an integer +solution), then √mp − 1 < α < √mp − 1 + 1. +Hence mp − 1 < α2 and (α − 1)2 < mp − 1. Since α < n − 1, mp ≤ +α2 ≤ α2 + n − α < n + α − 2 + mp < mp + p. Since +(k − 1)2 = (2n − α)2 = (n − α + m)p + α2 + n − α − mp, +rp ((k − 1)2) = α2 + n − α − mp. Finally, from k − n − 2 = n − α − 1 < +rp ((k − 1)2) < n + α − 2 = 3n − k − 1, we conclude that k ∈ D[−−). □ +16 + +Theorem 4.4 Let y, z the last two elements of C[−−). For k ∈ C[−−) − {y, z}, +consider m = ⌊(k − 1)2/p⌋ and u0 = u0(k) the first integer less than or equal +to n + 2 such that x = u0 satisfies +(m + 1) p ≤ (k + x − 1)2 + 1. +(4.14) +Then, the function f : C[−−) − {y, z} −→ D[−−) defined by +f(k) = kf = 2n + 2 − u0 − k, +is well-defined and bijective. +Proof. First, we will prove that f is well-defined. It is not hard to check +that u0 = +�� +(m + 1)p +� ++2−k. The definition of u0 implies that u0, u0+1, ... +satisfy Inequality 4.14 but u0 − 1 does not. Hence u0 satisfies +4n + mp ≤ k2 + 2k(u0 − 1) + u2 +0 − 2u0 + 3, +(4.15) +k2 + 2k(u0 − 2) + u2 +0 − 4u0 + 6 < 4n + mp. +(4.16) +If k2 = mp + 3n + 3k − 2 for 2 ≤ k ≤ n + 2 and we define x0 = 2n − k + 1, +then x2 +0 = (n+ m−k + 2)p + 1. Therefore p divides (x0 −1)(x0 + 1), however +since n ≥ 3 and 2 ≤ k ≤ n + 2, we have that +1 ≤ 2n − k = x0 − 1 < x0 + 1 = 2n + 2 − k ≤ 4n − 2, +which is impossible. Therefore +k2 ̸= mp + 3n + 3k − 2. +(4.17) +Observe that u0 ≥ 1 as x = 0 does not satisfy Inequality 4.14. Also u0 ≤ n+2 +as (k + n + 1)2 ≥ (k − 1)2 + (n + 2)2 ≥ mp + p − 1. +To shorten notation, define + + + + + + + +mf += +n + 2 − k − u0 + m, +∆ += +rp ((k − 1)2) , +∆f += +rp ((kf − 1)2) , +wf += +∆ + k(2u0 − 1) − p + n + u2 +0 − 3u0 + 1. +To check that f is well-defined, we need to verify that n + 3 ≤ kf ≤ 2n +and kf − n − 2 ≤ ∆f < 3n − kf − 1. It is not hard to check that +(kf − 1)2 = mf · p + wf. +17 + +Notice that ∆ = rp ((k − 1)2) = (k − 1)2 − mp ≤ 3n − k − 1. From Inequal- +ity 4.17, ∆ ≥ 3n − k. Therefore, +wf +≥ +3n − k + +k(2u0 − 1) − p + n + u2 +0 − 3u0 + 1, += +k(2u0 − 2) + u2 +0 − 3u0 + 2, +≥ +u2 +0 − 3u0 + 2 = (u0 − 2)(u0 − 1) ≥ 0. +From Inequality 4.16, +wf += +k2 + 2k(u0 − 2) + u2 +0 − 4u0 + 6 − 4n − mp + k + u0 + n − 3, +< +k + u0 + n − 3 ≤ 4n − 1. +This shows that wf = rp ((kf − 1)2) . Since 3n − kf − 1 = n − 3 + u0 + +k, wf < 3n − kf − 1. Since kf − n − 2 = n − u0 − k, Inequality 4.15 implies +that wf ≥ kf −n−2. Therefore kf −n−2 ≤ rp ((kf − 1)2) < 3n−kf −1. By +Lemma 4.2, x = n−k −1 satisfies Inequality 4.14, hence 1 ≤ u0 ≤ n−k −1. +Therefore n + 3 ≤ kf ≤ 2n. This proves that kf ∈ D[−−) and thus f is well +defined. +Consider k0 and k1 such that kf = f(k0) = f(k1). Take u0, u1, m0 and m1 +such that m0 = ⌊(k0 − 1)2/p⌋ , m1 = ⌊(k1 − 1)2/p⌋ and u0, u1 are the first +integers such that Inequality 4.14 holds with k = k0 and k = k1 respectively. +Now f(k0) = f(k1) implies that u0 + k0 = u1 + k1. Since +mf = n + 2 − k0 − u0 + m0 = n + 2 − k1 − u1 + m1, +we conclude m0 = m1. Since +wf = (k0+u0−1)2−u0−m0p−p+n+1 = (k1+u1−1)2−u1−m1p−p+n+1, +u0 = u1 and consequently k0 = k1. Therefore f is injective. +Take now kf ∈ D[−−). Let mf = ⌊(kf − 1)2/p⌋. Then kf − n − 2 ≤ ∆f < +3n − kf − 1. Notice that x = 0 satisfies the inequality +0 ≤ (kf + x − 1)2 + 1 − mfp − px. +(4.18) +Let v0 the first integer greater than or equal to 1 such that x = v0 does not +satisfy Inequality 4.18. Hence v0 = x satisfies the equivalent inequalities +(kf + x − 1)2 + 1 − mfp − px < 0, +(4.19) +∆f + 2kfx + (x − 1)2 − px < 0. +18 + +Consider k = 2n + 2 − v0 − kf, m = n − kf + mf and w = ∆f + kf(2v0 − +1) + n + (v2 +0 − 3v0 + 1) − v0p + p. Then +(k − 1)2 += +(n + 1 − kf + mf)p + k2 +f + kf(2v0 − 3) + n + 2 + v2 +0 − 3v0 − v0p − mp, += +(n − kf + mf)p + ∆f + kf(2v0 − 1) + n + (v2 +0 − 3v0 + 1) − v0p + p, += +mp + w. +By Lemma 4.3, there is an integer m, 1 ≤ m ≤ (n2 − 4n + 5)/p such that +kf = 2n− +�√mp − 1 +� +. Also from the proof of Lemma 4.3 if α = 2n−kf + 1 +then mf = n − α + m. Take x0 = α − 1 = 2n − kf. Notice that x0 ≥ 1 as +√mp − 1 ≥ 1. Substituting x = x0 into (kf + x − 1)2 + 1 − mfp − px gives us +(2n − 1)2 + 1 − (n − α + m)p − p(α − 1) = n + 3 − mp. +Since m ≥ 1, n+3−mp < 0. Then x = x0 satisfies Inequality 4.19, therefore +v0 exists and v0 ≤ 2n−kf. Since x = v0 −1 satisfies Inequality 4.18, we have +that +0 ≤ ∆f + kf(2v0 − 2) + (v0 − 2)2 + p − v0p. +Hence w ≥ kf + n + v0 − 3 = 3n − k − 1. Since kf ≥ n + 3 and v0 ≥ 1 we +conclude w ≥ 0 and 2 ≤ k ≤ n − 2. +Since v0 satisfies Inequality 4.19, +w < n − kf − v0 + p = 5n − kf − v0 − 1 = 3n + k − 3 ≤ 4n − 1. +Thus 0 ≤ w < p and 3n − k − 1 ≤ w < 3n + k − 3. This implies that +w = rp ((k − 1)2) , m = ⌊(k − 1)2/p⌋ and hence k ∈ C[−−). Lemma 3.2 and the +proof of Lemma 4.2 implies that {y, z} is a subset of {n − 1, n, n + 1, n + 2}, +then k ∈ C[−−) − {y, z}. +Now we will find u0 = u0(k). Since ∆f ≥ kf − n − 2, +(k + v0 − 1)2 + 1 += +(2n + 1 − kf)2 + 1, += +(n − kf + mf + 1)p + ∆f + n − kf + 2, += +(m + 1)p + ∆f + n − kf + 2 ≥ (m + 1)p. +19 + +From ∆f < 3n − kf − 1 = p − n − kf, we obtain +(k + v0 − 2)2 + 1 += +(2n − kf)2 + 1, += +(n − kf + mf + 1)p + ∆f + kf + n − p, += +(m + 1)p + ∆f + kf + n − p < (m + 1)p. +Therefore u0 = v0 and f(k) = kf. Then f is surjective and hence bijec- +tive. +□ +Corollary 4.5 +��D[−−) +�� = ⌊(n2 − 4n + 5)/2⌋ and +��C≥ +�� = +��D< +��, +��C[−−) +�� = +��D[−−) +�� + 2, +��C< +�� = +��D≥ +�� + 1. +Proof. From Lemma 4.1, +��C≥ +�� = +��D< +��. From Theorem 4.4, +��C[−−) +�� = +��D[−−) +��+ +2. Since n + 1 = +��C< +�� + +��C[−−) +�� + +��C≥ +�� and n − 2 = +��D< +�� + +��D[−−) +�� + +��D≥ +�� we +have +n + 1 = +��C< +�� + +��D< +�� + +��D[−−) +�� + 2 = +��C< +�� + n − +��D≥ +��, +hence +��C< +�� = +��D≥ +�� + 1. To see that +��D[−−) +�� = ⌊(n2 − 4n + 5)/2⌋ it is enough +to see that all the k′s in Lemma 4.3 given by each m are all different, which +is the case as +� +mp − 1 + 1 < +� +(m + 1)p − 1. +□ +Theorem 4.6 Under the hypothesis of Theorem 4.4, +���{k ∈ Z | 2 ≤ k ≤ 4n − 1, Γ(k) ≥ p} +��� = 2n − 2. +Proof. Let JΓ = {k ∈ Z : 2 ≤ k ≤ 4n − 2, +Γ(k) ≥ p}. By Lemmas 2.10 +and 2.11, +JΓ ∩ [2, n + 2] += +C[−−) ∪ C≥, +JΓ ∩ [n + 3, 2n] += +D≥. +20 + +��JΓ ∩ [2n + 1, 3n − 2] +�� += +���{2n + 1 ≤ k ≤ 3n − 2 | Γ(k) ≥ p} +���, += +���{2n + 1 ≤ p + 2 − k ≤ 3n − 2 | Γ(p + 2 − k) ≥ p} +���, += +���{n + 3 ≤ k ≤ 2n | Γ(p + 2 − k) ≥ p} +��� = +��D[−−) +�� + +��D≥ +��. +��JΓ ∩ [3n − 1, 4n − 1] +�� += +���{3n − 1 ≤ k ≤ 4n − 1 | Γ(k) ≥ p} +���, += +���{3n − 1 ≤ p + 2 − k ≤ 4n − 1 | Γ(p + 2 − k) ≥ p} +���, += +���{2 ≤ k ≤ n + 2 | Γ(p + 2 − k) ≥ p} +��� = +��C≥ +��. +Using these identities and Corollary 4.5, we obtain +��JΓ +�� = +��C[−−) +�� + 2 +��C≥ +�� + +��D[−−) +�� + 2 +��D≥ +��, += +��C[−−) +�� + 2 +��C≥ +�� + +��C[−−) +�� − 2 + 2 +���C< +�� − 1 +� +, += 2 +���C< +�� + +��C[−−) +�� + +��C≥ +��� +− 4, += 2 · +���Z ∩ [2, n + 2] +��� − 4 = 2n − 2. +□ +The following corollary comes from proof of the previous theorem. +Corollary 4.7 Under the hypotheses of Theorem 4.4, +��� {k ∈ Z : 2 ≤ k ≤ 2n, +Γ(k) ≥ p} +��� = n, +��� {k ∈ Z : 2n + 1 ≤ k ≤ 4n, +Γ(k) ≥ p} +��� = n − 2. +21 + +5 +Sums involving rp +� +k2 − k + 2 − 3n +� +. +Consider Jn = +��� {k ∈ Z : 2 ≤ k ≤ n + 2, +Γ(k) ≥ p} +��� and we call Jn simply +the number of jumps. We will now develop formulas relating the term residues +of k2 − k + 2 − 3n modulus p. +Theorem 5.1 If n > 3 and p = 4n − 1 is prime then +p−1 +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p−1 +� +k=1 +rp +� +k2� ++ 3n − 2. +2n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +2n +� +k=1 +rp +� +k2� ++ n. +n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +n +� +k=1 +rp +� +k2� ++ n(n + 1) +2 +− p(Jn − 1 − M). +Proof. Notice that +rp (x + y) = +� rp (x) + rp (y) +if rp (x) + rp (y) < p, +rp (x) + rp (y) − p +if rp (x) + rp (y) ≥ p. +Recall that we defined k as a jump when Γ(k) = rp ((k − 1)2)+rp (k + 1 − 3n) ≥ +p. By Theorem 4.6, +22 + +4n−1 +� +k=2 +rp +� +k2 − k + 2 − 3n +� += +� +k:Γ(k)≥p +� +rp +� +(k − 1)2� ++ rp (k + 1 − 3n) − p +� ++ +� +k:Γ(k)

3. Charts 2 and 3 contain the +sum of the residues rp (k2 − k + 2 − 3n) in the special cases excluded in such +theorem. +Corollary 5.3 If n > 3 and p = 4n − 1 is prime then +23 + +n +p−1 +� +k=0 +rp +� +k2 − k + 2 − 3n +� +p−1 +� +k=0 +rp +� +k2� ++ p +1 +5 +5 +2 +21 +21 +3 +55 +55 +Table 2: Sum of residues rp (k2 − k + 2 − 3n) in special cases. +n +2n +� +k=0 +rp +� +k2 − k + 2 − 3n +� +2n +� +k=0 +rp +� +k2� ++ 2n + 1 +1 +5 +5 +2 +14 +14 +3 +32 +32 +Table 3: Sum of residues rp (k2 − k + 2 − 3n) in special cases. +p−1 +� +k=1 +�k2 − k + 2 − 3n +p +� += +p(p − 5) + 6 − n +3 +− 1 +p +p−1 +� +k=1 +rp +� +k2� +. +2n +� +k=1 +�k2 − k + 2 − 3n +p +� += +n(2n − 1)(4n − 7) +3p +− 1 +p +2n +� +k=1 +rp +� +k2� +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +(n + 1)(2n2 − 23n + 6) +3p +− 1 +p +n +� +k=1 +rp +� +k2� ++ Jn − M. +Proof. From x = p +� +x +p +� ++ rp (x) , we obtain that if y = �p−1 +k=1 +� +k2−k+2−3n +p +� +then +p−1 +� +k=1 +� +k2 − k + 2 − 3n +� += py + +p−1 +� +k=1 +rp +� +k2 − k + 2 − 3n +� +. +From Theorem 5.1, +(p − 1)p(2p − 1) +6 +− (p − 1)p +2 ++ (p − 1)(2 − 3n) = py + +p−1 +� +k=1 +rp +� +k2� ++ 3n − 2, +then +p · p · (p − 5) + 6 − n +3 += py + +p−1 +� +k=1 +rp +� +k2� +. +24 + +The result follows. The proofs of the other two sums are done similarly. +□ +The purpose of the following lemma and remark is to find a formula for +�n +k=0 +� +k2−k+2−3n +p +� +. +Lemma 5.4 Let Qm = 1 + √4mp + 12n − 7 +2 +as in Notation 2.2. Then +⌊Qm⌋ = max{k ∈ N : k2 − k + 2 − 3n ≤ mp}. +Proof. +Clearly ⌊Qm⌋ ≤ Qm < ⌊Qm⌋+1 and since Qm is the non-negative root of +x2−x+2−3n = mp, k0 = ⌊Qm⌋ satisfies 0 ≤ k0 ≤ Qm and k2 +0 −k0+2−3n ≤ +mp. Hence k1 = ⌊Qm⌋ + 1 satisfies 0 ≤ k1 and k2 +1 − k1 + 2 − 3n > mp. If +k0 = 0 then k1 = 1 and hence p ≤ mp < 2−3n which is impossible, therefore +k0 ∈ N. +□ +Observation 5.5 By Lemma 2.5, there are no integers k, m, n such that +⌊Qm⌋ = Qm, i.e. k2 − k + 2 − 3n ̸= mp regardless of k, m, n. +Theorem 5.6 Let n > 3 and p = 4n − 1 prime. Then +n +� +k=1 +�k2 − k + 2 − 3n +p +� += (M − 1)n − +M−1 +� +m=0 +⌊Qm⌋ . +Proof. Since n > 3, 1 ≤ M. Define, tm = ⌊Qm⌋ and +Hm = + + + + + +� +1, ..., t0 +� +if m = 0, +� +tm−1 + 1, tm−1 + 2, ..., tm +� +if 1 ≤ m ≤ M − 1, +� +tM−1 + 1, tM−1 + 2, ..., n +� +if m = M. +By Lemma 5.4, k = ⌊Qm⌋ satisfies k2 − k + 2 − 3n ≤ mp and from +Observation 5.5, k2 −k + 2 −3n < mp. Therefore by Lemma 5.4, for k ∈ Hm +we have +(m − 1)p < k2 − k + 2 − 3n < mp. +For such k necessarily ⌊(k2 − k + 2 − 3n)/p⌋ = m − 1. +25 + +Notice that {H0, H1, ..., HM} is a partition of {1, 2..., n} as QM−1 < n < +QM (see Lemma 2.5 in [3]). Therefore +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +M +� +m=0 +n +� +k=0 +k∈Hm +�k2 − k + 2 − 3n +p +� +, += +M +� +m=0 +(m − 1)|Hm|, += +−t0 + 0(t1 − t0) + 1(t2 − t1) + · · · , ++(M − 2)(tM−1 − tM−2) + (M − 1)(n − tM−1), += +− +M−1 +� +m=0 +⌊Qm⌋ + (M − 1)n. +□ +Corollary 5.7 If n > 3 and p = 4n − 1 is prime then +n +� +k=0 +�k2 +p +� ++ +n +� +k=1 +�k2 − k + 2 − 3n +p +� += n(n − 5) +6 ++ M − 1 +2p +p−1 +� +k=1 +rp +� +k2� +. +Proof. From [6] (page 253) we have +M +� +m=1 +⌊Rm⌋ = Mn − +n +� +k=0 +�k2 +p +� +. +(5.20) +Corollary 2 in [3] states +1 +2 +p−1 +� +k=1 +rp +� +k2� += p +� M +� +m=1 +⌊Rm⌋ + +M−1 +� +m=0 +⌊Qm⌋ +� +− Mp(2n − 1) + p · (n2 + n) +6 +. +(5.21) +Using Theorem 5.6 and Equation 5.20 in Equation 5.21 we obtain +1 +2 +p−1 +� +k=1 +rp +� +k2� += +p +� +(2M − 1)n − +n +� +k=1 +�k2 +p +� +− +n +� +k=1 +�k2 − k + 2 − 3n +p +�� +−Mp(2n − 1) + p · (n2 + n) +6 +. +26 + +The result now follows. +□ +Compare Corollary 5.7 with Theorem 2.2 (case p = 4n + 1) in [4] which +can be rewritten as +n +� +k=0 +�k2 +p +� ++ +n +� +k=1 +�k2 + k + 1 − n +p +� += (n + 3)(n + 2) +6 ++M − 1 +2p +p−1 +� +k=1 +rp +� +k2� ++un. +Corollary 5.8 If n > 3 and p = 4n − 1 is prime then +M +� +m=1 +⌊Rm⌋ − +M−1 +� +m=0 +⌊Qm⌋ = Jn − M − 1. +Proof. From Lemmas 2.10, 4.3 and 3.5 we obtain +Jn += +|C[−−)| + |C≥| = |D[−−)| + 2 + |C≥|, += +M0 + 2 + +M0 +� +m=0 +(ℓm − km) , += +M + 1 + +M0 +� +m=0 +(⌊Rm+1⌋ − ⌊Qm⌋) , += +M + 1 + +M +� +m=1 +⌊Rm⌋ − +M−1 +� +m=0 +⌊Qm⌋ . +□ +Compare Corollary 5.8 with Theorem 5.3 (case p = 4n + 1) in [4] +M +� +m=1 +⌊Rm⌋ − +M +� +m=0 +⌊Sm⌋ = jn + 2 − n − un. +27 + +6 +Class Number Identities +In this section, we establish some identities involving the class number h = +h(−p) of the imaginary quadratic field Q(√−p) when p is of the form p = +4n−1. These identities are based on the previous formulas we have developed +in previous sections. +In [3], we have +h += +(2M + 1)(2n − 1) − 2 +� M +� +m=1 +⌊Rm⌋ − +M−1 +� +m=0 +⌊Qm⌋ +� +− n2 + n +3 +. +h += +p − 1 +2 +− 1 +p +p−1 +� +k=1 +rp +� +k2� +. +From Corollaries 5.3, 5.7 and �p−1 +k=1 rp (k2) = 2 �2n +k=1 rp (k2) − 2n we obtain +Corollary 6.1 +p−1 +� +k=1 +�k2 − k + 2 − 3n +p +� += +h + 16n2 − 35n + 15 +3 +. +2n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +2 + 4n2 − 14n + 9 +6 +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +4 + Jn +2 + n2 − 17n − 3 +12 +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +2 + M + n2 − 11n + 3 +6 +− +n +� +k=1 +�k2 +p +� +. +n +� +k=1 +�k2 − k + 2 − 3n +p +� += +h +2 + +M +� +m=1 +⌊Rm⌋ + M(1 − n) + n2 − 11n + 3 +6 +. +From Corollary 5.8, we have +28 + +Corollary 6.2 +Jn +2 + M(n − 1) += +h +4 + +M +� +m=1 +⌊Rm⌋ + n2 − 5n + 9 +12 +. +−Jn +2 + Mn += +h +4 + +M−1 +� +m=0 +⌊Qm⌋ + n2 − 5n − 3 +12 +. +Jn +2 + +n−1 +� +k=1 +�k2 +p +� += +h +4 + n2 − 5n + 9 +12 +. +From Theorem 5.1, we conclude +Corollary 6.3 +p−1 +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p · (2n − h) − n − 1. +2n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p · (2n − h) + 1 +2 +. +n +� +k=1 +rp +� +k2 − k + 2 − 3n +� += +p +4(3n + 2 − 2Jn − h) − (n + 1)(n − 1) +4 +. +Finally, combining this last formula with Theorem 5.1 we have +Corollary 6.4 +n +� +k=1 +rp +� +k2� += p +4(2Jn + 2n − 3 − 4M − h) + n(n + 1) +4 +. +The numerical data we have allow us to pose the following +Conjecture 6.5 Consider all n such that p = 4n−1 a prime number. Then +lim +n→∞ +Jn +n = 3 +8. +29 + +Conjecture 6.6 Consider all n such that p = 4n−1 a prime number. Then +lim +n→∞ +�M +m=1 ⌊Rm⌋ + �M−1 +m=0 ⌊Qm⌋ +Mp + 2n += 1 +3. +Also a good estimate of �M +m=1 ⌊Rm⌋ + �M−1 +m=0 ⌊Qm⌋ is ⌊(Mp + 2n)/3⌋ . +30 + +References +[1] H. Cohen, A course in computational algebraic number theory, vol. 138 +of Graduate Text in Mathematics., Springer-Verlag, New York, 1993. +[2] P. G. L. Dirichlet, Beweis des satzes, dass jede unbegrenzte arith- +metische progression, deren erstes glied und differenz ganze zahlen ohne +gemeinschaftlichen factor sind, unendlich viele primzahlen enth¨alt. ab- +handlungen der k¨oniglich preussischen akademie der wissenschaften von, +Abhandlungen der K¨oniglich Preussischen Akademie der Wissenschaften +von, (1837), pp. 45—-81. +[3] J. Garcia, A computable formula for the class number of the imaginary +quadratic field Q(√−p), p = 4n − 1, Electronic Research Archive, 29 +(2021), pp. 3853–3865. +[4] +, Sums involving quadratic residues modulus a prime of the form +p = 4n + 1., 2022. +[5] W. Narkiewicz, Elementary And Analytic Theory Of Algebraic Num- +bers, Springer, 2004. +[6] C. Zeller, Ueber Summen von gr¨ossten Ganzen bei arithmetische Rei- +hen., Nachricten von der K. Gesselschaft der Wissenschaften und der +Georg- Augusts Universitat, May 14, 1879. +31 + diff --git a/XdE1T4oBgHgl3EQfJQP0/content/tmp_files/load_file.txt b/XdE1T4oBgHgl3EQfJQP0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e8e4c970868a50729ba10285fc96c553035fabd --- /dev/null +++ b/XdE1T4oBgHgl3EQfJQP0/content/tmp_files/load_file.txt @@ -0,0 +1,685 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf,len=684 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='02951v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='NT] 8 Jan 2023 Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities Jorge Garcia January 10, 2023 Abstract A formula for the sum of quadratic residues modulus a prime p = 4n − 1 is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new con- cepts based on these roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' A number of formulas for the sum of the quadratic residues are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We finalize the paper by obtaining several identities involving h(−p) the class number of the imaginary quadratic field Q(√−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 1 Introduction Consider a prime p = 4n − 1 and 1 ≤ k ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By rp (k2) , we denote the remainder of k2 when we divide by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We call this number rp (k2) the quadratic residue of k2 modulus p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' When we add all these residues we obtain the sum of quadratic residues relative to the prime number p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' There is a complicated formula for such sum, p−1 � k=1 rp � k2� = �p 2 � − p · h(−p), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1) where h(−p) is the class number of the imaginary quadratic field Q(√−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Important formulas for the class number when the prime is of the form p = 4n − 1 > 3 include Dirichlet class number formula 1 h(−p) = √p 2π ∞ � r=1 χ(r) r , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2) where χ is the Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Formula 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Another formula that involves the Kro- necker symbol � p r � can be found in [5] (Corollary 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 428).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally a formula developed by Cohen [1] [Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='16] provides the class number too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The main purpose of this paper is to obtain some identities for this num- ber h(−p) by computing the quadratic residues in a different manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Here we provide a summary of the main formulas obtained in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Here Qk = � 1 + √4kp + 12n − 7 � /2, M = ⌊n2/p⌋ and Jn is the number of jumps (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' M � k=1 �� kp � − M−1 � k=0 ⌊Qk⌋ = Jn − M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 �k2 − k + 2 − 3n p � + M−1 � k=0 ⌊Qk⌋ = (M − 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' p−1 � k=1 rp � k2 − k + 2 − 3n � = p−1 � k=1 rp � k2� + 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2n � k=1 rp � k2 − k + 2 − 3n � = 2n � k=1 rp � k2� + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 rp � k2 − k + 2 − 3n � = n � k=1 rp � k2� + n(n + 1) 2 − p(Jn − 1 − M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' p−1 � k=1 �k2 − k + 2 − 3n p � = p(p − 5) + 6 − n 3 − 1 p p−1 � k=1 rp � k2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2n � k=1 �k2 − k + 2 − 3n p � = n(2n − 1)(4n − 7) 3p − 1 p 2n � k=1 rp � k2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 �k2 − k + 2 − 3n p � = (n + 1)(2n2 − 23n + 6) 3p − 1 p n � k=1 rp � k2� + Jn − M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2 This is a summary of the identities involving the class number h = h(−p) found on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Jn 2 + M(n − 1) = h 4 + M � k=1 �� kp � + n2 − 5n + 9 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' −Jn 2 + Mn = h 4 + M−1 � k=0 ⌊Qk⌋ + n2 − 5n − 3 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Jn 2 + n−1 � k=1 �k2 p � = h 4 + n2 − 5n + 9 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' p−1 � k=1 rp � k2 − k + 2 − 3n � = p · (2n − h) − n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2n � k=1 rp � k2 − k + 2 − 3n � = p · (2n − h) + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 rp � k2 − k + 2 − 3n � = p 4(3n + 2 − 2Jn − h) − (n + 1)(n − 1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 rp � k2� = p 4(2Jn + 2n − 3 − 4M − h) + n(n + 1) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' p−1 � k=1 �k2 − k + 2 − 3n p � = h + 16n2 − 35n + 15 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2n � k=1 �k2 − k + 2 − 3n p � = h 2 + 4n2 − 14n + 9 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 �k2 − k + 2 − 3n p � = h 4 + Jn 2 + n2 − 17n − 3 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 �k2 − k + 2 − 3n p � = h 2 + M(1 − n) + n2 − 11n + 3 6 + M � k=1 �� kp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 3 In Section 2 we provide the main definitions and the notation used in the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Here, we develop the concepts of jump and total residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We also identify when these jumps occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' In Section 3 we organize the jumps on six different sets and we state the main lemmas that will be used to count the jumps which is done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It is here where we define bijective functions among different pairs of sets to compute their cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' In Section 5 we obtain the sums of quadratic residues of terms of the form k2 − k + 2 − 3n where k ranges on the different intervals [1, 4n − 2], [1, 2n] or [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' In this section we also count to total amount of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally in Section 6 we establish several identities involving the class number h(−p) of the imaginary quadratic field Q(√−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' These identities are based on the sums found in previous sections and in some of these identities the jumps quantity appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Whereas in [4] we computed several sums of quadratic residues when p = 4n + 1, in this paper, we perform a similar but different analysis when p = 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It is on this paper where the class number is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' On some occasions we present both formulas (p = 4n − 1 and p = 4n + 1) for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2 Sum of Quadratic Residues and Jumps Consider p = 4n − 1 a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It will be understood that when we write n, we mean a natural number n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1 Let q be a positive integer and x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By rq(x) we denote the remainder of x when we divide by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence rq(x) ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', q − 1} satisfies x = m · q + rq(x), for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly, m = ⌊x/q⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The following notation found in [3] will be useful during the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 4 Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 For m ∈ Z, p = 4n − 1 prime and m ≥ 0 we denote Qm = 1 2 + 1 2 � 1 + 4 [(m + 1)p − n − 1] = 1 2 + 1 2 � 4mp + 3p − 4 , Rm = √mp , M = �n2 p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' In [3], we obtained a theorem that involves the sum of quadratic residues when p = 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' For reference purposes, we write here such theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 Let p = 4n − 1 be prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Using Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 we have 1 2 p−1 � k=1 rp � k2� = p � M � m=1 ⌊Rm⌋ + M−1 � m=0 ⌊Qm⌋ � − Mp(2n − 1) + p · (n2 + n) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' A concept arises naturally when we study the term Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This term is the positive root of the quadratic polynomial x2 − x + 2 − 3n − mp which is the same as (x − 1)2 + x + 1 − 3n − mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4 Let p = 4n − 1 be a prime and 0 ≤ k ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The total residue of k is defined and denoted by Γ(k) = rp � (k − 1)2� + rp (k + 1 − 3n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We also say that k is a jump if its total residue is p or more, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' if Γ(k) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' What is the importance of the jumps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Firstly, in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 (see [3]), a key technique is adding rp (k2) and rp ((2n − k)2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It happens that when k ∈ (Rm, Qm] this sum is constant, but as soon as k exceeds Qm, we need to subtract p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore knowing when we need to subtract p is key to comprehend better the formula in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Secondly, the amount of jumps in the interval [2, n+2] will allow us to establish a formula to compute the terms �M−1 m=0 ⌊Qm⌋ , �M m=1 ⌊Rm⌋ as well as �n k=0 rp (k2) as a function of n, h(−p) and the number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This is achieved in Corollaries 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The following three lemmas allow us to identify some jumps and when they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' For notation purposes we define Z0 = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5 Let p = 4n − 1 be a prime and m ∈ Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then ⌊Qm⌋ < Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Assume there is j ∈ Z such that j = � 1 + � 4mp + 12n − 7 � /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then j2 − j + 2 − 3n = mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Taking x0 = 2n − j gives x2 0 = 4n2 − n + mp + j − 4nj + n + 3n − 2 = (n + m − j + 1)p − 1, hence x2 0 ≡ −1 (mod p) which contradicts Fermat Little Theorem as p = 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6 By using the same argument, there is no integer k with k2 − 3k + 4 − 3n = mp, else taking k − 1 = j we obtain j2 − j + 2 − 3n = mp which leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7 Let p = 4n − 1 be a prime with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Let m ∈ Z0 with 0 ≤ m ≤ �n2 − 4n + 5 p � and km = 1 + �1 + √4mp + 12n − 7 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then (i) 3 ≤ km ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (ii) 1 + √4mp + 12n − 7 < 2km < 3 + √4mp + 12n − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iii) km is a jump and (iv) 3n − km − 1 < rp � (km − 1)2� = (km − 1)2 − mp ≤ 3n + km − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (i) Notice that 4mp + 12n − 7 ≤ 4n2 + 4n + 1, hence km ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since 2 ≤ n, 9 ≤ 4mp + 12n − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore 3 ≤ km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (ii) Notice that km is strictly above the positive root of x2−x+2−3n−mp, hence k2 m − km + 2 − 3n > mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3) 6 Now km ≤ n + 2 ≤ 3n and 2 ≤ n imply (km−1)2−mp = k2 m−km+2−3n−mp+3n−km−1 ≥ 3n−km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4) From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5, km < � 3 + √4mp + 12n − 7 � /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The other inequal- ity is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iii) From (ii), km is less than the positive root of x2 − 3x + 4 − mp − 3n, hence k2 m − 3km + 4 − mp − 3n ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore (km − 1)2 − mp ≤ 3n + km − 4 ≤ 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5) It is impossible that (km−1)2−mp = 4n, otherwise (km−1)2−1 = (m+ 1)p, hence p would divide (km − 2)km which forces km = 0 or km = 2, which contradicts km ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence (km − 1)2 − mp ≤ 4n − 1, however, if (km − 1)2 − mp = 4n − 1, then km = 1 which is again, impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then (km − 1)2 − mp < p and hence rp ((km − 1)2) = (km − 1)2 − mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now Γ(km) = rp ((km − 1)2) + rp (km + 1 − 3n) = (km − 1)2 − mp + km + 1 − 3n − p as km ≤ n + 2 ≤ 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now Inequality 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 implies Γ(km) = k2 m − km + 2 − 3n − mp + p ≥ p, hence km is a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iv) To finish the proof, we observe that from Inequalities 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5 we obtain 3n − km ≤ rp � (km − 1)2� = (km − 1)2 − mp ≤ 3n + km − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ The following lemma allows us to find more jumps based on the ones found in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='8 Consider km be the jump in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7 and k ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If km < k ≤ 1 + �√mp + p − 1 � then k is a jump and 3n + k − 3 < rp � (k − 1)2� = (k − 1)2 − mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly km < k implies mp ≤ (km − 1)2 < (k − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since k ≤ �√mp + p − 1 � , (k −1)2 ≤ mp+p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence rp ((k − 1)2) = (k −1)2 −mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 7 Since k ≤ n+2, rp (k + 1 − 3n) = k+1−3n−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We know that k is strictly above the positive root of x2 −x+2−3n−mp, hence k2 −k +2−3n > mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This implies that Γ(k) =rp � (k − 1)2� + rp (k + 1 − 3n) = (k − 1)2 − mp + k + 1 − 3n + p =k2 − k + 2 − 3n − mp + p ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' and then k is a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7, � 1 + √4mp + 12n − 7 � /2 < km ≤ k−1, hence � 3 + √4mp + 12n − 7 � /2 < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore 0 < k2 − 3k + 4 − mp − 3n, then 3n + k − 3 < (k − 1)2 − mp = rp � (k − 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='9 Note that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7, the jumps km satisty 3n − km − 1 < rp ((k − 1)2) ≤ 3n + km − 4 and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='8, the jumps k > km satisfy 3n + k − 3 < rp ((k − 1)2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now if k is not a jump and 2 ≤ k ≤ n + 2, then necessarily rp ((k − 1)2) < 3n − k − 1 as the following lemma shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The following two lemmas are about the total residues and will help us to count the total amount of jumps in the interval [1, 4n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We will only prove the fist one as the proof of the second one is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10 Let n ≥ 2 and 2 ≤ k ≤ n + 2 and p = 4n − 1 prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (a) If rp ((k − 1)2) < 3n − k − 1 then Γ(k) < p and Γ(p + 2 − k) < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) If 3n−k −1 ≤ rp ((k − 1)2) < 3n+ k −3 then Γ(p + 2 −k) < p ≤ Γ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (c) If 3n + k − 3 ≤ rp ((k − 1)2) then p ≤ Γ(p + 2 − k) and p ≤ Γ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Note that −(4n − 1) ≤ k + 1 − 3n ≤ −1, therefore rp (k + 1 − 3n) = k + 1 − 3n + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Similarly, k ≤ n + 2 implies 0 ≤ p + 3 − k − 3n ≤ p − 1 and hence rp (k2) p + 3 − k − 3n = p + 3 − k − 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 8 (a) Clearly Γ(k) =rp � (k − 1)2� + rp (k + 1 − 3n) , =rp � (k − 1)2� + p + k + 1 − 3n < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Γ(p + 2 − k) =rp � (p − (k − 1))2� + rp (p + 3 − k − 3n) , =rp � (k − 1)2� + p + 3 − k − 3n < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) Here Γ(k) = rp ((k − 1)2) + p + k + 1 − 3n ≥ p and Γ(p + 2 − k) = rp ((k − 1)2) + p + 3 − k − 3n < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (c) Finally Γ(k) = rp ((k − 1)2) + p + k + 1 − 3n ≥ p and Γ(p + 2 − k) = rp ((k − 1)2) + p + 3 − k − 3n ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Note that 2 ≤ k ≤ n + 2 implies 0 ≤ p + 2 − k ≤ p, hence Γ(p + 2 − k) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also when k = 0 or k = 1, rp (k2) < 3n − k − 1 and Γ(k) < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then Γ(p + 2 − k) is not defined as p + 2 − k > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='11 Let n ≥ 2, p = 4n − 1 prime and n + 3 ≤ k ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (a) If rp ((k − 1)2) < k − n − 2 then Γ(k) < p and Γ(p + 2 − k) < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) If k − n − 2 ≤ rp ((k − 1)2) < 3n − k − 1 then Γ(k) < p ≤ Γ(p + 2 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (c) If 3n − k − 1 ≤ rp ((k − 1)2) then p ≤ Γ(k) and Γ(p + 2 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (very similar to the one of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=') □ 3 Splitting the Jumps For future reference we define the following sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 9 Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1 For p = 4n − 1 prime, denote C< = � k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', n + 2} : rp � (k − 1)2� < 3n − k − 1 � , C[−−) = � k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', n + 2} : rp � (k − 1)2� ∈ [3n − k − 1, 3n + k − 3) � , C≥ = � k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', n + 2} : rp � (k − 1)2� ≥ 3n + k − 3 � , D< = � k ∈ {n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', 2n} : rp � (k − 1)2� < k − n − 2 � , D[−−) = � k ∈ {n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', 2n} : rp � (k − 1)2� ∈ [k − n − 2, 3n − k − 1) � , D≥ = � k ∈ {n + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', 2n} : rp � (k − 1)2� ≥ 3n − k − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 Let ℓ ∈ Z and p = 4n − 1 prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (a) Let ℓ ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ + 2 or n = 4ℓ + 3 then n − 3, n − 1, n + 1 ∈ C[−−) and n − 2, n, n + 2 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) Let ℓ ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ + 1 or n = 4ℓ + 4 then n − 3, n − 1, n + 1 ∈ C< and n − 2, n, n + 2 ∈ C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Table 1 summarizes the residues of (k − 1)2 for k = n − 3, n − 2, n − 1, n, n + 1 and n + 2 given the four different cases for n which are 4ℓ + 1, 4ℓ + 2, 4ℓ + 3 and 4ℓ + 4, ℓ ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We only verify the first column of Table 1, that is, we will compute the residues of (k − 1)2 when k = n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4ℓ − 3)2 = (16ℓ + 3)(ℓ − 2) + 5ℓ + 15, 0 ≤ 5ℓ + 15 ≤ 16ℓ + 2, (4ℓ − 2)2 = (16ℓ + 7)(ℓ − 2) + 9ℓ + 18, 0 ≤ 9ℓ + 18 ≤ 16ℓ + 6, (4ℓ − 1)2 = (16ℓ + 11)(ℓ − 2) + 13ℓ + 23, 0 ≤ 13ℓ + 23 ≤ 16ℓ + 10, (4ℓ)2 = (16ℓ + 15)(ℓ − 1) + ℓ + 15, 0 ≤ ℓ + 15 ≤ 16ℓ + 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' For a given n, k, denote Ik n = [3n−k −1, 3n+k −3) and ∆k n = rp ((k − 1)2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (a) Consider n = 4ℓ+2 and k = n−3 then Ik n = [8ℓ+6, 16ℓ+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' According to Table 1, ∆k n = 9ℓ + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We observe that if ℓ ≥ 3, ∆k n ∈ Ik n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Similarly, if k = n − 1, Ik n = [8ℓ + 4, 16ℓ + 4) and ∆k n = 9ℓ + 7 ∈ Ik n for ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k = n + 1, Ik n = [8ℓ + 2, 16ℓ + 6) and ∆k n = 9ℓ + 4 ∈ Ik n for ℓ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence for ℓ ≥ 6, n − 3, n − 1, n + 1 ∈ C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Notice that if k = n − 2, ∆k n = ℓ + 8 /∈ [8ℓ + 5, 16ℓ + 3) = Ik n when ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Likewise, if k = n, ∆k n = ℓ + 1 /∈ [8ℓ + 3, 16ℓ + 5) = Ik n when ℓ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 10 k n − 3 n − 2 n − 1 n n + 1 n + 2 n = 4ℓ + 1 p = 16ℓ + 3 5ℓ + 15 13ℓ + 10 5ℓ + 4 13ℓ + 3 5ℓ + 1 13ℓ + 4 n = 4ℓ + 2 p = 16ℓ + 7 9ℓ + 18 ℓ + 8 9ℓ + 7 ℓ + 1 9ℓ + 4 ℓ + 2 n = 4ℓ + 3 p = 16ℓ + 11 13ℓ + 23 5ℓ + 11 13ℓ + 12 5ℓ + 4 13ℓ + 9 5ℓ + 5 n = 4ℓ + 4 p = 16ℓ + 15 ℓ + 15 9ℓ + 16 ℓ + 4 9ℓ + 9 ℓ + 1 9ℓ + 10 Table 1: Residues of (k − 1)2 when k = n − 3, n − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', n + 2 for the different cases of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally, if k = n + 2, ∆k n = ℓ + 2 /∈ [8ℓ + 1, 16ℓ + 7) = Ik n when ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore n − 2, n, n + 2 /∈ C[−−) when ℓ ≥ 1, in fact, n − 2, n, n + 2 ∈ C< for ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The case n = 4ℓ + 3 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) This case is done analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 If n > 3 either Γ(n + 2) < p ≤ Γ(n + 1) or Γ(n + 1) < p ≤ Γ(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, if n = 4ℓ + 2 or n = 4ℓ + 3 and ℓ ≥ 6, then n + 1 ∈ C[−−) and n + 2 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10, Γ(n + 2) < p ≤ Γ(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, if n = 4ℓ + 1 or n = 4ℓ + 4 and ℓ ≥ 4, then n + 1 ∈ C< and n + 2 ∈ C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10, Γ(n + 1) < p ≤ Γ(n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We only need to verify the cases n = 5, 6, 8, 11, 12, 15 and 18 as the cases n = 4, 7, 9, 10, 13, 16 and 19 do not give prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (i) If n = 5 then n + 2 ∈ C[−−) = {5, 7} and n + 1 ∈ C< = {2, 3, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (ii) If n = 6 then n + 1 ∈ C[−−) = {5, 7} and n + 2 ∈ C< = {2, 3, 4, 6, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 11 (iii) If n = 8 then n+2 ∈ C[−−) = {6, 8, 10} and n+1 ∈ C< = {2, 3, 4, 5, 7, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iv) If n = 11 then n + 1 ∈ C[−−) = {7, 10, 12} and n + 2 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (v) If n = 12 then n + 2 ∈ C[−−) = {7, 10, 12, 14} and n + 1 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (vi) If n = 15 then n + 1 ∈ C[−−) = {8, 11, 14, 16} and n + 2 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (vii) If n = 18 then n + 1 ∈ C[−−) = {8, 12, 15, 17, 19} and n + 2 ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' An application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10 gives us the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Observe that when n = 3, C[−−) = {4, 5}, C< = {2, 3}, in this case both Γ(n + 1) = 16 and Γ(n + 2) = 13 are greater than p = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ The following two lemmas allow us to compute specifically the cardinality of C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4 Let n > 3, p = 4n − 1 prime and M0 = ⌊(n2 − 4n + 5)/p⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Define ℓm = 1 + �√mp + p − 1 � and consider km defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Let 2 ≤ k ≤ n + 2 and 0 ≤ m ≤ M0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k < k0, k > ℓM0 or ℓm < k ≤ km+1 then k /∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also if k < k0 or ℓm < k < km+1 then k is not a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider k < k0 = 1+ � (1 + √12n − 7)/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then (2k−1)2 < 12n−7 from which (k − 1)2 < 3n − k − 1 ≤ 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence rp ((k − 1)2) = (k − 1)2 < 3n − k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10, k ∈ C< and k is not a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider k > ℓM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Notice that ℓM0 = 1 + �√Mp � ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, either k ∈ C[−−) or k ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k = km+1, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7 (iv), k ∈ C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally consider ℓm < k < km+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then 1 + � (m + 1)p < k < 1 + � 4(m + 1)p + 12n − 7 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence (m + 1)p < (k − 1)2 < (m + 1)p + 3n − k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore rp ((k − 1)2) = (k−1)2−(m+1)p < 3n−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, k is not a jump and k ∈ C<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ 12 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5 Consider n, p and M0 as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then k ∈ C≥ if and only if there is m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', M0} such that km < k ≤ 1 + �√mp + p − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='8, if km < k ≤ ℓm then k is a jump and rp ((k − 1)2) > 3n + k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Note that 2 ≤ k0 < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since m ≤ M0 we have ℓm ≤ n + 2, hence 2 ≤ k ≤ n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore k ∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Conversely, let k ∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then 2 ≤ k ≤ n+2 and rp ((k − 1)2) ≥ 3n+k−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Observe that [2, n + 2] = [2, k0] ∪ (k0, ℓ0] ∪ (ℓ0, k1] ∪ (k1, ℓ1] ∪ · · · ∪ (kM0, ℓM0] ∪ (ℓM0, n + 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k < k0, k > ℓM0 or ℓm < k ≤ km+1, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4, k /∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This forces k to be in (km, ℓm] for some m ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', M0}, which is what we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ 4 Counting the Jumps In this section we will relate the jumps in the different sets C<, C[−−), C≥, D<, D[−−) and D≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We will compute different cardinalities when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' In this section we consider n > 3 and p = 4n − 1 a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1 The function f : C≥ −→ D< defined by f(k) = 2n + 2 − k, is well-defined and bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider k ∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5 and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='9, there is m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=', M0} with M0 = ⌊(n2 − 4n + 5)/p⌋ such that km < k ≤ 1 + �√mp + p − 1 � and 3n + k − 3 ≤ (k − 1)2 − mp = rp ((k − 1)2) , where km is the jump given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence 0 ≤ k2 − 3k − 3n + 4 − mp, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6) k2 − 2k + 1 − mp ≤ 4n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7) 13 Let kf = f(k) = 2n + 2 − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We will first prove that kf ∈ D<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now (kf − 1)2 = (n − k + m + 2)p + (k − 1)2 − mp + n − k + 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Let w = (k − 1)2 − mp + n − k + 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6, w ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6, k2 − 3k + 4 − 3n − mp ̸= 0, hence w ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Note that in the case of equality in Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7, we would have (k−1)2 = mp + p − 1, hence the congruency x2 ≡ −1 (mod p) would have a solution, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore k2 − 2k + 1 − mp < 4n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='8) From Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7, w = k2 − 2k + 1 − mp + 2 − k − 3n < n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='9) Clearly n − k ≤ 4n − 2, hence w < p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore rp ((kf − 1)2) = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='9, 0 ≤ w < n−k, this forces k ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also from Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='9, since kf − n − 2 = n − k, we conclude that rp ((kf − 1)2) = w < kf − n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally, 2 ≤ k ≤ n − 1 implies n + 3 ≤ kf ≤ 2n, hence f is well defined as kf ∈ D<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Take now �k ∈ D<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then n + 3 ≤ �k ≤ 2n and rp � (�k − 1)2� < �k − n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10) Consider k = 2n + 2 − �k and �m ∈ Z with �m ≤ (�k − 1)2 < �m · p + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='11) Then 2 ≤ k ≤ n − 1 and (k − 1)2 = (n − �k + �m) · p + (�k − 1)2 − �mp + n − �k + 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider m = n − �k + �m and u = (�k − 1)2 − �mp + n − �k + 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since �k ≤ 5n, Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='11 implies 0 ≤ n − �k + 1 − p ≤ (k − 1)2 − �mp + n − �k + 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='12) Also from Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10, we have (k − 1)2 − �mp + n − �k + 1 − p < p − 1, therefore 0 ≤ u < p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence rp ((k − 1)2) = (k − 1)2 − mp = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='12 implies u ≥ 5n − �k ≥ 5n − 1 − �k = 3n + k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore rp ((k − 1)2) ≥ 3n + k − 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' k ∈ C≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly f(k) = �k, then f is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ 14 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 Let y, z be the last two elements in C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k ∈ C[−−) − {y, z} then mp + p ≤ (n − 2)2 + 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13) where m = m(k) = ⌊(k − 1)2/p⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (a) Case n = 4ℓ + 2 or n = 4ℓ + 3, ℓ ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, k ∈ C[−−) − {y, z} implies k ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence m ≤ ℓ − 2 = ⌊(n − 4)2/p⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ + 2 then mp + p ≤ (ℓ − 1)(16ℓ + 7) ≤ 16ℓ2 + 1 = (n − 2)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ+3 then mp+p ≤ (ℓ−1)(16ℓ+11) ≤ 16ℓ2+8ℓ+2 = (n−2)2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (b) Case n = 4ℓ + 1 or n = 4ℓ + 4, ℓ ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, k ∈ C[−−) − {y, z} implies k ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ + 1 then m ≤ ⌊(n − 3)2/p⌋ = ℓ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence mp + p ≤ (ℓ − 1)(16ℓ + 3) ≤ 16ℓ2 − 8ℓ + 2 = (n − 2)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If n = 4ℓ+4 then m ≤ ⌊(n − 3)2/p⌋ = ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence mp+p ≤ ℓ·(16ℓ+15) ≤ 16ℓ2 + 16ℓ + 5 = (n − 2)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (c) The only left cases are n = 5, 6, 8, 11, 12, 15 and 18 as the choices n = 4, 7, 9, 10, 13, 14, 16, 19, 22, 23 do not provide prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (i) If n = 5 or 6 then C[−−) has only two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (ii) If n = 8 then C[−−) = {6, 8, 10} and k = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore m(k) = 0 = ⌊(k − 1)2/p⌋ = ⌊25/31⌋ clearly satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iii) If n = 11 then C[−−) = {7, 10, 12} and k = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore m(k) = 0 clearly satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (iv) If n = 12 then C[−−) = {7, 10, 12, 14}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k = 10, then m(k) = 1 = ⌊(k − 1)2/p⌋ = ⌊81/47⌋ clearly satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13 as 94 ≤ 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly m(7) also does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (v) If n = 15 then C[−−) = {8, 11, 14, 16}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k = 11, then m = 1 = ⌊(k − 1)2/p⌋ = ⌊100/59⌋ clearly satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13 as 118 ≤ 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly m(8) also does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (vi) If n = 18 then n + 1 ∈ C[−−) = {8, 12, 15, 17, 19}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' If k = 15, then m(k) = 2 = ⌊(k − 1)2/p⌋ = ⌊196/71⌋ clearly satisfies Inequal- ity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13 as 213 ≤ 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly m(8), m(12) also satisfy Inequal- ity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 15 □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 k ∈ D[−−) if and only if there is an integer m with 1 ≤ m ≤ ⌊(n2 − 4n + 5)/p⌋ and k = 2n − �� mp − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' ⇒) Let k ∈ D[−−) and define α = 2n + 1 − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence 1 ≤ α ≤ n − 2 and (k − 1)2 = 4n2 + α2 − 4nα = (n − α + m)p + α2 + n − α − mp, where m satisfies mp ≤ α2+n−α < (m+1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore rp ((k − 1)2) = α2+n−α−mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since k−n−2 ≤ rp ((k − 1)2) < 3n−k−1, mp−1 ≤ α2 and (α − 1)2 < mp − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence √mp − 1 ≤ α < √mp − 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since x2 ≡ −1 (mod p) has no solution, √mp − 1 is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore α = �√mp − 1 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then k = 2n − �√mp − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since 0 ≤ α2 + n − α, m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now (α − 1)2 < mp − 1 implies 1 ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly mp ≤ α2 + 1 ≤ (n − 2)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore m ≤ (n2 − 4n + 5)/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' ⇐) Consider an integer m with 1 ≤ m ≤ (n2 − 4n + 5)/p and k = 2n − �√mp − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since mp − 1 ≤ (n − 2)2, n + 2 ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly n + 2 = k leads us to an integer solution of x2 ≡ −1 (mod p), therefore n + 3 ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also 1 ≤ m implies k ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider α = �√mp − 1 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then √mp − 1 ≤ α < √mp − 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Clearly α ̸= √mp − 1 + 1 (otherwise x2 ≡ −1 (mod p) has an integer solution), then √mp − 1 < α < √mp − 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence mp − 1 < α2 and (α − 1)2 < mp − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since α < n − 1, mp ≤ α2 ≤ α2 + n − α < n + α − 2 + mp < mp + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since (k − 1)2 = (2n − α)2 = (n − α + m)p + α2 + n − α − mp, rp ((k − 1)2) = α2 + n − α − mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Finally, from k − n − 2 = n − α − 1 < rp ((k − 1)2) < n + α − 2 = 3n − k − 1, we conclude that k ∈ D[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ 16 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4 Let y, z the last two elements of C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' For k ∈ C[−−) − {y, z}, consider m = ⌊(k − 1)2/p⌋ and u0 = u0(k) the first integer less than or equal to n + 2 such that x = u0 satisfies (m + 1) p ≤ (k + x − 1)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='14) Then, the function f : C[−−) − {y, z} −→ D[−−) defined by f(k) = kf = 2n + 2 − u0 − k, is well-defined and bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' First, we will prove that f is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It is not hard to check that u0 = �� (m + 1)p � +2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' The definition of u0 implies that u0, u0+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' satisfy Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='14 but u0 − 1 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence u0 satisfies 4n + mp ≤ k2 + 2k(u0 − 1) + u2 0 − 2u0 + 3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='15) k2 + 2k(u0 − 2) + u2 0 − 4u0 + 6 < 4n + mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='16) If k2 = mp + 3n + 3k − 2 for 2 ≤ k ≤ n + 2 and we define x0 = 2n − k + 1, then x2 0 = (n+ m−k + 2)p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore p divides (x0 −1)(x0 + 1), however since n ≥ 3 and 2 ≤ k ≤ n + 2, we have that 1 ≤ 2n − k = x0 − 1 < x0 + 1 = 2n + 2 − k ≤ 4n − 2, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore k2 ̸= mp + 3n + 3k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='17) Observe that u0 ≥ 1 as x = 0 does not satisfy Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also u0 ≤ n+2 as (k + n + 1)2 ≥ (k − 1)2 + (n + 2)2 ≥ mp + p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' To shorten notation, define \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 mf = n + 2 − k − u0 + m, ∆ = rp ((k − 1)2) , ∆f = rp ((kf − 1)2) , wf = ∆ + k(2u0 − 1) − p + n + u2 0 − 3u0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' To check that f is well-defined, we need to verify that n + 3 ≤ kf ≤ 2n and kf − n − 2 ≤ ∆f < 3n − kf − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' It is not hard to check that (kf − 1)2 = mf · p + wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 17 Notice that ∆ = rp ((k − 1)2) = (k − 1)2 − mp ≤ 3n − k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Inequal- ity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='17, ∆ ≥ 3n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore, wf ≥ 3n − k + +k(2u0 − 1) − p + n + u2 0 − 3u0 + 1, = k(2u0 − 2) + u2 0 − 3u0 + 2, ≥ u2 0 − 3u0 + 2 = (u0 − 2)(u0 − 1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='16, wf = k2 + 2k(u0 − 2) + u2 0 − 4u0 + 6 − 4n − mp + k + u0 + n − 3, < k + u0 + n − 3 ≤ 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This shows that wf = rp ((kf − 1)2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since 3n − kf − 1 = n − 3 + u0 + k, wf < 3n − kf − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since kf − n − 2 = n − u0 − k, Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='15 implies that wf ≥ kf −n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore kf −n−2 ≤ rp ((kf − 1)2) < 3n−kf −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2, x = n−k −1 satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='14, hence 1 ≤ u0 ≤ n−k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore n + 3 ≤ kf ≤ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This proves that kf ∈ D[−−) and thus f is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider k0 and k1 such that kf = f(k0) = f(k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Take u0, u1, m0 and m1 such that m0 = ⌊(k0 − 1)2/p⌋ , m1 = ⌊(k1 − 1)2/p⌋ and u0, u1 are the first integers such that Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='14 holds with k = k0 and k = k1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now f(k0) = f(k1) implies that u0 + k0 = u1 + k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since mf = n + 2 − k0 − u0 + m0 = n + 2 − k1 − u1 + m1, we conclude m0 = m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since wf = (k0+u0−1)2−u0−m0p−p+n+1 = (k1+u1−1)2−u1−m1p−p+n+1, u0 = u1 and consequently k0 = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Take now kf ∈ D[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Let mf = ⌊(kf − 1)2/p⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then kf − n − 2 ≤ ∆f < 3n − kf − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Notice that x = 0 satisfies the inequality 0 ≤ (kf + x − 1)2 + 1 − mfp − px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='18) Let v0 the first integer greater than or equal to 1 such that x = v0 does not satisfy Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence v0 = x satisfies the equivalent inequalities (kf + x − 1)2 + 1 − mfp − px < 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='19) ∆f + 2kfx + (x − 1)2 − px < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 18 Consider k = 2n + 2 − v0 − kf, m = n − kf + mf and w = ∆f + kf(2v0 − 1) + n + (v2 0 − 3v0 + 1) − v0p + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then (k − 1)2 = (n + 1 − kf + mf)p + k2 f + kf(2v0 − 3) + n + 2 + v2 0 − 3v0 − v0p − mp, = (n − kf + mf)p + ∆f + kf(2v0 − 1) + n + (v2 0 − 3v0 + 1) − v0p + p, = mp + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3, there is an integer m, 1 ≤ m ≤ (n2 − 4n + 5)/p such that kf = 2n− �√mp − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Also from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 if α = 2n−kf + 1 then mf = n − α + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Take x0 = α − 1 = 2n − kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Notice that x0 ≥ 1 as √mp − 1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Substituting x = x0 into (kf + x − 1)2 + 1 − mfp − px gives us (2n − 1)2 + 1 − (n − α + m)p − p(α − 1) = n + 3 − mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since m ≥ 1, n+3−mp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then x = x0 satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='19, therefore v0 exists and v0 ≤ 2n−kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since x = v0 −1 satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='18, we have that 0 ≤ ∆f + kf(2v0 − 2) + (v0 − 2)2 + p − v0p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Hence w ≥ kf + n + v0 − 3 = 3n − k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since kf ≥ n + 3 and v0 ≥ 1 we conclude w ≥ 0 and 2 ≤ k ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since v0 satisfies Inequality 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='19, w < n − kf − v0 + p = 5n − kf − v0 − 1 = 3n + k − 3 ≤ 4n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Thus 0 ≤ w < p and 3n − k − 1 ≤ w < 3n + k − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' This implies that w = rp ((k − 1)2) , m = ⌊(k − 1)2/p⌋ and hence k ∈ C[−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 and the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='2 implies that {y, z} is a subset of {n − 1, n, n + 1, n + 2}, then k ∈ C[−−) − {y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Now we will find u0 = u0(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since ∆f ≥ kf − n − 2, (k + v0 − 1)2 + 1 = (2n + 1 − kf)2 + 1, = (n − kf + mf + 1)p + ∆f + n − kf + 2, = (m + 1)p + ∆f + n − kf + 2 ≥ (m + 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 19 From ∆f < 3n − kf − 1 = p − n − kf, we obtain (k + v0 − 2)2 + 1 = (2n − kf)2 + 1, = (n − kf + mf + 1)p + ∆f + kf + n − p, = (m + 1)p + ∆f + kf + n − p < (m + 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Therefore u0 = v0 and f(k) = kf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Then f is surjective and hence bijec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5 ��D[−−) �� = ⌊(n2 − 4n + 5)/2⌋ and ��C≥ �� = ��D< ��, ��C[−−) �� = ��D[−−) �� + 2, ��C< �� = ��D≥ �� + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1, ��C≥ �� = ��D< ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4, ��C[−−) �� = ��D[−−) ��+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Since n + 1 = ��C< �� + ��C[−−) �� + ��C≥ �� and n − 2 = ��D< �� + ��D[−−) �� + ��D≥ �� we have n + 1 = ��C< �� + ��D< �� + ��D[−−) �� + 2 = ��C< �� + n − ��D≥ ��, hence ��C< �� = ��D≥ �� + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' To see that ��D[−−) �� = ⌊(n2 − 4n + 5)/2⌋ it is enough to see that all the k′s in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='3 given by each m are all different, which is the case as � mp − 1 + 1 < � (m + 1)p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6 Under the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4, ���{k ∈ Z | 2 ≤ k ≤ 4n − 1, Γ(k) ≥ p} ��� = 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Let JΓ = {k ∈ Z : 2 ≤ k ≤ 4n − 2, Γ(k) ≥ p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='10 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='11, JΓ ∩ [2, n + 2] = C[−−) ∪ C≥, JΓ ∩ [n + 3, 2n] = D≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 20 ��JΓ ∩ [2n + 1, 3n − 2] �� = ���{2n + 1 ≤ k ≤ 3n − 2 | Γ(k) ≥ p} ���, = ���{2n + 1 ≤ p + 2 − k ≤ 3n − 2 | Γ(p + 2 − k) ≥ p} ���, = ���{n + 3 ≤ k ≤ 2n | Γ(p + 2 − k) ≥ p} ��� = ��D[−−) �� + ��D≥ ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' ��JΓ ∩ [3n − 1, 4n − 1] �� = ���{3n − 1 ≤ k ≤ 4n − 1 | Γ(k) ≥ p} ���, = ���{3n − 1 ≤ p + 2 − k ≤ 4n − 1 | Γ(p + 2 − k) ≥ p} ���, = ���{2 ≤ k ≤ n + 2 | Γ(p + 2 − k) ≥ p} ��� = ��C≥ ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Using these identities and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='5, we obtain ��JΓ �� = ��C[−−) �� + 2 ��C≥ �� + ��D[−−) �� + 2 ��D≥ ��, = ��C[−−) �� + 2 ��C≥ �� + ��C[−−) �� − 2 + 2 ���C< �� − 1 � , = 2 ���C< �� + ��C[−−) �� + ��C≥ ��� − 4, = 2 · ���Z ∩ [2, n + 2] ��� − 4 = 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' □ The following corollary comes from proof of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='7 Under the hypotheses of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='4, ��� {k ∈ Z : 2 ≤ k ≤ 2n, Γ(k) ≥ p} ��� = n, ��� {k ∈ Z : 2n + 1 ≤ k ≤ 4n, Γ(k) ≥ p} ��� = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 21 5 Sums involving rp � k2 − k + 2 − 3n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Consider Jn = ��� {k ∈ Z : 2 ≤ k ≤ n + 2, Γ(k) ≥ p} ��� and we call Jn simply the number of jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' We will now develop formulas relating the term residues of k2 − k + 2 − 3n modulus p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='1 If n > 3 and p = 4n − 1 is prime then p−1 � k=1 rp � k2 − k + 2 − 3n � = p−1 � k=1 rp � k2� + 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 2n � k=1 rp � k2 − k + 2 − 3n � = 2n � k=1 rp � k2� + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' n � k=1 rp � k2 − k + 2 − 3n � = n � k=1 rp � k2� + n(n + 1) 2 − p(Jn − 1 − M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Notice that rp (x + y) = � rp (x) + rp (y) if rp (x) + rp (y) < p, rp (x) + rp (y) − p if rp (x) + rp (y) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' Recall that we defined k as a jump when Γ(k) = rp ((k − 1)2)+rp (k + 1 − 3n) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE1T4oBgHgl3EQfJQP0/content/2301.02951v1.pdf'} +page_content=' 22 4n−1 � k=2 rp � k2 − k + 2 − 3n � = � k:Γ(k)≥p � rp � (k − 1)2� + rp (k + 1 − 3n) − p � + � k:Γ(k)

0 is +a function ξ : [0, T] → Rn that admits derivative at each point in [0, T] such that, for all t ∈ [0, T], +it holds true that ξ(t) ∈ X and ˙ξ(t) = f(ξ(t)) + dt for some ∥dt∥ < δ. Notably, symbol d in +Equation (1) is interpreted as a non-deterministic disturbance that at any time can take any possible +value within the bound provided by δ. +Let the sets X0 ⊂ X be a region of initial states and XB ⊂ X be a region of bad states. We say that a +trajectory ξ defined over time horizon T is initialised if ξ(0) ∈ X0; additionally, we say that it is safe +if ξ(t) ̸∈ XB for all t ∈ [0, T]; dually, we say that it is unsafe if ξ(t) ∈ XB for some t ∈ [0, T]. The +safety verification question for consists of determining whether all initialised trajectories are safe. If +this is the case, then we say that the model is safe with respect to X0 and XB. If there exist at least +one initialised trajectory that is unsafe, then we say that the model is unsafe. +We tackle safety verification by abstraction, that is, we construct an abstract dynamical model that +captures all behaviours of the concrete nonlinear model. This implies that if the abstract model is safe +then the concrete model is necessarily safe too, and we can thus apply a verification procedure over the +abstraction to determine whether the concrete model is safe. Notably, the converse may not hold: lack +of safety of the abstract model does not carry over to the concrete model, because our abstraction is +an overapproximation. We ultimately obtain a sound (but not complete) safety verification procedure. +Our approach synthesises an abstract dynamical model defined in terms a feed-forward neural network +with ReLU activation functions and endowed with a bounded non-deterministic disturbance. This +can be seen as a neural ODEs [33] augmented with an additive non-deterministic drift that ensures +the abstract model to overapproximate the concrete model. To the best of our knowledge, this is the +first work to consider neural ODEs with non-deterministic semantics. +Our feed-forward neural network consists of an n-dimensional input layer y0, k hidden layers +y1, . . . , yk with dimensions h1, . . . , hk respectively, and an n-dimensional output layer yk+1. Each +hidden or output layer with index i are respectively associated matrices of weights Wi ∈ Rhi×hi−1 +and a vectors of biases bi ∈ Rhi. Upon a valuation of the input layer, the value of every subsequent +hidden layer is given by the following equation: +yi = ReLU(Wiyi−1 + bi). +(2) +Whereas many activation functions exist, we focus our study on ReLU activation functions, applying +function max{x, 0} to every element x ∈ R of its hi-dimensional argument. Finally, the value of the +output layer is given by the affine map yk+1 = Wk+1xk + bk+1. Altogether, the network results in a +function N whose output is N(x) = yk+1 for every given input y0 = x. +Definition 2 (Neural Abstraction). Let F be a dynamical model given by function f : Rn → Rn and +disturbance radius δ ≥ 0 and let X ⊆ Rn be a region of interest. A feed-forward neural network +N : Rn → Rn defines a neural abstraction of F with error bound ϵ > 0 over X, if it holds true that +∀x ∈ X : ∥f(x) − N(x)∥ ≤ ϵ − δ. +(3) +Then, the neural abstraction consists of the dynamical model A defined by N and disturbance ϵ, +whose dynamics are given by the following neural ODE with bounded additive disturbances: +˙x = N(x) + d, +∥d∥ ≤ ϵ, +x ∈ X. +(4) +Theorem 1 (Soundness of Neural Abstractions). If A is a neural abstraction of a dynamical system +F over a region of interest X ⊆ Rn, then every trajectory of F is also a trajectory of A. +4 + +Abstraction +Synthesis +Learner +Certifier +Counterexample scex +Candidate N +Valid neural +abstraction +A +F, X, ϵ +F, S, ϵ +Safety +Verification +F, X, S, ϵ +X, X0, XB +Figure 2: Architecture for the safety verification of nonlinear dynamical models using neural abstrac- +tions. The inputs to our architecture are a concrete model F and its domain of interest X, a finite set +of initial datapoints S, a desired approximation error ϵ, and regions of initial X0 and bad states XB. +Proof of Theorem 1. Let ξ be a trajectory of F and T be the time horizon over which ξ is defined. +Then, let t ∈ [0, T]. By definition of trajectory we have that (i) ξ(t) ∈ X and there exists dt s.t. +(ii) ∥dt∥ ≤ δ and (iii) ˙ξ(t) = f(ξ(t))+dt. By (i) and condition (3) we have that ∥f(ξ(t))−N(ξ(t))∥+ +δ ≤ ϵ. Then, by (ii) we have that ∥f(ξ(t)) − N(ξ(t))∥ + ∥dt∥ ≤ ϵ which, by triangle inequality, +implies that ∥f(ξ(t)) + dt − N(ξ(t))∥ ≤ ϵ. Using (iii), we rewrite it into ∥ ˙ξ(t) − N(ξ(t))∥ ≤ ϵ. +Finally, we define d′ +t = ˙ξ(t) − N(ξ(t)). As a result, we have that ∥d′ +t∥ ≤ ϵ and ˙ξ(t) = N(ξ(t)) + d′ +t +which, together with (i), shows that ξ is a trajectory of A. +Corollary 1. Let X0 ⊂ X be a region of initial states and XB ⊂ X and region of bad states. It holds +true that if A is safe with respect to X0 and XB then also F is safe with respect to X0 and XB. +Proof of Corollary 1. By Theorem 1, if there exists an initialised trajectory of F that is unsafe, then +the same is an initialised trajectory of A that is unsafe. The statement follows by contraposition. +Remark 1 (Existence of Neural Abstractions). Let F be a dynamical model defined by function +f and disturbance radius δ ≥ 0, and let X ⊆ Rn be a domain of interest. A neural abstraction +of F with arbitrary error bound ϵ > 0 over X exists if a neural network that approximates f with +error bound ϵ − δ (cf. condition (3)) exists over the same domain. In this work, we do not prescribe +conditions on either width or depth of the network to ensure existence of a neural abstraction. Such +conditions are given by universal approximation theorems for neural networks with ReLU activation +functions, which have been developed in seminal work in machine learning [25,42,63,74,90,93]. +Altogether, we define the neural abstraction of a non-linear dynamical system F as a neural ODE with +an additive disturbance A that approximates the dynamics while also accounting for the approximation +error. Notably, we place no assumptions on the vector field f. In particular, Theorem 1 does not +require f to be Lipschitz continuous: the soundness of a neural abstraction relies on condition (3), +whose certification we offload to an SMT solver (cf. Section 3.2). The resulting neural abstraction is +to a hybrid automaton with affine dynamics and non-deterministic disturbance (cf. Section 4), which +does not rely on the Picard-Lindelof theorem to ensure uniqueness or existence of a solutions. +3 +Formal Synthesis of Neural Abstractions +Our approach to abstraction synthesis follows two phases—a learning phase and a certification phase— +that alternate each other in a CEGIS loop [1,3,43,78,101,108,109] (cf. Figure 2, left). Our learning +phase trains the parameters of a neural network N to approximate the system dynamics over a finite +set of samples S ⊂ X of the domain of interest. Learning uses gradient descent algorithms, which can +possibly scale to large amounts of samples. Then, our certification phase either confirms the validity +of condition (3) or produces a counterexample which we use to sample additional states and repeat the +loop. Certification is based on SMT solving, which reasons symbolically over the continuous domain +X and assures soundness. As a consequence, when certification confirms condition (3) formally valid, +then as per Theorem 1 our neural abstraction A is a sound overapproximation of the concrete model +F and is thus passed to safety verification (cf. Figure 2, right). +Neural networks have been used in the past as representations of formal certificates for the correctness +of systems such as Lyapunov neural networks, neural barrier certificates, neural ranking functions +5 + +and supermartingales [1, 2, 4, 31, 32, 44, 67, 89, 97, 111, 117–119]. In the present work, we use +neural networks for the first time as abstractions, and we instantiate this idea in safety verification +of nonlinear models. We shall now present the components of our abstraction synthesis procedure: +learner (cf. Section 3.1) and certifier (cf. Section 3.2). +3.1 +Learning Phase +As with many machine learning-based algorithm, learning neural abstractions hinges on the loss +function used as part of the gradient descent scheme for optimising parameters. The task is that of a +regression problem, so the choice of loss function to be minimised is simple, namely, +L = +� +s∈S +∥f(s) − N(s)∥2, +(5) +where ∥ · ∥2 represents the 2 − norm of its input, and S ⊂ X is a finite set of data points that are +sampled from the domain of interest. In other words, the neural abstractions are synthesised using a +scheme based on gradient descent to find the parameters that minimise the mean square error over S. +The main inputs to the learning procedure are the vector field f of the concrete dynamical model, +an initial set of points S sampled uniformly from the domain of interest X. Additional parameters +include the hyper-parameters for the learning scheme such as the learning rate, and a stopping +criterion for the learning procedure. For the latter, there are two possible options: a target error which +all data points must satisfy, or a bound on the value of the loss function. +If a target error smaller than ϵ − δ is provided, this is when all points in the data set S satisfy the +specification (3) and certification subsequently check that this generalises over the entire X. If an +alternative loss-based stopping criterion is provided, then an error bound on the approximation is +estimated using the maximum approximation error over the data set S for use in certification. This +estimated bound is conservative, i.e., greater than the maximum, to allow for successful certification +to be more likely. +After learning, the network N is translated to symbolic form and passed to the certification block, +which checks condition (3) as described in Section 3.2. The certifier either determines condition (3) +valid, and thus the CEGIS loop terminates, or computes a counterexample that falsifies the condition. +The counterexample is returned to the learning procedure and augmented by sampling for additional +points nearby in order to maximise the efficiency of learning and the overall synthesis. +3.2 +Certification Phase +The purpose of the certification is to check that at no point in the domain of interest X is the maximum +error greater than the upper bound ϵ − δ, as per the specification in condition (3). Therefore, the +certifier is provided with the negation of the specification, namely +∃x: x ∈ X ∧ ∥f(x) − N(x)∥ > ϵ − δ +� +�� +� +φ +. +(6) +The certifier seeks an assignment scex of the variable x such that the quantifier-free formula φ +is satisfiable, namely that the specified bound is violated. If this search is successful, then the +network N has not achieved the specified accuracy over X, and is thus not a valid neural abstraction. +The corresponding assignment scex forms the counterexample that is provided back to the learner +(the machine learning procedure from Section 3.1). Alternatively, if no assignment is found then +specification (3) is proven valid; network N and error bound ϵ are then passed to the safety verification +procedure (cf. Section 4). +Certification of the accuracy of the neural abstractions is performed by an SMT solver. Several +options exist for the selection of the SMT solver, with the requirement that the solver should reason +over quantifier-free nonlinear real arithmetic formulae [57,64]. This is because the vector field f may +contain nonlinear terms. In our experiments, we employ dReal [64], which supports polynomial and +non-polynomial terms such as transcendental functions like trigonometric or exponential ones. +A successful verification process allows for the full abstraction to be constructed using the achieved +error ϵ and neural network N. CEGIS has been shown to perform well and terminate successfully +across a wide variety of problems. We demonstrate the robustness of our procedure in Appendix B. +6 + +x +y +X1 +X2 +X3 +˙x = f1(x) +x ∈ X1 +˙x = f3(x) +x ∈ X3 +˙x = f2(x) +x ∈ X2 +x ∈ X1 +x ∈ X3 +x ∈ X2 +x ∈ X3 +Figure 3: A hybrid automaton corresponding to a state-space partitioning. Each of the three discrete +modes corresponds to a unique partition Xi and vector field fi(x). Discrete transitions are denoted by +the edges of the directed graph with a transition between two modes if the corresponding partitions +Xi and Xj are adjacent and a trajectory from fi ‘crosses’ the corresponding partition. +4 +Safety Verification of Neural Abstractions +Neural abstractions are dynamical models expressed in terms of neural ODEs with additive distur- +bances (cf. Equation 4). Corollary 1 ensures the fact for which concluding that a neural abstraction is +safe suffices to assert that the concrete dynamical model is also safe. Consequently, once a neural +ODE is formally proven to be an abstraction for the concrete dynamical model, which is entirely +delegated to our synthesis procedure (cf. Section 3), our definition of neural abstractions enables any +procedure for the safety verification of neural ODEs with disturbances to be a valid safety verification +procedure for the corresponding dynamical model. +Safety verification approaches for dynamical systems controlled by neural networks solve a similar +problem [18,54,75,77,106,113,114,116], yet with a subtle difference: neural network controllers take +control actions at discrete points in time. Instead, neural ODEs characterise dynamics over continuous +time. Some procedures for the direct verification of neural ODEs have been introduced very recently, +and this currently an area under active development [68, 69, 95]. Yet, existing approaches do not +consider the case of a neural ODE with a non-deterministic drift. Therefore, in order to obtain a +verification procedure for neural abstractions, we build upon the observation that a neural ODEs +with ReLU activation functions and non-deterministic drift defines a hybrid automaton with affine +dynamics. +Hybrid automata (cf. Figure 3) model the interaction between continuous dynamical systems and +finite-state transition systems [71,115]. A hybrid automaton consists of a finite set of variables and a +finite graph, whose vertices we call discrete modes and edges we call discrete transitions. Every mode +is associated with an invariant condition and a flow condition over the variables, which determine the +continuous dynamics of the systems on the specific mode. Every discrete transition is associated with +a guard condition, which determines the effect on discrete transitions between modes. While we refer +the reader to seminal work for a general definition of hybrid automata [71], we present a translation +from neural abstractions to hybrid automata. +4.1 +Translation of Neural Abstractions Into Hybrid Automata +We begin with the observation that each neuron within a given hidden layer of a neural network with +ReLU activation functions induces a hyperplane in the vector space associated with the previous layer +This hyperplane results in two half-spaces, one corresponding to the neuron being active and one to it +being inactive. For the jth neuron in the ith layer, these two halfspaces are respectively the two parts +of the hyperplane given by +{yi−1 | Wi,jyi−1 + bi,j = 0}, +(7) +where Wi,j is the jth row of the weight matrix Wi and bi,j is the jth element of the bias bi (cf. +Section 2). Therefore, every combinatorial configuration of the neural network defines an intersection +of halfspaces that defines a polyhedral region in the vector space of the input neurons. Moreover, every +7 + +configuration also defines a linear function from input to output neurons. The space of configurations +thus defines a partitioning of the input space, where each region is associated with an affine function. +A neural abstraction casts into a hybrid automaton, where every mode is determined by a configuration +of the hidden neurons and each of these configurations induces a system of affine ODEs (cf. Figure 3). +Discrete Modes +We represent a configuration of a neural network as a sequence C = (c1, . . . , ck) +of Boolean vectors c1 ∈ {0, 1}h1, . . . , ck ∈ {0, 1}hk, where k denotes the number of hidden layers +and h1, . . . , hk denote the number neurons in each of them (cf. Section 2). Every vector ci represents +the configuration of the neurons at the ith hidden later, and the jth element of ci represent the +activation status of the jth neuron at the ith later, which equals to 1 is the neuron is active and 0 if it +is inactive. Every mode of the hybrid automaton corresponds to exactly one configuration of neurons. +Invariant Conditions +We define the invariant of each mode as a restriction of the domain of +interest to a region XC ⊆ X, which denotes the maximal set of states that enables configuration C. +To construct XC, we define a higher-dimensional polyhedron on the space of valuation of the neurons +that enable configuration C, i.e., +YC = +� +(y0, . . . , yk) +��� ∧k +i=1yi = diag(ci)(Wiyi−1 + bi)∧ +diag(2ci − 1)(Wiyi−1 + bi) ≥ 0 +� +. +(8) +Note that diag(v) denotes the square diagonal matrix whose diagonal takes its coefficients from +vector v; in our case, this results in a square diagonal matrix whose coefficients are either 0 or 1. +Then, we project YC onto the input neurons y0, denoted YC ↾y0. Since the input neurons y0 are +equivalent to the state variables of the dynamical model, the invariant condition of mode C results in +XC = (YC ↾y0) ∩ X. +(9) +A projection can be computed using the Fourier-Motzkin algorithm or by projecting the vertices +of the polyhedron in a double description method. However, even though this is effective in our +experiments, it has worst-case exponential time complexity. A polynomial time construction can +be obtained by propagating halfspaces backwards along the network, similarly to methods used in +abstraction-refinement [29,60]. We outline the alternative construction in Appendix C.1. +Flow Conditions +The dynamics of each mode C can be seen itself as a dynamical system with +bounded disturbance: +˙x = ACx + bC + d, +∥d∥ ≤ ϵ, +x ∈ XC. +(10) +The matrix AC ∈ Rn×n and the vector of drifts bC ∈ Rn determine the linear ODE of the mode, +whereas ϵ > 0 is the error bound derived from the neural abstraction. +The coefficients of the system are given by the weights and biases of the neural network as follows: +AC = Wk+1 +�k +i=1 diag(ci)Wi, +(11) +bC = bk+1 + �k +i=1(Wk+1 +�k +j=i+1 diag(cj)Wj) diag(ci)bi. +(12) +Discrete Transitions and Guard Conditions +A discrete transition exists between any two given +modes if the two polyhedra that define their invariant conditions share a facet and the dynamics pass +through at some point along the facet. This can be checked by considering the sign of the Lie derivative +between the dynamics and the corresponding facet, that is, the inner product between the dynamics +and the normal vector to the facet. In practice, we take a faster but more conservative approach by +considering that a transition exists between two modes when the corresponding polyhedral regions +share at least a vertex. The guard condition of a discrete transition is simply the invariant of the +destination mode. +4.2 +Enumeration of Feasible Modes +A given configuration C exists in the hybrid automaton if and only if the corresponding set XC, which +is a convex polyhedron in Rn, is nonempty; this consists of verifying that the linear program (LP) +constructed from the polyhedron is feasible. Finding all modes of the hybrid automaton therefore +consists of solving 2H linear programs, where H = h1 + · · · + hk is the total number of hidden +neurons in the network. However, this exponential scaling with the number of neurons is limiting +8 + +in terms of network size. Therefore, we propose an approach that works very well in practice to +determine all valid neuron configurations. +The approach relies on the observation that within a bounded polyhedron P, a given neuron has two +modes (ReLU enabled or disabled) only if the induced hyperplane intersects P. If it does not, only +one of the two possible half-spaces contributes to any possible active configuration, and the other +neuron mode can be disregarded. Therefore, this approach involves iterating through each neuron in +turn and constructing two LPs—one for each halfspace intersected with the domain of interest X. If +only one LP is valid, we can fix the neuron to this mode, i.e., from this point onward only consider +the intersection with the halfspace corresponding to the feasible LP, and construct a new polyhedron +from the intersection of X and the feasible half-space. +In short, we consider the neurons of the network as a binary tree, with the branches representing +the enabled and disabled state of this neuron. We perform a depth-first tree search through this +tree by intersecting with the corresponding half-spaces. Upon reaching an end node, we store this +configuration (branches taken) and revert back to the most recent unexplored branch and continue. +We include a more detailed description of this algorithm in Appendix C.2. This approach is inspired +by that presented in [23], which similarly enumerates through the path of neurons using sets to +determine the output range of a network. +5 +Experimental Results +5.1 +Safety Verification Using Neural Abstractions +We benchmark the results obtained by the safety verification algorithm proposed in Section 4 against +Flow* [35] (available under GPL), which is a mature tool for computing reachable regions of +hybrid automata. It relies on computing flowpipes, i.e., sets of reachable states across time, which +are propagated from a given set of initial states. The flowpipes are generated from Taylor series +approximations of the model’s vector field in (1), over subsequent discrete time steps. Crucially, +the use of a higher-order Taylor series, or of smaller time steps, leads to more precise computation +of reachable sets. Since Flow*, like SpaceEx (available under GPLv3) is able to calculate over- +approximations of flowpipes, it is suitable for use in safety verification, and is a state-of-the-art tool +for verifying safety of nonlinear models. +Making a fair comparison around metrics for accuracy between Flow* and SpaceEx is challenging, +as they represent flowpipes differently [22,38]. We ask them to perform safety verification for a given +pair of initial and bad states, on a collection of different nonlinear models. These models, and their +parameters, are detailed in Appendix A. As described in Section 2, the task of safety verification +consists of ensuring that no trajectory starting within the set of initial states enters the set of bad +states, within a given time horizon. +Our setup is as follows. Firstly, for a given benchmark model we define a finite time horizon T, +a region of initial states X0 and a region of bad states XB. Then, we run flowpipe computations +with Flow* using high-order Taylor models. Similarly we run the procedure described in Section 3, +and construct a hybrid automaton as described in Section 4 to perform flowpipe computations using +SpaceEx. We present the results in Table 1. In the table, we show the Taylor model order (TM) +and time step used within Flow*, as well as the structure of the neural networks used for neural +abstractions. For example, we denote a network with two hidden layers with h1 neurons in the first +layer and h2 neurons in the second hidden layer as [h1, h2]. We note that while Flow*, much like +SpaceEx, can perform flowpipe computation on the constructed hybrid automaton, it is not specialised +to linear models like SpaceEx is and in practice struggles with the number of modes. +Notably, Flow* is unable to handle the two models that do not exhibit local Lipschitz continuity. Flow* +constructs Taylor models that incorporate the derivatives of the dynamics: as expected, unbounded +derivatives will cause issues for this approach. Meanwhile, Ariadne [24] a is an alternative tool +for over-approximating flowpipes of nonlinear systems. While Ariadne does not explicitly require +Lipschitz continuity, it is also unable to perform analysis on tools with nth root terms at zero, due to +numerical instability. Instead, our abstraction method works directly on the dynamics themselves, +rather than their derivatives, in order to construct simpler, abstract models that are amenable to be +verified. By formally quantifying how different an abstract model is through the approximation error, +we are able to formally perform safety verification on such challenging concrete models. +9 + +Table 1: Comparison of safety verification between Flow* and the combination of Neural Abstractions +plus SpaceEx. Here, T: time horizon, TM: Taylor model order, δ: time-step, t: total computation time +(better times denoted by bold), W: network neural structure, M: total number of modes in resulting +hybrid automaton, Blw: blowup in the error before T is reached, and -: no results unobtainable. +Model +T +Flow* +Neural Abstractions +TM +δ +Safety Ver. +t +W +M +Safety Ver. +t +Jet Engine +1.5 +10 +0.1 +Yes +1.3 +[10, 16] +8 +Yes +215 +Steam Governor +2.0 +10 +0.1 +Yes +62 +[12] +29 +Yes +219 +Exponential +1.0 +30 +0.05 +Blw +1034 +[14, 14] +12 +Yes +308 +Water Tank +2.0 +- +- +No +- +[12] +6 +Yes +49 +Non-Lipschitz 1 +1.4 +- +- +No +- +[10] +12 +Yes +19 +Non-Lipschitz 2 +1.5 +- +- +No +- +[12, 10] +32 +Yes +59 +Notice that we additionally outperform Flow* on a Lipschitz-continuous model (Exponential in Table +1), where the composition of functions that make up the model’s dynamics result in high errors in +Flow* before the flowpipe can be calculated across the given time horizon. We highlight that despite +relying on affine approximations (i.e., 1st order models), neural abstractions are able to compete with, +and even outperform, methods that use much higher order functions (10th and 30th in the benchmarks) +for approximation. +5.2 +Limitations +Our approach is limited in terms of scalability, both with regards to the dimension of the models and +to the size of the utilised neural networks. The causes of this limitation are twofold: firstly we are +bound by the computational complexity of SMT solving - known to be NP-hard - which can struggle +with complex formaulae with many variables. The certification step requires the largest amount of +time (cf. Appendix B), indicating that improvements in the verification of neural networks can lead +to a large performance increase for our abstractions. +Secondly, we are limited in terms of the complexity of our abstractions by SpaceEx. While SpaceEx +is a highly efficient implementation of LGG [88], the presence of a large number of discrete modes +poses a significant computational challenge. It future work, we hope to investigate the balance +between abstraction complexity and accuracy. The efficacy of neural abstraction on further tools for +hybrid automata with affine dynamics also remains to be investigated [6,24,28,107]. +6 +Conclusion +We have proposed a novel technique that leverages the approximation capabilities of neural networks +with ReLU activation functions to synthesise formal abstractions of dynamical models. By combining +machine learning and SMT solving algorthms in a CEGIS loop, our method computes abstract neural +ODEs with non-determinism that overapproximate the concrete nonlinear models. This guarantees the +property for which safety of the abstract model carries over to the concrete model. Our method casts +these neural ODEs into hybrid automata with affine dynamics, which we have verified using SpaceEx. +We have demonstrated that our method is not only comparable to Flow* in safety verification on +existing nonlinear benchmarks, but also shows superior effectiveness on models that do not exhibit +local Lipschitz continuity, which is a hard problem in formal verification. Yet, our experiments are +limited to low-dimensional models and scalability remains an open challenge. Our approach has +advanced the state of the art in terms of expressivity, which is the first step toward obtaining a general +and efficient verifier based on neural abstraction. Obtaining scalability to higher dimensions will +require a synergy of efficient SMT solvers for neural networks and safety verification of neural ODEs, +which are both novel and actively researched questions in formal verification [68,69,76,79,92,95,114]. +Acknowledgements +We thank the anonymous reviewers for their helpful suggestions. Alec was supported by the EPSRC +Centre for Doctoral Training in Autonomous Intelligent Machines and Systems (EP/S024050/1). +10 + +References +[1] Abate, A., Ahmed, D., Edwards, A., Giacobbe, M., Peruffo, A.: FOSSIL: a software tool for +the formal synthesis of Lyapunov functions and barrier certificates using neural networks. In: +HSCC. pp. 24:1–24:11. ACM (2021) +[2] Abate, A., Ahmed, D., Giacobbe, M., Peruffo, A.: Formal synthesis of Lyapunov neural +networks. IEEE Control. Syst. Lett. 5(3), 773–778 (2021) +[3] Abate, A., David, C., Kesseli, P., Kroening, D., Polgreen, E.: Counterexample guided inductive +synthesis modulo theories. In: CAV (1). 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If you ran experiments... +(a) Did you include the code, data, and instructions needed to reproduce the main experi- +mental results (either in the supplemental material or as a URL)? [Yes] The code and +data generation will be part of the supplementary material. Reproducing the results +will be possible from this but is not the intention of the authors. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they +were chosen)? [Yes] The hyper-parameters for the learning procedure are chosen +heuristically, but we include the relevant configuration files in the supplementary +material. +(c) Did you report error bars (e.g., with respect to the random seed after running experi- +ments multiple times)? [N/A] +(d) Did you include the total amount of compute and the type of resources used (e.g., type +of GPUs, internal cluster, or cloud provider)? [Yes] See Table 1. +4. 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[N/A] +17 + +A +Benchmark Nonlinear Dynamical Models +For each dynamical model, we report the vector field f : Rn → Rn and the spatial domain X +over which the abstraction is performed and which, unless otherwise stated, is taken to be the +hyper-rectangle [−1, 1]n. +Water Tank +� +� +� +˙x = 1.5 − √x +X0 = [0, 0.01] +XB = {x|x ≥ 2} +(13) +Jet Engine [17] +� +� +� +� +� +� +� +˙x = −y − 1.5x2 − 0.5x3 − 0.1, +˙y = 3x − y, +X0 = [0.45, 0.50] × [−0.60, −0.55] +XB = [0.3, 0.35] × [0.5, 0.6] +(14) +Steam Governor [110] +� +� +� +� +� +� +� +� +� +� +� +� +� +˙x = y, +˙y = z2 sin(x) cos(x) − sin(x) − 3y, +˙z = −(cos(x) − 1), +X0 = [0.70, 0.75] × [−0.05, 0.05] × [0.70, 0.75] +XB = [0.5, 0.6] × [−0.4, −0.3] × [0.7, 0.8] +(15) +Exponential +� +� +� +� +� +� +� +˙x = − sin(exp(y3 + 1)) − y2 +˙y = −x, +X0 = [0.45, 0.5] × [0.86, 0.91] +XB = [0.3, 0.4] × [0.5, 0.6] +(16) +Non-Lipschitz Vector Field 1 (NL1) +� +� +� +� +� +� +� +� +� +� +� +� +� +˙x = y +˙y = √x +X = [0, 1] × [−1, 1], +X0 = [0, 0.05] × [0, 0.1] +XB = [0.35, 0.45] × [0.1, 0.2] +(17) +Non-Lipschitz Vector Field 2 (NL2) +� +� +� +� +� +� +� +˙x = x2 + y +˙y = +3√ +x2 − x, +X0 = [−0.025, 0.025] × [−0.9, −0.85] +XB = [−0.05, 0.05] × [−0.8, −0.7] +(18) +B +Additional Experimental Results and Figures +B.1 +Experimental Comparison Against Affine Simplical Meshes +In this section, we present some supplementary empirical results on neural abstractions. Firstly, we +note that hybridisation-based abstraction of nonlinear models have been studied previously, such as +in [16], which describes a type of hybridisation-based abstractions that is similar to those constructed +in this work. The approach relies first on partitioning the state space using a simplicial mesh grid, and +18 + +Table 2: A comparison between abstractions constructed using an affine simplicial mesh and neural +abstractions. Here, W represents the neural structure used for neural abstraction, NP : total number of +partitions, ϵ: the calculated upper bound on the approximation error, ¯ +NP : average (mean) number of +partitions, ¯ϵ: average (mean) approximation error bound, ϵ+ : the maximum approximation error, ϵ−: +the minimum approximation error, Success Ratio: the ratio of repeated experiments that terminated +successfully (i.e., an error of 0.5 was reached within the first timeout of 300s). Note, we only +include successful experiments when calculating the average, min and max (since no error exists for +unsuccessful experiments). All reported errors use the 2-norm. +Benchmark +Affine Simplicial Mesh +Neural Abstractions +Np +ϵ +W +¯ +NP +¯ϵ +ϵ+ +ϵ− +Success Ratio +Jet Engine +8 +1.33 +[10] +9 +0.11 +0.22 +0.040 +1.0 +32 +0.33 +[10, 10] +27 +0.077 +0.17 +0.040 +1.0 +128 +0.083 +[15, 15] +61 +0.058 +0.071 +0.053 +1.0 +Steam +24 +3.58 +[10] +27 +0.27 +0.37 +0.21 +1.0 +192 +0.89 +[20] +236 +0.18 +0.27 +0.15 +1.0 +Exponential +8 +13.7 +[10] +9 +0.29 +0.40 +0.22 +0.5 +32 +3.44 +[20] +30 +0.19 +0.22 +0.13 +0.9 +128 +0.86 +[20, 20] +75 +0.15 +0.22 +0.071 +1.0 +then allowing the dynamics in each mesh to be calculated from an affine interpolation between the +vertices of the simplex. This affine simplicial mesh (ASM) based approach constructs abstractions +of the same expressivity as neural abstractions (first order approximations) with partitions defined +by affine inequalities. An approximation-error bound for ASM can be calculated for systems which +have bounded second order derivatives using the model dynamics and the size of each simplex (all +simplices are assumed to be the same size), as described in [16]. In Table 2 we compare between +abstractions constructed using an affine simplicial mesh and neural abstractions. We run our procedure +to synthesise certified abstractions using selected network structures and an initial target error of +0.5. If a successful abstraction is synthesised, we reduce the error by some multiplicative factor and +repeat. This iterative procedure continues until no success is reached within a time of 300s. We report +the results from 10 repeated experiments over different initial random seeds for neural abstractions, +reporting the average (mean), minimum and maximum results obtained. In contrast, we report the +approximation-error bound for ASM for different numbers of partitions. +The results reported in Table 2 illustrate that neural abstractions outperform ASM based abstractions +in terms of error for similar numbers of partitions. Furthermore, neural abstractions generally require +significantly fewer partitions for significantly lower approximation-error bounds. In practice this +means neural abstractions will outperform ASM-based abstractions for safety verification both in +terms of speed and accuracy. We also note the success ratio of our experiments, i.e., the ratio of all +experiments which achieve an approximation-error bound of 0.5 or less. These results suggest that in +general or procedure is robust and terminates successfully with high probability for reasonable target +errors. +We note that since ASM based abstractions are constructive and are able to deterministically increase +the number partitions and consequently reduce the error, for very large numbers of partitions they +would achieve lower errors than neural abstractions. However, in practice these abstractions would +be too large in complexity to use with SpaceEx for safety verification. +B.2 +Computation Run-time Profiling +In Table 3 we show a breakdown of the runtimes of our procedure shown in the main text. In +particular, we present the total time spent during learning, certification of the abstraction and finally +in safety verification. +19 + +Table 3: Breakdown of the timings shown in Table 1. Shown are the timings in the constituent +component shown in Figure 2: time spent during learning, time spent during certification of the +neural abstraction, and time spent during safety verification. Remaining time is spent in overheads, +such as converting from neural network to hybrid automaton. +Model +Learner +Certifier +Safety Verification +Jet Engine +19 +194 +1.8 +Steam Governor +42 +177 +0.5 +Exponential +27 +278 +3.3 +Water-tank +48 +0.001 +0.05 +Non-Lipschitz 1 +13 +0.50 +5.5 +Non-Lipschitz 2 +31 +15 +5.1 +C +Improved Translation from Neural Abstractions to Hybrid Automata +C.1 +Computing Invariant Conditions +Invariant conditions are computed from the configuration of a neural network denoted as the sequence +C = (c1, . . . , ck) of Boolean vectors c1 ∈ {0, 1}h1, . . . , ck ∈ {0, 1}hk, where k denotes the number +of hidden layers and h1, . . . , hk denote the number neurons in each of them (cf. Section 2). Every +vector ci represents the configuration of the neurons at the ith hidden later, and its jth element ci,j +represents the activation status of the jth neuron at the ith layer. Every mode of the hybrid automaton +corresponds to exactly one configuration of neurons. In turn, every configuration of neurons C +restricts the neural network N into a linear function. More precisely, we inductively define the linear +restriction at the ith hidden layer as follows: +N (i) +C (x) = diag(ci)(WiN (i−1) +C +(x) + bi), for i = 1, . . . , k, +N (0) +C (x) = x. +(19) +We define the invariant of each mode as a restriction of the domain of interest to a region XC ⊆ X, +which denotes the maximal set of states that enables configuration C. To construct XC, we begin +with the observation that the activation configuration ci at every ith hidden layer induces a halfspace +on the vector space of the previous layer of the neural network. Then, the pre-image of this +halfspace backward along the previous layers of the linear restriction of the network characterises +a corresponding halfspace on its input neurons. Since the input neurons are equivalent to the state +variables of the dynamical model, the halfspace induced by layer i projected onto state variables x is +H(i) +C = pre-image of {yi−1 | diag(2ci − 1)(Wiyi−1 + bi) ≥ 0} +� +�� +� +halfspace induced by ith layer onto (i − 1)th layer +under N (i−1) +C +(20) +The pre-image of a set Y under a function g is defined as {x | g(x) ∈ Y} and can be generally +computed by quantifier elimination or, in the linear case, double description methods. However, these +methods have worst-case exponential time complexity. To obtain XC efficiently, we can leverage the +fact that the pre-image of any halfspace {y | cTy ≤ d} under any affine function g(x) = Ax+b equals +to the set {x | cTy ≤ d ∧ y = Ax + b}, which in turn defines the halfspace {x | cTAx ≤ d − cTb}. +Therefore, since N (i−1) +C +is an affine function, every halfspace can be projected backward through the +affine functions N (i−1) +C +, . . . , N (1) +C +using O(k) linear algebra operations. Finally, the entire invariant +condition for configuration C is defined as the following polyhedron: +XC = ∩{H(i) +C | i = 1, . . . , k} ∩ X. +(21) +An invariant condition thus results in a polyhedron defined as the intersection of k halfspaces together +with the constrains that define the domain of interest. Notably, under the definition in this appendix, +the dynamics of mode C given in Equation 10 correspond to the affine dynamical model +˙x = N (k+1) +C +(x) + d, +∥d∥ ≤ ϵ, +x ∈ XC, +(22) +whose dynamics are governed by the affine function +N (k+1) +C +(x) = Wk+1N (k) +C (x) + bk+1. +(23) +20 + +N1 +N2 +X = ∅ +N3 +N3 +End +End +End +X = ∅ +C = (1, 0, 1) +C = (1, 1, 1) +C = (1, 0, 0) +X ← X ∩ h+ +1 , +X ̸= ∅ +X ← X ∩ h− +1 +X ← X ∩ h+ +2 , +X ̸= ∅ +X ← X ∩ h+ +3 , +X ̸= ∅ +X ← X ∩ h− +3 +X ← X ∩ h− +2 , +X ̸= ∅ +X ← X ∩ h+ +3 +X ← X ∩ h− +3 +Figure 4: Example Tree search to determine the active configurations for a neural network consisting +of a single hidden layer with 3 neurons. Here, h+ +i denotes the positive half-space ({x : wix+bi ≥ 0}) +and h− +i denotes the negative half-space ({x : wix + bi ≤ 0}) of the ith neuron; wi represents the ith +row of the weight matrix corresponding to the hidden layer, and bi represents the ith element of the +bias vector of the hidden layer. Notably, when the set X becomes empty, it is no longer necessary to +continue along that path. Once we reach the end of the tree, we have an active configuration C, and +backtrack to the last node that was not fully explored. +C.2 +Enumerating Feasible Modes +Determining whether a mode C exists in the hybrid automaton amounts to determining the linear +program (LP) associated to polyhedron XC is feasible. Finding all modes therefore consists of +solving 2H linear programs, where H = h1 + · · · + hk is the total number of neurons. This scales +exponentially in the number of neurons. Here, we elaborate on the tree search algorithm described in +Section 4.2 using a diagram; the purpose of this algorithm is to efficiently determine all active neuron +configurations within a bounded domain of interest X. +We consider an example tree in Figure 4, which depicts an example search for a neural network with +a single hidden layer consisting of three neurons. The tree illustrates the construction of XC through +repeated intersections of half-spaces as paths are taken through the tree structure. Nodes represent +each neuron, labelled Ni, i = 1, 2, 3 and each edge represents one of two possible half-spaces for the +neuron it leaves from (ReLU enabled, solid line, and disabled, dashed line). This approach allows +us to prune neurons and overall solve significantly fewer linear programs than simply enumerating +through all possible configurations. +21 + diff --git a/atFJT4oBgHgl3EQf8S2B/content/tmp_files/load_file.txt b/atFJT4oBgHgl3EQf8S2B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29e040c73159be8d9c662d25e59f0193a98ea6a5 --- /dev/null +++ b/atFJT4oBgHgl3EQf8S2B/content/tmp_files/load_file.txt @@ -0,0 +1,1525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf,len=1524 +page_content='Neural Abstractions Alessandro Abate∗ Department of Computer Science University of Oxford, UK Alec Edwards∗ Department of Computer Science University of Oxford, UK Mirco Giacobbe∗ School of Computer Science University of Birmingham, UK Abstract We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Neural net- works have extensively been used before as approximators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' in this work, we make a step further and use them for the first time as abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For a given dynamical model, our method synthesises a neural network that overapproximates its dynam- ics by ensuring an arbitrarily tight, formally certified bound on the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For this purpose, we employ a counterexample-guided inductive synthesis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We show that this produces a neural ODE with non-deterministic distur- bances that constitutes a formal abstraction of the concrete model under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This guarantees a fundamental property: if the abstract model is safe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', free from any initialised trajectory that reaches an undesirable state, then the concrete model is also safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By using neural ODEs with ReLU activation functions as abstractions, we cast the safety verification problem for nonlinear dynamical models into that of hybrid automata with affine dynamics, which we verify using SpaceEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We demonstrate that our approach performs comparably to the mature tool Flow* on existing benchmark nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We additionally demonstrate and that it is effective on models that do not exhibit local Lipschitz continuity, which are out of reach to the existing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 1 Introduction Dynamical models describe processes that are ubiquitous in science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' They are widely used to model the behaviour of cyber-physical system designs, whose correctness is crucial when they are deployed in safety-critical domains [10,13,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' To guarantee that a dynamical model satisfies a safety specification, simulations are useful but insufficient because they are inherently non-exhaustive and they suffer from numerical errors, which may leave unsafe behaviours unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Formal verification of continuous dynamical models tackles the question of determining with formal certainty whether every possible behavior of the model satisfies a safety specification [45, 51, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In this paper, we present a method to combine machine learning and symbolic reasoning for a sound and effective safety verification of nonlinear dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The formal verification problem for continuous-time and hybrid dynamical models is unsolvable in general and, even for models with linear dynamics, complete procedures are available under stringent conditions [11, 12, 72, 86, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For most practical models that contain nonlinear terms [81, 105], ∗The authors are listed alphabetically 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='11683v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='LO] 27 Jan 2023 methods for formal verification with soundness guarantees involve laborious safety and reachability procedures whose efficacy can only be demonstrated in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Formal verification of nonlinear models require ingenuity, and has involved sophisticated analysis techniques such as mathematical relaxations [27,34–36,48,103,104], abstract interpretation [52,53,56,82,96], constraint solving [19, 39, 85], and discrete abstractions [5, 9, 30, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notwithstanding recent progress, both scalability and expressivity remain open challenges for nonlinear models: the largest model used in the annual competition has 7 variables [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In addition, existing formal approaches rely on symbolic reasoning techniques that explicitly leverage the structure of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This results in verification procedures that are bespoke to restricted classes of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For example, it is common for formal verification procedures to require the input model to be Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Yet, dynamical models with vector fields that violate this assumption are abundant in literature, and a wide variety of models of natural phenomena are non-Lipschitz, from fluid dynamics to n-body orbits and chaotic systems, as well as in engineering, from electrical circuits and hydrological systems [50,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our approach makes progress in expressivity, showing that using neural networks as abstractions of dynamical systems enables an effective formal verification of nonlinear dynamical models, including models that do not exhibit local Lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Abstraction is a standard process in formal verification that aims at translating the model under analysis—the concrete model—into a model that is simpler to analyse—the abstract model—such that verification results from the abstract model carry over to the concrete model [20, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In verification of systems with continuous time and space, an abstraction usually consists of a partitioning of the state space of the concrete model into a finite set of regions that define the states of an abstract, finite-state machine with a corresponding behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our method follows an approach that constructs abstract, finite-state machines whose states are augmented with continuous linear dynamics and non-deterministic drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finite-state machines with continuous, possibly non-deterministic dynamics are known as hybrid automata [71], and the process of abstracting dynamical nonlinear models into hybrid automata is called hybridisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' this process has been widely applied in formal verification [8,15,16,21,46,58,65,73,91,94,99,100,102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Hybridising involves partitioning the state space and computing a local overapproximation of the concrete model within each region of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Common approaches for hybridisation partition the state space by tuning the granularity of rectangular or simplicial meshes, until a desired approximation error is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This may yield abstract hybrid automata that are too large in the number of discrete states to be effectively verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, modern tools for the verification of hybrid automata are designed for models that rarely have over hundred discrete states [7], while arbitrary meshes grow exponentially as the granularity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Explosion in discrete states has been mitigated using deductive approaches that construct an appropriate partitioning from the expressions that define the concrete model and, unlike our method, rely on syntactic restrictions [14,26,30,47,70,80,83,84,98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We propose an inductive approach to abstraction that combines the tasks of partitioning the state space and overapproximating the dynamics into the single task of training a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We leverage the approximation capability of neural networks with ReLU activation functions to partition the state space into arbitrary polyhedral regions, where each region and local affine approximation correspond to a combinatorial configuration of the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We show that this ultimately enables verifying nonlinear dynamical models using efficient safety verifiers for hybrid automata with affine dynamics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our abstraction procedure synthesises abstract models by alternating a learner, which proposes candidate abstractions, and a certifier, which formally assures (or disproves) their validity, in a counterexample-guided inductive synthesis (CEGIS) loop [108,109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' First, the learner uses gradient descent to train a neural network that approximates the concrete model over a finite set of sample observations of its dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' then, the certifier uses satisfiability modulo theories (SMT) to check the validity of an upper-bound on the approximation error over the entire continuous domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If the latter disproves the bound, then it produces a counterexample which its added to the set of samples and the loop is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If it certifies the bound, then the procedure returns a neural network approximation and a sound upper-bound on the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Altogether, neural network and error bound define a neural ODE with bounded additive non-determinism that overapproximates the concrete model, which we call a neural abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We demonstrate the efficacy of our method over multiple dynamical models from a standard bench- mark set for the verification of nonlinear systems [66], as well as additional locally non-Lipschitz 2 1 0 1 1 0 1 Concrete nonlinear system x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' ˙x ReLU Neural abstraction 1 0 1 1 0 1 Abstract hybrid automaton Flowpipe propagation Abstraction synthesis Model translation Safety verification Figure 1: Overview of our workflow on a non-Lipschitz dynamical model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 5, NL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The concrete dynamics are abstracted by a neural ODE with ReLU activation functions and a certified upper-bound on the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This characterises a polyhedral partitioning and defines a hybrid automaton with affine dynamics and additive non-deterministic drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Flowpipe propagation is finally performed through a region of non-Lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' models, and compare our approach with Flow*, the state-of-the-art verification tool for nonlinear models [34,35,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We instantiate our approach on top of SpaceEx [62], which is a state-of-the-art tool specialised to linear hybrid models [59,61,88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We evaluate both approaches in safety verifi- cation using flowpipe propagation, which computes the set of reachable states from a given set of initial states up to a given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our experiments demonstrate that our approach performs comparably with Flow* for Lipschitz continuous model, and succeeds with non-Lipschitz models that are out of range for Flow* and violate the working assumptions of many verification tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' These outcomes suggest that neural abstractions are a promising technology, also in view of recent results on direct methods for the safety verification for neural ODEs [68,69,95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We summarise our contributions in the following points: we introduce the novel idea of leveraging neural networks to represent abstractions in formal verification, and we instantiate it in safety verification of nonlinear dynamical models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' we present a CEGIS procedure for the synthesis of neural ODEs that formally overapproxi- mate the dynamics of nonlinear models, which we call neural abstractions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' we define a translation from neural abstractions defined using ReLU activation functions to hybrid automata with affine dynamics and additive non-determinism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' we implement our approach2 and demonstrate its comparable performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' the state-of- the-art tool Flow* in safety verification of Lipschitz-continuous models, and even superior efficacy on models that do not exhibit local Lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We consider there to be no significant negative societal impact of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 2The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='com/aleccedwards/neural-abstractions-nips22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='52 Neural Abstractions of Dynamical Models We study the formal verification question of whether an n-dimensional, continuous-time, autonomous dynamical model with possibly uncertain (bounded) disturbances, considered within a region of interest, is safe with respect to a region of bad states when initialised from a region of initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Definition 1 (Dynamical Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A dynamical model F defined over a region of interest X ⊆ Rn consists of a nonlinear function f : Rn → Rn and a possibly null disturbance radius δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Its dynamics are given by the system of nonlinear ODEs ˙x = f(x) + d, ∥d∥ ≤ δ, x ∈ X, (1) where ∥ · ∥ denotes a norm operator (unless explicitly stated, we assume the norm operator to be given the same semantics across the paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A trajectory of F defined over time horizon T > 0 is a function ξ : [0, T] → Rn that admits derivative at each point in [0, T] such that, for all t ∈ [0, T], it holds true that ξ(t) ∈ X and ˙ξ(t) = f(ξ(t)) + dt for some ∥dt∥ < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, symbol d in Equation (1) is interpreted as a non-deterministic disturbance that at any time can take any possible value within the bound provided by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Let the sets X0 ⊂ X be a region of initial states and XB ⊂ X be a region of bad states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We say that a trajectory ξ defined over time horizon T is initialised if ξ(0) ∈ X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' additionally, we say that it is safe if ξ(t) ̸∈ XB for all t ∈ [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' dually, we say that it is unsafe if ξ(t) ∈ XB for some t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The safety verification question for consists of determining whether all initialised trajectories are safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If this is the case, then we say that the model is safe with respect to X0 and XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If there exist at least one initialised trajectory that is unsafe, then we say that the model is unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We tackle safety verification by abstraction, that is, we construct an abstract dynamical model that captures all behaviours of the concrete nonlinear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This implies that if the abstract model is safe then the concrete model is necessarily safe too, and we can thus apply a verification procedure over the abstraction to determine whether the concrete model is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, the converse may not hold: lack of safety of the abstract model does not carry over to the concrete model, because our abstraction is an overapproximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We ultimately obtain a sound (but not complete) safety verification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our approach synthesises an abstract dynamical model defined in terms a feed-forward neural network with ReLU activation functions and endowed with a bounded non-deterministic disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This can be seen as a neural ODEs [33] augmented with an additive non-deterministic drift that ensures the abstract model to overapproximate the concrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' To the best of our knowledge, this is the first work to consider neural ODEs with non-deterministic semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our feed-forward neural network consists of an n-dimensional input layer y0, k hidden layers y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , yk with dimensions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , hk respectively, and an n-dimensional output layer yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Each hidden or output layer with index i are respectively associated matrices of weights Wi ∈ Rhi×hi−1 and a vectors of biases bi ∈ Rhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Upon a valuation of the input layer, the value of every subsequent hidden layer is given by the following equation: yi = ReLU(Wiyi−1 + bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (2) Whereas many activation functions exist, we focus our study on ReLU activation functions, applying function max{x, 0} to every element x ∈ R of its hi-dimensional argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finally, the value of the output layer is given by the affine map yk+1 = Wk+1xk + bk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Altogether, the network results in a function N whose output is N(x) = yk+1 for every given input y0 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Definition 2 (Neural Abstraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Let F be a dynamical model given by function f : Rn → Rn and disturbance radius δ ≥ 0 and let X ⊆ Rn be a region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A feed-forward neural network N : Rn → Rn defines a neural abstraction of F with error bound ϵ > 0 over X, if it holds true that ∀x ∈ X : ∥f(x) − N(x)∥ ≤ ϵ − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (3) Then, the neural abstraction consists of the dynamical model A defined by N and disturbance ϵ, whose dynamics are given by the following neural ODE with bounded additive disturbances: ˙x = N(x) + d, ∥d∥ ≤ ϵ, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (4) Theorem 1 (Soundness of Neural Abstractions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If A is a neural abstraction of a dynamical system F over a region of interest X ⊆ Rn, then every trajectory of F is also a trajectory of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 4 Abstraction Synthesis Learner Certifier Counterexample scex Candidate N Valid neural abstraction A F, X, ϵ F, S, ϵ Safety Verification F, X, S, ϵ X, X0, XB Figure 2: Architecture for the safety verification of nonlinear dynamical models using neural abstrac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The inputs to our architecture are a concrete model F and its domain of interest X, a finite set of initial datapoints S, a desired approximation error ϵ, and regions of initial X0 and bad states XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Let ξ be a trajectory of F and T be the time horizon over which ξ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, let t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By definition of trajectory we have that (i) ξ(t) ∈ X and there exists dt s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (ii) ∥dt∥ ≤ δ and (iii) ˙ξ(t) = f(ξ(t))+dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By (i) and condition (3) we have that ∥f(ξ(t))−N(ξ(t))∥+ δ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, by (ii) we have that ∥f(ξ(t)) − N(ξ(t))∥ + ∥dt∥ ≤ ϵ which, by triangle inequality, implies that ∥f(ξ(t)) + dt − N(ξ(t))∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Using (iii), we rewrite it into ∥ ˙ξ(t) − N(ξ(t))∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finally, we define d′ t = ˙ξ(t) − N(ξ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' As a result, we have that ∥d′ t∥ ≤ ϵ and ˙ξ(t) = N(ξ(t)) + d′ t which, together with (i), shows that ξ is a trajectory of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Let X0 ⊂ X be a region of initial states and XB ⊂ X and region of bad states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' It holds true that if A is safe with respect to X0 and XB then also F is safe with respect to X0 and XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By Theorem 1, if there exists an initialised trajectory of F that is unsafe, then the same is an initialised trajectory of A that is unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The statement follows by contraposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Remark 1 (Existence of Neural Abstractions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Let F be a dynamical model defined by function f and disturbance radius δ ≥ 0, and let X ⊆ Rn be a domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A neural abstraction of F with arbitrary error bound ϵ > 0 over X exists if a neural network that approximates f with error bound ϵ − δ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' condition (3)) exists over the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In this work, we do not prescribe conditions on either width or depth of the network to ensure existence of a neural abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Such conditions are given by universal approximation theorems for neural networks with ReLU activation functions, which have been developed in seminal work in machine learning [25,42,63,74,90,93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Altogether, we define the neural abstraction of a non-linear dynamical system F as a neural ODE with an additive disturbance A that approximates the dynamics while also accounting for the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, we place no assumptions on the vector field f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In particular, Theorem 1 does not require f to be Lipschitz continuous: the soundness of a neural abstraction relies on condition (3), whose certification we offload to an SMT solver (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The resulting neural abstraction is to a hybrid automaton with affine dynamics and non-deterministic disturbance (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 4), which does not rely on the Picard-Lindelof theorem to ensure uniqueness or existence of a solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 3 Formal Synthesis of Neural Abstractions Our approach to abstraction synthesis follows two phases—a learning phase and a certification phase— that alternate each other in a CEGIS loop [1,3,43,78,101,108,109] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Figure 2, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our learning phase trains the parameters of a neural network N to approximate the system dynamics over a finite set of samples S ⊂ X of the domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Learning uses gradient descent algorithms, which can possibly scale to large amounts of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, our certification phase either confirms the validity of condition (3) or produces a counterexample which we use to sample additional states and repeat the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Certification is based on SMT solving, which reasons symbolically over the continuous domain X and assures soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' As a consequence, when certification confirms condition (3) formally valid, then as per Theorem 1 our neural abstraction A is a sound overapproximation of the concrete model F and is thus passed to safety verification (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Figure 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Neural networks have been used in the past as representations of formal certificates for the correctness of systems such as Lyapunov neural networks, neural barrier certificates, neural ranking functions 5 and supermartingales [1, 2, 4, 31, 32, 44, 67, 89, 97, 111, 117–119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In the present work, we use neural networks for the first time as abstractions, and we instantiate this idea in safety verification of nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We shall now present the components of our abstraction synthesis procedure: learner (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1) and certifier (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Learning Phase As with many machine learning-based algorithm, learning neural abstractions hinges on the loss function used as part of the gradient descent scheme for optimising parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The task is that of a regression problem, so the choice of loss function to be minimised is simple, namely, L = � s∈S ∥f(s) − N(s)∥2, (5) where ∥ · ∥2 represents the 2 − norm of its input, and S ⊂ X is a finite set of data points that are sampled from the domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In other words, the neural abstractions are synthesised using a scheme based on gradient descent to find the parameters that minimise the mean square error over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The main inputs to the learning procedure are the vector field f of the concrete dynamical model, an initial set of points S sampled uniformly from the domain of interest X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Additional parameters include the hyper-parameters for the learning scheme such as the learning rate, and a stopping criterion for the learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For the latter, there are two possible options: a target error which all data points must satisfy, or a bound on the value of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If a target error smaller than ϵ − δ is provided, this is when all points in the data set S satisfy the specification (3) and certification subsequently check that this generalises over the entire X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If an alternative loss-based stopping criterion is provided, then an error bound on the approximation is estimated using the maximum approximation error over the data set S for use in certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This estimated bound is conservative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', greater than the maximum, to allow for successful certification to be more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' After learning, the network N is translated to symbolic form and passed to the certification block, which checks condition (3) as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The certifier either determines condition (3) valid, and thus the CEGIS loop terminates, or computes a counterexample that falsifies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The counterexample is returned to the learning procedure and augmented by sampling for additional points nearby in order to maximise the efficiency of learning and the overall synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 Certification Phase The purpose of the certification is to check that at no point in the domain of interest X is the maximum error greater than the upper bound ϵ − δ, as per the specification in condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, the certifier is provided with the negation of the specification, namely ∃x: x ∈ X ∧ ∥f(x) − N(x)∥ > ϵ − δ � �� � φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (6) The certifier seeks an assignment scex of the variable x such that the quantifier-free formula φ is satisfiable, namely that the specified bound is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If this search is successful, then the network N has not achieved the specified accuracy over X, and is thus not a valid neural abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The corresponding assignment scex forms the counterexample that is provided back to the learner (the machine learning procedure from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Alternatively, if no assignment is found then specification (3) is proven valid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' network N and error bound ϵ are then passed to the safety verification procedure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Certification of the accuracy of the neural abstractions is performed by an SMT solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Several options exist for the selection of the SMT solver, with the requirement that the solver should reason over quantifier-free nonlinear real arithmetic formulae [57,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This is because the vector field f may contain nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In our experiments, we employ dReal [64], which supports polynomial and non-polynomial terms such as transcendental functions like trigonometric or exponential ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A successful verification process allows for the full abstraction to be constructed using the achieved error ϵ and neural network N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' CEGIS has been shown to perform well and terminate successfully across a wide variety of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We demonstrate the robustness of our procedure in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 6 x y X1 X2 X3 ˙x = f1(x) x ∈ X1 ˙x = f3(x) x ∈ X3 ˙x = f2(x) x ∈ X2 x ∈ X1 x ∈ X3 x ∈ X2 x ∈ X3 Figure 3: A hybrid automaton corresponding to a state-space partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Each of the three discrete modes corresponds to a unique partition Xi and vector field fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Discrete transitions are denoted by the edges of the directed graph with a transition between two modes if the corresponding partitions Xi and Xj are adjacent and a trajectory from fi ‘crosses’ the corresponding partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 4 Safety Verification of Neural Abstractions Neural abstractions are dynamical models expressed in terms of neural ODEs with additive distur- bances (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Corollary 1 ensures the fact for which concluding that a neural abstraction is safe suffices to assert that the concrete dynamical model is also safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Consequently, once a neural ODE is formally proven to be an abstraction for the concrete dynamical model, which is entirely delegated to our synthesis procedure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 3), our definition of neural abstractions enables any procedure for the safety verification of neural ODEs with disturbances to be a valid safety verification procedure for the corresponding dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Safety verification approaches for dynamical systems controlled by neural networks solve a similar problem [18,54,75,77,106,113,114,116], yet with a subtle difference: neural network controllers take control actions at discrete points in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Instead, neural ODEs characterise dynamics over continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Some procedures for the direct verification of neural ODEs have been introduced very recently, and this currently an area under active development [68, 69, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Yet, existing approaches do not consider the case of a neural ODE with a non-deterministic drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, in order to obtain a verification procedure for neural abstractions, we build upon the observation that a neural ODEs with ReLU activation functions and non-deterministic drift defines a hybrid automaton with affine dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Hybrid automata (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Figure 3) model the interaction between continuous dynamical systems and finite-state transition systems [71,115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A hybrid automaton consists of a finite set of variables and a finite graph, whose vertices we call discrete modes and edges we call discrete transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every mode is associated with an invariant condition and a flow condition over the variables, which determine the continuous dynamics of the systems on the specific mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every discrete transition is associated with a guard condition, which determines the effect on discrete transitions between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' While we refer the reader to seminal work for a general definition of hybrid automata [71], we present a translation from neural abstractions to hybrid automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Translation of Neural Abstractions Into Hybrid Automata We begin with the observation that each neuron within a given hidden layer of a neural network with ReLU activation functions induces a hyperplane in the vector space associated with the previous layer This hyperplane results in two half-spaces, one corresponding to the neuron being active and one to it being inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For the jth neuron in the ith layer, these two halfspaces are respectively the two parts of the hyperplane given by {yi−1 | Wi,jyi−1 + bi,j = 0}, (7) where Wi,j is the jth row of the weight matrix Wi and bi,j is the jth element of the bias bi (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, every combinatorial configuration of the neural network defines an intersection of halfspaces that defines a polyhedral region in the vector space of the input neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Moreover, every 7 configuration also defines a linear function from input to output neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The space of configurations thus defines a partitioning of the input space, where each region is associated with an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A neural abstraction casts into a hybrid automaton, where every mode is determined by a configuration of the hidden neurons and each of these configurations induces a system of affine ODEs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Discrete Modes We represent a configuration of a neural network as a sequence C = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , ck) of Boolean vectors c1 ∈ {0, 1}h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , ck ∈ {0, 1}hk, where k denotes the number of hidden layers and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , hk denote the number neurons in each of them (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every vector ci represents the configuration of the neurons at the ith hidden later, and the jth element of ci represent the activation status of the jth neuron at the ith later, which equals to 1 is the neuron is active and 0 if it is inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every mode of the hybrid automaton corresponds to exactly one configuration of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Invariant Conditions We define the invariant of each mode as a restriction of the domain of interest to a region XC ⊆ X, which denotes the maximal set of states that enables configuration C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' To construct XC, we define a higher-dimensional polyhedron on the space of valuation of the neurons that enable configuration C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', YC = � (y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , yk) ��� ∧k i=1yi = diag(ci)(Wiyi−1 + bi)∧ diag(2ci − 1)(Wiyi−1 + bi) ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (8) Note that diag(v) denotes the square diagonal matrix whose diagonal takes its coefficients from vector v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' in our case, this results in a square diagonal matrix whose coefficients are either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, we project YC onto the input neurons y0, denoted YC ↾y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Since the input neurons y0 are equivalent to the state variables of the dynamical model, the invariant condition of mode C results in XC = (YC ↾y0) ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (9) A projection can be computed using the Fourier-Motzkin algorithm or by projecting the vertices of the polyhedron in a double description method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' However, even though this is effective in our experiments, it has worst-case exponential time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' A polynomial time construction can be obtained by propagating halfspaces backwards along the network, similarly to methods used in abstraction-refinement [29,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We outline the alternative construction in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Flow Conditions The dynamics of each mode C can be seen itself as a dynamical system with bounded disturbance: ˙x = ACx + bC + d, ∥d∥ ≤ ϵ, x ∈ XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (10) The matrix AC ∈ Rn×n and the vector of drifts bC ∈ Rn determine the linear ODE of the mode, whereas ϵ > 0 is the error bound derived from the neural abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The coefficients of the system are given by the weights and biases of the neural network as follows: AC = Wk+1 �k i=1 diag(ci)Wi, (11) bC = bk+1 + �k i=1(Wk+1 �k j=i+1 diag(cj)Wj) diag(ci)bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (12) Discrete Transitions and Guard Conditions A discrete transition exists between any two given modes if the two polyhedra that define their invariant conditions share a facet and the dynamics pass through at some point along the facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This can be checked by considering the sign of the Lie derivative between the dynamics and the corresponding facet, that is, the inner product between the dynamics and the normal vector to the facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In practice, we take a faster but more conservative approach by considering that a transition exists between two modes when the corresponding polyhedral regions share at least a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The guard condition of a discrete transition is simply the invariant of the destination mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 Enumeration of Feasible Modes A given configuration C exists in the hybrid automaton if and only if the corresponding set XC, which is a convex polyhedron in Rn, is nonempty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' this consists of verifying that the linear program (LP) constructed from the polyhedron is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finding all modes of the hybrid automaton therefore consists of solving 2H linear programs, where H = h1 + · · · + hk is the total number of hidden neurons in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' However, this exponential scaling with the number of neurons is limiting 8 in terms of network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, we propose an approach that works very well in practice to determine all valid neuron configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The approach relies on the observation that within a bounded polyhedron P, a given neuron has two modes (ReLU enabled or disabled) only if the induced hyperplane intersects P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If it does not, only one of the two possible half-spaces contributes to any possible active configuration, and the other neuron mode can be disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, this approach involves iterating through each neuron in turn and constructing two LPs—one for each halfspace intersected with the domain of interest X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If only one LP is valid, we can fix the neuron to this mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', from this point onward only consider the intersection with the halfspace corresponding to the feasible LP, and construct a new polyhedron from the intersection of X and the feasible half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In short, we consider the neurons of the network as a binary tree, with the branches representing the enabled and disabled state of this neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We perform a depth-first tree search through this tree by intersecting with the corresponding half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Upon reaching an end node, we store this configuration (branches taken) and revert back to the most recent unexplored branch and continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We include a more detailed description of this algorithm in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This approach is inspired by that presented in [23], which similarly enumerates through the path of neurons using sets to determine the output range of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 5 Experimental Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Safety Verification Using Neural Abstractions We benchmark the results obtained by the safety verification algorithm proposed in Section 4 against Flow* [35] (available under GPL), which is a mature tool for computing reachable regions of hybrid automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' It relies on computing flowpipes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', sets of reachable states across time, which are propagated from a given set of initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The flowpipes are generated from Taylor series approximations of the model’s vector field in (1), over subsequent discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Crucially, the use of a higher-order Taylor series, or of smaller time steps, leads to more precise computation of reachable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Since Flow*, like SpaceEx (available under GPLv3) is able to calculate over- approximations of flowpipes, it is suitable for use in safety verification, and is a state-of-the-art tool for verifying safety of nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Making a fair comparison around metrics for accuracy between Flow* and SpaceEx is challenging, as they represent flowpipes differently [22,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We ask them to perform safety verification for a given pair of initial and bad states, on a collection of different nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' These models, and their parameters, are detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' As described in Section 2, the task of safety verification consists of ensuring that no trajectory starting within the set of initial states enters the set of bad states, within a given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our setup is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Firstly, for a given benchmark model we define a finite time horizon T, a region of initial states X0 and a region of bad states XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, we run flowpipe computations with Flow* using high-order Taylor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Similarly we run the procedure described in Section 3, and construct a hybrid automaton as described in Section 4 to perform flowpipe computations using SpaceEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We present the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In the table, we show the Taylor model order (TM) and time step used within Flow*, as well as the structure of the neural networks used for neural abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For example, we denote a network with two hidden layers with h1 neurons in the first layer and h2 neurons in the second hidden layer as [h1, h2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We note that while Flow*, much like SpaceEx, can perform flowpipe computation on the constructed hybrid automaton, it is not specialised to linear models like SpaceEx is and in practice struggles with the number of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, Flow* is unable to handle the two models that do not exhibit local Lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Flow* constructs Taylor models that incorporate the derivatives of the dynamics: as expected, unbounded derivatives will cause issues for this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Meanwhile, Ariadne [24] a is an alternative tool for over-approximating flowpipes of nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' While Ariadne does not explicitly require Lipschitz continuity, it is also unable to perform analysis on tools with nth root terms at zero, due to numerical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Instead, our abstraction method works directly on the dynamics themselves, rather than their derivatives, in order to construct simpler, abstract models that are amenable to be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By formally quantifying how different an abstract model is through the approximation error, we are able to formally perform safety verification on such challenging concrete models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 9 Table 1: Comparison of safety verification between Flow* and the combination of Neural Abstractions plus SpaceEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Here, T: time horizon, TM: Taylor model order, δ: time-step, t: total computation time (better times denoted by bold), W: network neural structure, M: total number of modes in resulting hybrid automaton, Blw: blowup in the error before T is reached, and -: no results unobtainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Model T Flow* Neural Abstractions TM δ Safety Ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' t W M Safety Ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' t Jet Engine 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Yes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='3 [10, 16] 8 Yes 215 Steam Governor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Yes 62 [12] 29 Yes 219 Exponential 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05 Blw 1034 [14, 14] 12 Yes 308 Water Tank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 No [12] 6 Yes 49 Non-Lipschitz 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='4 No [10] 12 Yes 19 Non-Lipschitz 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 No [12, 10] 32 Yes 59 Notice that we additionally outperform Flow* on a Lipschitz-continuous model (Exponential in Table 1), where the composition of functions that make up the model’s dynamics result in high errors in Flow* before the flowpipe can be calculated across the given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We highlight that despite relying on affine approximations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', 1st order models), neural abstractions are able to compete with, and even outperform, methods that use much higher order functions (10th and 30th in the benchmarks) for approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 Limitations Our approach is limited in terms of scalability, both with regards to the dimension of the models and to the size of the utilised neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The causes of this limitation are twofold: firstly we are bound by the computational complexity of SMT solving - known to be NP-hard - which can struggle with complex formaulae with many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The certification step requires the largest amount of time (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Appendix B), indicating that improvements in the verification of neural networks can lead to a large performance increase for our abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Secondly, we are limited in terms of the complexity of our abstractions by SpaceEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' While SpaceEx is a highly efficient implementation of LGG [88], the presence of a large number of discrete modes poses a significant computational challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' It future work, we hope to investigate the balance between abstraction complexity and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The efficacy of neural abstraction on further tools for hybrid automata with affine dynamics also remains to be investigated [6,24,28,107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 6 Conclusion We have proposed a novel technique that leverages the approximation capabilities of neural networks with ReLU activation functions to synthesise formal abstractions of dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' By combining machine learning and SMT solving algorthms in a CEGIS loop, our method computes abstract neural ODEs with non-determinism that overapproximate the concrete nonlinear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This guarantees the property for which safety of the abstract model carries over to the concrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our method casts these neural ODEs into hybrid automata with affine dynamics, which we have verified using SpaceEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We have demonstrated that our method is not only comparable to Flow* in safety verification on existing nonlinear benchmarks, but also shows superior effectiveness on models that do not exhibit local Lipschitz continuity, which is a hard problem in formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Yet, our experiments are limited to low-dimensional models and scalability remains an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Our approach has advanced the state of the art in terms of expressivity, which is the first step toward obtaining a general and efficient verifier based on neural abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Obtaining scalability to higher dimensions will require a synergy of efficient SMT solvers for neural networks and safety verification of neural ODEs, which are both novel and actively researched questions in formal verification [68,69,76,79,92,95,114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Acknowledgements We thank the anonymous reviewers for their helpful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Alec was supported by the EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines and Systems (EP/S024050/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 10 References [1] Abate, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Ahmed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Edwards, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Giacobbe, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Musau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Johnson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Koutsoukos, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' : Safety verification of cyber-physical systems with reinforcement learning control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 18(5s), 105:1–105:22 (2019) [114] Tran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=': Synthesizing barrier certificates using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In: HSCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 25:1–25:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' ACM (2020) [118] Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Zeng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Woodcock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=': Learning safe neural network controllers with barrier certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Formal Aspects Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 33(3), 437–455 (2021) [119] Zhou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Quartz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Sterck, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=': Neural Lyapunov control of unknown nonlinear systems with stability guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In: NeurIPS (2022) 16 Checklist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] Please see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] See 1 (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] See 2 (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] See 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] The code and data generation will be part of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Reproducing the results will be possible from this but is not the intention of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', data splits, hyperparameters, how they were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] The hyper-parameters for the learning procedure are chosen heuristically, but we include the relevant configuration files in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', with respect to the random seed after running experi- ments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', type of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] See Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] We have cited all used tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] See Section 5 (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [Yes] The code will be included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (a) Did you include the full text of instructions given to participants and screenshots, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' [N/A] 17 A Benchmark Nonlinear Dynamical Models For each dynamical model, we report the vector field f : Rn → Rn and the spatial domain X over which the abstraction is performed and which, unless otherwise stated, is taken to be the hyper-rectangle [−1, 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Water Tank � � � ˙x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 − √x X0 = [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='01] XB = {x|x ≥ 2} (13) Jet Engine [17] � � � � � � � ˙x = −y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5x2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5x3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1, ˙y = 3x − y, X0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='50] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='60, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='55] XB = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='35] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='6] (14) Steam Governor [110] � � � � � � � � � � � � � ˙x = y, ˙y = z2 sin(x) cos(x) − sin(x) − 3y, ˙z = −(cos(x) − 1), X0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='75] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='75] XB = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='6] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='3] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='8] (15) Exponential � � � � � � � ˙x = − sin(exp(y3 + 1)) − y2 ˙y = −x, X0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='86, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='91] XB = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='4] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='6] (16) Non-Lipschitz Vector Field 1 (NL1) � � � � � � � � � � � � � ˙x = y ˙y = √x X = [0, 1] × [−1, 1], X0 = [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05] × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1] XB = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='45] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2] (17) Non-Lipschitz Vector Field 2 (NL2) � � � � � � � ˙x = x2 + y ˙y = 3√ x2 − x, X0 = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='025] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='9, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='85] XB = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='8, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='7] (18) B Additional Experimental Results and Figures B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Experimental Comparison Against Affine Simplical Meshes In this section, we present some supplementary empirical results on neural abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Firstly, we note that hybridisation-based abstraction of nonlinear models have been studied previously, such as in [16], which describes a type of hybridisation-based abstractions that is similar to those constructed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The approach relies first on partitioning the state space using a simplicial mesh grid, and 18 Table 2: A comparison between abstractions constructed using an affine simplicial mesh and neural abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Here, W represents the neural structure used for neural abstraction, NP : total number of partitions, ϵ: the calculated upper bound on the approximation error, ¯ NP : average (mean) number of partitions, ¯ϵ: average (mean) approximation error bound, ϵ+ : the maximum approximation error, ϵ−: the minimum approximation error, Success Ratio: the ratio of repeated experiments that terminated successfully (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 was reached within the first timeout of 300s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Note, we only include successful experiments when calculating the average, min and max (since no error exists for unsuccessful experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' All reported errors use the 2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Benchmark Affine Simplicial Mesh Neural Abstractions Np ϵ W ¯ NP ¯ϵ ϵ+ ϵ− Success Ratio Jet Engine 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='33 [10] 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='33 [10, 10] 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='083 [15, 15] 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='053 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 Steam 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='58 [10] 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='89 [20] 236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 Exponential 8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='7 [10] 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='44 [20] 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='9 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='86 [20, 20] 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='071 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='0 then allowing the dynamics in each mesh to be calculated from an affine interpolation between the vertices of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This affine simplicial mesh (ASM) based approach constructs abstractions of the same expressivity as neural abstractions (first order approximations) with partitions defined by affine inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' An approximation-error bound for ASM can be calculated for systems which have bounded second order derivatives using the model dynamics and the size of each simplex (all simplices are assumed to be the same size), as described in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In Table 2 we compare between abstractions constructed using an affine simplicial mesh and neural abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We run our procedure to synthesise certified abstractions using selected network structures and an initial target error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' If a successful abstraction is synthesised, we reduce the error by some multiplicative factor and repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This iterative procedure continues until no success is reached within a time of 300s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We report the results from 10 repeated experiments over different initial random seeds for neural abstractions, reporting the average (mean), minimum and maximum results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In contrast, we report the approximation-error bound for ASM for different numbers of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The results reported in Table 2 illustrate that neural abstractions outperform ASM based abstractions in terms of error for similar numbers of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Furthermore, neural abstractions generally require significantly fewer partitions for significantly lower approximation-error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In practice this means neural abstractions will outperform ASM-based abstractions for safety verification both in terms of speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We also note the success ratio of our experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=', the ratio of all experiments which achieve an approximation-error bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' These results suggest that in general or procedure is robust and terminates successfully with high probability for reasonable target errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We note that since ASM based abstractions are constructive and are able to deterministically increase the number partitions and consequently reduce the error, for very large numbers of partitions they would achieve lower errors than neural abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' However, in practice these abstractions would be too large in complexity to use with SpaceEx for safety verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 Computation Run-time Profiling In Table 3 we show a breakdown of the runtimes of our procedure shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In particular, we present the total time spent during learning, certification of the abstraction and finally in safety verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 19 Table 3: Breakdown of the timings shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Shown are the timings in the constituent component shown in Figure 2: time spent during learning, time spent during certification of the neural abstraction, and time spent during safety verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Remaining time is spent in overheads, such as converting from neural network to hybrid automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Model Learner Certifier Safety Verification Jet Engine 19 194 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='8 Steam Governor 42 177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 Exponential 27 278 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='3 Water-tank 48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='05 Non-Lipschitz 1 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='5 Non-Lipschitz 2 31 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 C Improved Translation from Neural Abstractions to Hybrid Automata C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='1 Computing Invariant Conditions Invariant conditions are computed from the configuration of a neural network denoted as the sequence C = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , ck) of Boolean vectors c1 ∈ {0, 1}h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , ck ∈ {0, 1}hk, where k denotes the number of hidden layers and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , hk denote the number neurons in each of them (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every vector ci represents the configuration of the neurons at the ith hidden later, and its jth element ci,j represents the activation status of the jth neuron at the ith layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Every mode of the hybrid automaton corresponds to exactly one configuration of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' In turn, every configuration of neurons C restricts the neural network N into a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' More precisely, we inductively define the linear restriction at the ith hidden layer as follows: N (i) C (x) = diag(ci)(WiN (i−1) C (x) + bi), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , k, N (0) C (x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (19) We define the invariant of each mode as a restriction of the domain of interest to a region XC ⊆ X, which denotes the maximal set of states that enables configuration C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' To construct XC, we begin with the observation that the activation configuration ci at every ith hidden layer induces a halfspace on the vector space of the previous layer of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Then, the pre-image of this halfspace backward along the previous layers of the linear restriction of the network characterises a corresponding halfspace on its input neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Since the input neurons are equivalent to the state variables of the dynamical model, the halfspace induced by layer i projected onto state variables x is H(i) C = pre-image of {yi−1 | diag(2ci − 1)(Wiyi−1 + bi) ≥ 0} � �� � halfspace induced by ith layer onto (i − 1)th layer under N (i−1) C (20) The pre-image of a set Y under a function g is defined as {x | g(x) ∈ Y} and can be generally computed by quantifier elimination or, in the linear case, double description methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' However, these methods have worst-case exponential time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' To obtain XC efficiently, we can leverage the fact that the pre-image of any halfspace {y | cTy ≤ d} under any affine function g(x) = Ax+b equals to the set {x | cTy ≤ d ∧ y = Ax + b}, which in turn defines the halfspace {x | cTAx ≤ d − cTb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Therefore, since N (i−1) C is an affine function, every halfspace can be projected backward through the affine functions N (i−1) C , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , N (1) C using O(k) linear algebra operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finally, the entire invariant condition for configuration C is defined as the following polyhedron: XC = ∩{H(i) C | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' , k} ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (21) An invariant condition thus results in a polyhedron defined as the intersection of k halfspaces together with the constrains that define the domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, under the definition in this appendix, the dynamics of mode C given in Equation 10 correspond to the affine dynamical model ˙x = N (k+1) C (x) + d, ∥d∥ ≤ ϵ, x ∈ XC, (22) whose dynamics are governed by the affine function N (k+1) C (x) = Wk+1N (k) C (x) + bk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' (23) 20 N1 N2 X = ∅ N3 N3 End End End X = ∅ C = (1, 0, 1) C = (1, 1, 1) C = (1, 0, 0) X ← X ∩ h+ 1 , X ̸= ∅ X ← X ∩ h− 1 X ← X ∩ h+ 2 , X ̸= ∅ X ← X ∩ h+ 3 , X ̸= ∅ X ← X ∩ h− 3 X ← X ∩ h− 2 , X ̸= ∅ X ← X ∩ h+ 3 X ← X ∩ h− 3 Figure 4: Example Tree search to determine the active configurations for a neural network consisting of a single hidden layer with 3 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Here, h+ i denotes the positive half-space ({x : wix+bi ≥ 0}) and h− i denotes the negative half-space ({x : wix + bi ≤ 0}) of the ith neuron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' wi represents the ith row of the weight matrix corresponding to the hidden layer, and bi represents the ith element of the bias vector of the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Notably, when the set X becomes empty, it is no longer necessary to continue along that path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Once we reach the end of the tree, we have an active configuration C, and backtrack to the last node that was not fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 Enumerating Feasible Modes Determining whether a mode C exists in the hybrid automaton amounts to determining the linear program (LP) associated to polyhedron XC is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Finding all modes therefore consists of solving 2H linear programs, where H = h1 + · · · + hk is the total number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This scales exponentially in the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Here, we elaborate on the tree search algorithm described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content='2 using a diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' the purpose of this algorithm is to efficiently determine all active neuron configurations within a bounded domain of interest X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' We consider an example tree in Figure 4, which depicts an example search for a neural network with a single hidden layer consisting of three neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' The tree illustrates the construction of XC through repeated intersections of half-spaces as paths are taken through the tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' Nodes represent each neuron, labelled Ni, i = 1, 2, 3 and each edge represents one of two possible half-spaces for the neuron it leaves from (ReLU enabled, solid line, and disabled, dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' This approach allows us to prune neurons and overall solve significantly fewer linear programs than simply enumerating through all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFJT4oBgHgl3EQf8S2B/content/2301.11683v1.pdf'} diff --git a/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/2301.04535v1.pdf.txt b/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/2301.04535v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..786ef82729ea1a162a88e6cc7ca3a0b1845fa1e2 --- /dev/null +++ b/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/2301.04535v1.pdf.txt @@ -0,0 +1,1297 @@ +1 +Few-shot Learning for Cross-Target Stance +Detection by Aggregating Multimodal Embeddings +Parisa Jamadi Khiabani, Arkaitz Zubiaga +Queen Mary University of London, UK +{p.jamadikhiabani,a.zubiaga}@qmul.ac.uk +Abstract—Despite the increasing popularity of the stance detec- +tion task, existing approaches are predominantly limited to using +the textual content of social media posts for the classification, +overlooking the social nature of the task. The stance detection +task becomes particularly challenging in cross-target classifi- +cation scenarios, where even in few-shot training settings the +model needs to predict the stance towards new targets for which +the model has only seen few relevant samples during training. +To address the cross-target stance detection in social media by +leveraging the social nature of the task, we introduce CT-TN, +a novel model that aggregates multimodal embeddings derived +from both textual and network features of the data. We conduct +experiments in a few-shot cross-target scenario on six different +combinations of source-destination target pairs. By comparing +CT-TN with state-of-the-art cross-target stance detection models, +we demonstrate the effectiveness of our model by achieving +average performance improvements ranging from 11% to 21% +across different baseline models. Experiments with different +numbers of shots show that CT-TN can outperform other models +after seeing 300 instances of the destination target. Further, +ablation experiments demonstrate the positive contribution of +each of the components of CT-TN towards the final performance. +We further analyse the network interactions between social media +users, which reveal the potential of using social features for cross- +target stance detection. +Index Terms—stance detection, social media, multimodal, clas- +sification. +I. INTRODUCTION +I +N the information-driven world we now inhabit, a large +amount of opinion texts can be found on the Web. The +presence of this content is ever growing as social networking +platforms become increasingly popular, which according to +recent statistics are used by around 65% of American adults +[1], attracting a great deal of public attention [2]. However, +given the volume of content posted daily, monitoring the +opinions expressed in social media platforms remains a time- +consuming task which is not manageable without the support +of automated means [3]. Hence, there is a need to develop +novel methods for automated classification and processing of +these texts to determine the stance expressed in texts with the +aim of mining public opinion. +Stance classification is concerned with identifying a per- +son’s or a post’s standpoint towards a target [4], which is +generally classified as one of in favor of (supporting) or against +(opposing) the target in question [5], [6], [7], [4], [8]. Stance +classification from social media data is however a challenging +task [9], [10], given the diverse and informal nature of social +media data. Despite recent progress in stance classification, +there is still substantial room for improvement, particularly +when it comes to enabling generalisability of classifiers to +deal with new targets [11]. This is the case in cross-target +stance classification; where a classifier has seen training data +associated with particular targets but needs to predict the +stance towards new targets, of which the model has seen +no or few instances. An ability to deal with new targets is +however important, given the evolving nature of the targets +intended for the analysis. For example, an interest for mining +stances towards US President Donald Trump can eventually +shift towards Joe Biden as the new president takes office. +Previous research in cross-target stance classification has +introduced approaches that leverage the textual content of +posts, generally by using transfer learning strategies [12], [10]. +Previous research has however been limited to the sole use of +the textual content of posts to determine the stance, hence +overlooking the potential of other features inherent to social +media. Following the intuition of the theory of homophily, +which suggests that like-minded users will tend to follow +each other and like each other’s posts, we therefore propose +further digging into network features for cross-target stance +detection. Our objective here is both to demonstrate that +network information can be uniquely valuable to enhance the +cross-target stance detection task as well as to propose a novel +methodology that effectively does so. We propose a novel +method, CT-TN, which encapsulates text and network features +through a proposed architecture that aggregates both feature +types for improved stance classification. Our aim here, in line +with much of the previous work, is to use a small number +of instances from the destination target in the training phase +through few-shot learning. +To the best of our knowledge, CT-TN is the first multimodal +architecture for cross-target stance classification which com- +bines text and network features. By experimenting on six dif- +ferent combinations of source and destination targets in a few- +shot cross-target stance detection scenario, we demonstrate the +effectiveness of CT-TN to consistently outperform two state- +of-the-art cross-target stance detection models as well as a +state-of-the-art pre-trained language model, RoBERTa. +Contributions. Through our study, we make the following +key contributions: +• We propose CT-TN (Cross-Target Text-Net) model, a +model that encapsulates multimodal embeddings by inte- +grating state-of-the-art text and graph embedding strate- +gies for the cross-target stance classification task. +arXiv:2301.04535v1 [cs.CL] 11 Jan 2023 + +2 +• We investigate the effectiveness of CT-TN in the few-shot +cross-target stance detection task by using the P-Stance +dataset, one of the largest stance datasets which enables +experimenting with combinations of six different source +and destination target pairs. +• We perform ablation experiments to investigate the im- +pact of the different components that form CT-TN, as- +sessing whether and the extent to which each of them is +contributing positively to improved performance of the +model. +• In addition to our initial experiments with 400 shots +from the destination target used for training, we further +investigate scenarios where fewer shots are available, +investigating its impact on the CT-TN model, and as- +sessing how many shots the model needs to perform +competitively. +We find that our model can consistently outperform state- +of-the-art text-based cross-target stance detection models such +as TGA-Net and CrossNet. Ablation experiments with CT- +TN alternatives demonstrates the contribution of its different +components. While we demonstrate the effectiveness of CT- +TN, we also observe that it becomes less effective when we +dramatically reduce the number of shots used during training, +suggesting that the contribution of carefully integrated network +features becomes useful after 300+ shots, but is less reliable +when fewer shots (100 or 200) are available. +Paper structure. The remainder of this paper is organised +as follows: Section II discusses work related to ours looking +at the challenges of cross-target stance detection as well as +the use of multimodal embeddings for stance detection. We +introduce our proposed method CT-TN in Section III, followed +by the experiment settings which we describe in Section IV. +We present our experiments results in Section V as well as we +further discuss and delve into them in Section VI. We conclude +the paper in Section VII. +II. RELATED WORK +We discuss related work in two main directions: research +on cross-target stance detection and the use of multimodal +embeddings in stance detection. +A. Cross-target Stance Detection +Despite a substantial body of research in stance detection +in recent years [13], [5], [14], the more challenging task +of cross-target stance detection has received less attention. +One of the first approaches to cross-target stance detection +is Bicond [15], which combined multiple layers of LSTM +models in different settings encoding the texts from left to +right and from right to left. [16] developed CrossNet, which +added an Aspect Attention Layer to the Bicond model, which +enabled discovering domain-specific aspects for cross-target +stance inference, utilising self-attention to signal the core +parts of a stance-bearing sentence. Their model consists of +four main layers: Embedding Layer, Context Encoding Layer, +Aspect Attention Layer, and Prediction Layer. Their model +showed to outperform the Bicond model. +A different type of approach focused on transferable topic +modelling. This is the case of the VTN model [12]. This model +uses shared latent topics between two targets as transferable +knowledge in order to achieve model adaptation. The latent +topics are determined by using Neural variational inference +[17]. Another model by [10], called SEKT, proposed to +leverage external knowledge to perform stance detection across +targets. Still limited to processing textual content, they pro- +posed to generate a semantic-emotion heterogeneous graph +(SE-graph) which is fed to a GCN and a BiLSTM for the +classification. +One of the best-known models among those presented +recently is Topic-Grouped Attention (TGA), introduced by +[18], and is one of our key baseline models. The model +consists of four main phases: (i) Contextual Conditional +Encoding, i.e. using contextual emebddings like BERT to +embed a document and topic together, (ii) Generalized Topic +Representations (GTR), i.e. using Ward hierarchical clustering, +(iii) Topic-Grouped Attention, i.e. using learned scaled dot- +product attention (compute similarity scores), and (iv) Label +Prediction, i.e. feed-forward neural network to compute the +output probabilities. This model proved competitive in com- +parison with a range of other baseline models, showing greater +generalisability across different targets than other models. +However, existing approaches are limited to processing +the textual part of the posts only for the cross-target stance +detection. Here, we further design this research by proposing +the first model that leverages network features in addition to +text, through our proposed model CT-TN. To the best of our +knowledge, this is the first study that takes advantage of using +network/social information along with text into the cross- +target stance detection task. In our experiments, in addition +to CT-TN, we also experiment with TGA-Net and CrossNet +as competitive baseline models. +B. Multimodal Embeddings and Stance Detection +Text and Graph Embeddings. There has been a substantial +body of research in recent years in developing embedding +approaches that deal with either texts or graphs separately. +When +it +comes +to +text-based +embeddings, +non- +contextualised +representations +such +as +Word2vec +[19] +and GloVe [20] were soon followed by more sophisticated, +contextualised +representations +such +as +ELMo +[21] +and +OpenAI’s GPT [22]. The latter use unidirectional language +models in order to learn general language representations, +restricting the efficiency of the pre-trained models. +Recently, researchers spend more time on applying transfer +learning by fine-tuning large pre-trained language models for +downstream NLP/NLU tasks, including a small number of +examples which achieves distinguish performance improve- +ment regarding these tasks. Despite the fact that pre-trained +language models are used for this approach, they suffer from a +main limitation which is needing huge corpora for pre-training +plus the high computational cost of days needed for training +[9]. +Transformer models [23], including the likes of BERT [24] +have more recently become the state-of-the-art models for text + +3 +representation in text classification. Models under this category +include BERT, RoBERTa [25], XLM [26] and XLM-R [27]. In +our work, we make use of the RoBERTa model as a component +of CT-TN, as well as a baseline model when used on its own. +When it comes to graph embeddings, different approaches +have been proposed which operate at the node, sub-graph or +different levels of granularity. These types of model include +DeepWalk [28], a method based on random walks. A more +popular method for generating embeddings from graphs is +Node2Vec [29], which consists of a flexible biased random +walk procedure to explore networks, being one of the first +Deep Learning attempts to learn from graph data. There are +similarities between Deepwalk [28] and Node2Vec [29] in that +they both maximise the probability of node co-occurrences in +sampled random-walks approaches across the graph. However +there is a difference based on how the random walks are +sampled. The former uses unbiased random walks, whereas +the latter biases the random walks using two random walk +hyperparameters return parameter (p) and in-out parameter +(q). The p parameter controls the likelihood of immediately +revisiting a vertex; and the q parameter q is responsible to +controlling the likelihood that the walk revisits a vertex’s one- +hop neighbourhood. Deepwalk can in fact be viewed as a +special case of node2vec where p=q=1. +In our work, we use PecanPy [30] as the component to +implement graph embeddings, which is an optimised imple- +mentation of Node2Vec that makes it more efficient thanks +to its paralellisation. We use PecanPy to extract embeddings +from three different types of network information: followers, +friends and likes. +Multimodal Embeddings in Stance Detection. Combining +textual embeddings with graph embeddings has been studied in +previous research, however barely in the context of stance de- +tection. This is the case of [31], who proposed an approach to +enrich a BERT transformer by incorporating knowledge graph +embeddings trained from Wikidata. Through experiments on a +book classification task, they showed that their method could +outperform other text-only baseline methods. +Despite the inherently social nature of the stance detection +task, the vast majority of the research has been limited to +textual features and embeddings. The main exception to this +is the work by [5], who demonstrated that social signals could +also be helpful to predict the stance expressed by a user, +suggesting that it could even be possible to predict the stance +of a user who has not posted anything, solely based on their +network. This work is however limited to in-target stance +detection. +To the best of our knowledge, no work has studied the +multimodal aggregation of textual and graph embeddings for +cross-target stance detection. Through the introduction of CT- +TN, we aim to propose the first approach that effectively +achieves this in the cross-target stance detection task. +III. METHODOLOGY +A. Problem Formulation +We formulate the stance detection task as that in which +each of the posts in a collection P = {p1, p2, ..., pn} has to +be classified into one of the stances S = {favor, against}, +where each post pi expresses a stance towards a target t. We +have a dataset where each post expresses a stance towards one +of the targets in a collection T = {t1, t2, ..., tm}. The cross- +target stance detection task consists in predicting the stance +expressed in posts referring to target ti, where the training data +is composed of posts referring to other targets excluding ti, +hence requiring a transfer of knowledge from a set of targets to +another. In the few-shot cross-target stance detection scenario, +however, we experiment with a small number of instances +referring to ti incorporated into the training data, to relax this +cross-target scenario. In our particular case, we experiment +with 400 instances (i.e. 400 shots) pertaining to ti incorporated +into the training data in the few-shot scenario; in subsequent +experiments, we test with smaller numbers of instances in the +few-shot scenario, namely 100, 200 and 300. +B. Proposed Method: CT-TN +The CT-TN architecture consists of three main types of +components: (i) text-based embedding generation and classifi- +cation, (ii) three instances of network encoding components for +graph-based embedding generation and classification, which +are used for feeding followers, friends and likes, and (iii) +output aggregation. The first two types of components are +executed in parallel to produce isolated stance predictions, +after which the final component aggregates the predictions for +final output. Figure 1 demonstrates general architecture of our +proposed model. In what follows we describe the specifics of +these three components. +Fig. 1. Architecture of the proposed model, CT-TN. +Component #1: Text-based classification: Contextual +Conditional Encoding. This component takes in the textual +content of the input posts using bidirectional conditional en- +coding (conditioning the document representation on the topic) +layer followed by a feed-forward neural network. Previous +works have shown the advantage of utilising contextual em- +beddings [24]. We embed the user generated text through the +RoBERTa language model [25] to embed a document and topic +jointly (768 dimension vector). RoBERTa can take as input +either one or two sentences, and uses the special token [SEP] + +Majority Voting +Predicted Labels (FAVOR/AGAINST) +Cassification +Cassification +Cassification +Cassification +Layer +Layer +Layer +Layer +FollowerGraph +Like Graph +Friend Graph +TextEmbedding +Embedding +Embedding +Embedding +Follower +Like +Friend +Tweet4 +to separate them. In order to input both the textual content +and target information associated with the post, we feed the +following to the model: “[CLS] + target + [SEP] + context”. +This component produces an output with its own prediction for +the stance of a particular post, as either supporting or opposing. +Component #2: Graph-based classification: Network +Encoding. The CT-TN model uses three different instances of +the network encoding model for graph-based classification, for +representing three types of inputs: followers, friends and likes. +To generate embeddings using the Node2Vec architecture [29], +we use the PecanPy implementation [30] which optimises its +performance. The node embeddings calculated using PecanPy +(128 dimension vector) can be used as feature vectors in +a downstream task such as node classification. In our case, +user IDs are considered as graph nodes and the relation- +ships between users (friends/followers/likes) are provided as +graph edges. Each of the three components implemented here +through network encodings produce their own predictions on +a particular post (supporting or opposing). +Component #3: Output aggregation. The final component +takes as input the predictions made by all the above compo- +nents, i.e. the text-based and three network-based components. +To aggregate the predictions of all these four components, +the “output aggregation” component implements a voting +ensemble (or a “majority voting“ strategy) which combines +the different predictions, ultimately choose the class with a +larger number of votes. +C. Model Hyperparameters +We use the RoBERTa base model (roberta-base-cased) as +our pre-trained language model, due to its improved perfor- +mance over similar transformer models such as BERT [32]. +It consists of 12 transformer layers, each of which adopts a +hidden state size of 768 and 12 attention headers. Training for +RoBERTa text embedding is performed with batch size b = +128, dropout probability d = 0.2, learning rate= 3e-5 (AdamW +optimiser) and 40 training epochs. While we trained graph +embedding models as follow: batch size b = 128, dropout +probability d = 0.2, learning rate= 1e-2 (SGD optimiser) and +100 training epochs. For model training, we use multi-class +cross-entropy loss. +While previous research has addressed the stance detection +task in both 3-class [33], [34] and 2-class [35], [36] settings, +our focus here is on the latter, while the extension of our +proposed model to 3-class is beyond the scope of this paper. +IV. EXPERIMENTS +Next, we provide the details of the dataset we use in +our research, as well as the baseline methods we compare +our method against, followed by experiment settings and +evaluation metrics used in our work. +A. Dataset +We chose to use the P-Stance dataset [11], given that it +is an order of magnitude larger than other publicly available +datasets and because it provides more than one target, as +required for our research in cross-target stance detection. The +P-Stance dataset is originally composed of tweets pertaining +to three different political figures as targets: “Donald Trump,” +“Joe Biden,” and “Bernie Sanders.” A sample of the P-Stance +dataset is provided as Table I. +The original P-Stance dataset, as published by the authors, +only contains the tweet texts associated with their stance +labels. This original dataset lacked the network information +that we needed. Upon request, the authors kindly shared 9,307 +tweet IDs, which we use to reconstruct and expand the dataset. +This includes retrieving full tweet metadata, from which we +can extract the user IDs, which would then allow us to retrieve +the user network. +Given the focus of our research in 2-class classification +(i.e. favor or against), we retrieve metadata for the tweets +associated with these categories. This led to 4,212 tweets with +available metadata. The resulting dataset has a distribution +of labels as shown in Table II, and a distribution in the +number of tweets per target as shown in Figure 2. While the +number of tweets across targets is very similar, we observe +some differences in the number of favor and against tweets, +with Donald Trump having the largest ratio of against tweets, +and Bernie Sanders having the largest ratio of favor tweets. +For these available tweets, we further complement the data +retrieval as described below. +Fig. 2. Distribution of targets in the P-Stance dataset. +Having the collection of tweet IDs, tweet metadata and +user IDs, we proceeded with the retrieval of additional data +including networks of the users (followers and friends) and +likes (tweets they liked from other users). We detail each of +these additional data collection steps next: +• Retrieval of followers: Followers include the set of users +who follow a particular user. For the user IDs in the +dataset, we retrieve the complete list of followers for each +user. This leads to a list of user IDs followed by each user, +which allows us to build a network of followers. +• Retrieval of friends: Friends constitute the set of users +followed by a user. Similar to the list of followers, this +provides a list of user IDs per user, with which we can +build a network. +• Retrieval of likes: For each user, we retrieve the tweets +they liked from others. Given that we are interested +in building networks of users, we obtain the user IDs +pertaining to the tweets liked by the users. This again +allows us to build a network, very similar to the friend / + +1600 +1400 +1200 +1000 +count +800 +600 +400 +200 +0 +DonaldTrump +JoeBiden +BernieSanders +Target5 +TABLE I +A SAMPLE OF THE P-STANCE DATASET. +Tweet +Target +Stance +How Joe Biden would make community college free and fix student loans via +@politico +Joe Biden +FAVOR +Glad our GREAT President called out the so called whistleblower. If there is a Senate +trail, they may call the whistleblower to testify. BTW Trump is not impeached until +crazy Nancy send the articles over to the Senate. Trump will not be convicted. Vote +Trump 2020 +Donald Trump +FAVOR +#Bernie Sanders says he’s ’one of the poorer members of the #UnitedStatesSenate’ +#BetOil is A Multimillionaire,#Warren has A 5 Million dollar Home,#Hillary HAS +several Mansions plus A Super Millionaire! Whats Your Point? +Bernie Sanders +AGAINST +TABLE II +THE STATISTICS OF THE RESULTING DATASET. +Target +Favor +Against +Avg. length +Donald Trump +519 +947 +34.7 +Joe Biden +702 +716 +33.7 +Bernie Sanders +776 +553 +31.5 +follower networks above, in this case based on the user +IDs whose tweets have been liked by each user. +Hence, for each user, we have four features: (i) the textual +content of their tweet, (ii) the network of followers, (iii) the +network of friends, and (iv) the network of likes. Each of these +is associated with a favor or against label, which we aim to +predict. +After aggregating all these four features, we end up with +a dataset of 4,144 tweets (posted by 3,871 distinct users) for +which we have all features available. +B. Baseline Methods +We evaluate and compare our model with several strong +baselines, including two of the main state-of-the-art cross- +target stance detection models as well as the widely-used +Transformer model RoBERTa: +• CrossNet [16]: This model is a variant of BiCond, which +leverages a self-attention layer to capture important words +in the input text. +• TGA-NET [18]: A new model has been proposed for +(few-shot) cross-target stance detection that implicitly +captures relationships between topics using generalized +topic representations. +• RoBERTa [25]: The method fine-tunes a pre-trained +BERT model to perform cross-target detection. Specif- +ically, we convert the given context and target to “[CLS] ++ target + [SEP] + context” structure for source and target +domain, respectively. +Note that all of the above baselines make use of the textual +content of the posts, as opposed to our proposed CT-TN also +incorporating network information. +C. Experiment Settings +Experiments for the proposed few-shot cross-target ap- +proach are conducted in 100-shot, 200-shot, 300-shot, and 400- +shot settings (e.g. injecting N samples of destination target +into (source-target based) train data and then predicting the +stance on the test data consist of only destination target) with +5 different random seeds: 24, 524, 1024, 1524, and 2024. Then +we average the 5 seeds’ results per shot. +D. Evaluation Metrics +In line with previous research in stance detection [15], [16], +[18], we also adopt the macro-averaged F1 score (MacFavg) +as the main metric to evaluate the performance in our ex- +periments. In our case with binary classification involving +the support and oppose classes, the resulting metric is the +arithmetic mean of the F1 scores for each class, as follows: +MacFavg = F1support + F1oppose +2 +(1) +where each of F1support and F1oppose is defined as follows: +F1c = 2 ∗ precisionc ∗ recallc +precisionc + recallc +(2) +V. RESULTS +We next present results of our proposed CT-TN model. +We first discuss results of the model compared to a set +of competitive baselines. We then delve into the results by +analysing the performance of ablated versions of the model, +and by looking at the impact of the number of shots used for +training. +A. CT-TN vs Baselines +Table III shows the results of CT-TN for the six com- +binations of source-destination targets under consideration, +compared with the baseline models RoBERTa, CrossNet and +TGA-Net. In addition to the results for each of the target pairs, +we also show the average performance of each model across +all pairs. +We observe that CT-TN consistently outperforms both cross- +target stance detection models, CrossNet and TGA-Net, when +we look at each target pair independently as well as at + +6 +TABLE III +MACRO-AVERAGED F1 SCORES FOR CT-TN VS BASELINE MODELS. +Source +Trump +Sanders +Sanders +Biden +Trump +Biden +Average +Test +Sanders +Trump +Biden +Sanders +Biden +Trump +RoBERTa +0.53 +0.59 +0.78 +0.66 +0.77 +0.62 +0.66 +CrossNet +0.46 +0.51 +0.69 +0.58 +0.6 +0.54 +0.56 +TGA-Net +0.57 +0.6 +0.69 +0.6 +0.69 +0.59 +0.62 +CT-TN +0.72 +0.8 +0.78 +0.73 +0.77 +0.82 +0.77 +TABLE IV +MACRO-AVERAGED F1 SCORES FOR THE FAVOR AND AGAINST CLASSES WITH CT-TN VS BASELINE MODELS. +Source +Trump +Sanders +Sanders +Biden +Trump +Biden +Average +Test +Sanders +Trump +Biden +Sanders +Biden +Trump +‘Favor’ class +RoBERTa +0.64 +0.5 +0.78 +0.72 +0.72 +0.47 +0.64 +CrossNet +0.4 +0.47 +0.68 +0.54 +0.6 +0.49 +0.53 +TGA-Net +0.55 +0.56 +0.7 +0.67 +0.65 +0.47 +0.6 +CT-TN +0.69 +0.78 +0.79 +0.74 +0.75 +0.78 +0.76 +‘Against’ class +RoBERTa +0.44 +0.68 +0.77 +0.59 +0.8 +0.75 +0.67 +CrossNet +0.5 +0.55 +0.7 +0.61 +0.63 +0.6 +0.6 +TGA-Net +0.59 +0.61 +0.65 +0.54 +0.73 +0.7 +0.64 +CT-TN +0.76 +0.81 +0.76 +0.75 +0.8 +0.84 +0.79 +the overall average absolute improvements of 0.21 and 0.15, +respectively. +CT-TN also performs remarkably better for than RoBERTa +for a number of target pairs, not least Trump-Sanders, Sanders- +Trump and Biden-Trump, with absolute improvements of +19%, 21% and 20% respectively. This improvement is more +modest for Biden-Sanders (7%), with similar performances +for the Sanders-Biden and Trump-Biden (0%) target-pairs. On +average, CT-TN still outperforms RoBERTa by 0.11, showing +that it is more consistent across targets and overall more +reliable. We believe that the strongest improvements of CT-TN +with respect to the baselines come particularly for targets with +significantly different ideology (i.e. those combining Trump +and Sanders, and Trump and Biden); this suggests that for +more distant targets, textual models may be more limited in +capturing these substantial linguistic differences, whereas a +network-based model generalises better in these situations. +We next delve into the performance scores of the models +broken down by category: favor and against. Table IV shows +the results for the favor and against categories. We see that the +improvement of the CT-TN model with respect to the baselines +is consistent across both classes, hence showing that CT-TN +provides a positive boost that impacts both classes positively. +The extent of the improvement across classes is also consistent +with the overall results shown above, as we see that the set of +target pairs where CT-TN achieves the highest improvement +matches those with the highest improvement in the overall +results, i.e. Trump-Sanders, Sanders-Trump and Biden-Trump. +Overall, CT-TN achieves improvements of 12% in both the +favor and against class over the second best model, RoBERTa. +B. Ablated versions of CT-TN +To evaluate the effectiveness of each of the text and network +components of CT-TN, we perform a set of ablation of +experiments with different sets of these components removed. +Table V shows the performance scores of the full CT-TN +model compared with ablated versions of the model. +We can see that the full CT-TN model achieves top perfor- +mance in five of the six source-destination target pairs, with the +exception of the Sanders-Trump pair where the use of likes +only outperforms the full model. For the rest of the target +pairs, the full CT-TN either outperforms all ablated models or +achieves the same performance as one of the ablated models. +Interestingly, however, even if some of the ablated models +perform at the same level as the full model on some occasions, +there is no consistency on the best ablated model across target +pairs. Given the uncertainty on the selection of the best ablated +model in each case, it is more reliable to use the full CT- +TN model instead, which is more consistent across all target +pairs. Indeed, this consistency is also demonstrated in the +highest average performance across targets, with an average +0.77 overall. Among the ablated models, those using the +likes feature show competitive performance, with the model +using only likes and the model combining likes, friends and +followers both achieving second-best performance with an +average of 0.76. This in turn suggests that, among the network +features, the likes are the most useful ones. +C. Reducing the number of shots +Experiments so far have relied on the use of 400 shots +associated with the destination target, showing competitive + +7 +TABLE V +MACRO-AVERAGED F1 ON FULL CT-TN VS ABLATED VERSIONS OF CT-TN. +LI: LIKE, FR: FRIENDS, FL: FOLLOWERS, RB: ROBERTA. +Source +Trump +Sanders +Sanders +Biden +Trump +Biden +Average +Test +Sanders +Trump +Biden +Sanders +Biden +Trump +Li +Fr +Fl +Rb +x +0.64 +0.5 +0.78 +0.72 +0.72 +0.47 +0.64 +x +0.71 +0.83 +0.74 +0.73 +0.74 +0.82 +0.76 +x +0.7 +0.79 +0.75 +0.71 +0.75 +0.81 +0.75 +x +0.69 +0.8 +0.74 +0.7 +0.74 +0.8 +0.75 +x +x +0.63 +0.79 +0.77 +0.7 +0.76 +0.74 +0.73 +x +x +0.63 +0.71 +0.76 +0.69 +0.77 +0.73 +0.72 +x +x +0.63 +0.73 +0.78 +0.69 +0.76 +0.71 +0.72 +x +x +x +0.71 +0.81 +0.75 +0.72 +0.75 +0.82 +0.76 +x +x +x +x +0.72 +0.8 +0.78 +0.73 +0.77 +0.82 +0.77 +TABLE VI +MACRO-AVERAGED F1 SCORES ON MODELS USING DIFFERENT NUMBERS OF TRAINING SHOTS (100-400) FROM THE DESTINATION TARGET. +Trump → Sanders +Sanders → Trump +#100 +#200 +#300 +#400 +#100 +#200 +#300 +#400 +RoBERTa +0.24 +0.28 +0.49 +0.53 +0.31 +0.35 +0.54 +0.59 +CrossNet +0.41 +0.46 +0.48 +0.46 +0.49 +0.45 +0.5 +0.51 +TGA-Net +0.4 +0.47 +0.55 +0.57 +0.43 +0.48 +0.58 +0.6 +CT-TN +0.44 +0.4 +0.68 +0.72 +0.3 +0.33 +0.78 +0.8 +Sanders → Biden +Biden → Sanders +#100 +#200 +#300 +#400 +#100 +#200 +#300 +#400 +RoBERTa +0.76 +0.77 +0.76 +0.78 +0.5 +0.47 +0.58 +0.66 +CrossNet +0.62 +0.61 +0.68 +0.69 +0.57 +0.56 +0.58 +0.58 +TGA-Net +0.61 +0.65 +0.69 +0.69 +0.57 +0.56 +0.59 +0.6 +CT-TN +0.59 +0.73 +0.77 +0.78 +0.38 +0.41 +0.73 +0.73 +Trump → Biden +Biden → Trump +#100 +#200 +#300 +#400 +#100 +#200 +#300 +#400 +RoBERTa +0.75 +0.76 +0.78 +0.77 +0.5 +0.5 +0.61 +0.62 +CrossNet +0.6 +0.58 +0.62 +0.6 +0.52 +0.56 +0.57 +0.54 +TGA-Net +0.67 +0.68 +0.69 +0.69 +0.49 +0.5 +0.56 +0.59 +CT-TN +0.74 +0.76 +0.77 +0.77 +0.38 +0.43 +0.8 +0.82 +performance. We are further interested in investigating how +CT-TN performs with fewer shots, as well as to assess the +number of shots the model needs to outperforms other baseline +models. +Table VI shows the results for varying numbers of shots, +ranging from 100 to 400. These results show a clear trend +where the CT-TN model becomes remarkably effective with +300 shots used for training, after which it starts to outperform +baseline models, generally by a margin. Conversely, results +also show that using 200 or fewer shots is insufficient for CT- +TN, where baseline models CrossNet and TGA-Net can per- +form better. Hence, CT-TN becomes especially reliable as the +number of shots increases; however, performance scores are +substantially lower for all models when the number of shots is +200 or fewer, hence suggesting that it is worth labelling some +more instances up to 300 to achieve a substantial performance +gain. +Figure 3 shows the performance of the full CT-TN model, +ablated models as well as baseline models with different +numbers of shots used for training. In addition to the re- +sults presented in Table VI, this figure enables additional +visual comparison by also incorporating ablated models. These +results reaffirm our previous observations, showing that it +is especially after 300 shots that CT-TN and its ablated +models become effective. All CT-TN based models achieve a +remarkable gain of performance from 200 to 300 shots, which +becomes less pronounced when shots are increased from 300 +to 400. +VI. DISCUSSION +Through our experiments, CT-TN has proven to be a +very competitive model achieving state-of-the-art performance +when it is given a moderate number of training data associated +with the destination target. In our case, we have seen that the + +8 +TABLE VII +SAMPLES WITH CORRECT PREDICTION ONLY BY CC-TN, WHERE BASELINE MODELS MISPREDICTED. EXAMPLES EXTRACTED FROM EXPERIMENTS FOR +THE BIDEN-TRUMP TARGET PAIR. +Tweet +Real label +CC-TN +RoBERTa +CrossNet +TGA-NET +Guess we will have to wait forever!! Were working with +heartless #CLOWNS her!! #TRUMP #Trump +FAVOR +FAVOR +AGAINST +AGAINST +AGAINST +What was Nancy Pelosi doing when realDonaldTrump +was putting the #Coronavirus task force together? Hand- +ing out impeachment pens. Voting #Trump and red down +the entire ticket! +FAVOR +FAVOR +AGAINST +AGAINST +AGAINST +People talk about the #GOP being the party of Lincoln +and Reagan, well add realDonaldTrump to it because he +is a game changing #POTUS with policy like this! +FAVOR +FAVOR +AGAINST +AGAINST +AGAINST +Fig. 3. The overall performance of 11 few-shot cross-target tasks. +model can outperform all other baselines with 300 instances +pertaining to the destination target. We are however interested +in further delving into the performance of CT-TN, which we +do next by looking at some of its correct predictions as well as +further investigating the structure of the network data it uses. +To better understand the benefits of CT-TN, we delve into +some of the examples where CT-TN made a correct prediction +and the baseline models made a wrong prediction. We show +some of these CT-TN’s correct predictions in Table VII. We +observe that these are indeed difficult to predict solely from +text for an automated model, not least because there are no +explicitly positive keywords, often requiring more complex +understanding of the text which is not trivial. In situations +like these, information derived from the network through CT- +TN can be particularly valuable to correct these otherwise +challenging predictions. +Looking at the network data, Figure 4 shows a visualisation +of the aggregate of follower, friend and like connections of +supporters of each of the political candidates in the dataset, i.e. +Bernie Sanders (blue), Joe Biden (green) and Donald Trump +(red). Interestingly, we can observe three clear clusters of +supporters of each candidate, with strong connections within +Fig. 4. +Network visualisation of followers, friends and likes for users +expressing supporting stance towards Bernie Sanders (purple), Joe Biden +(green) and Donald Trump (red). +clusters and fewer connections across clusters. Further, we can +also observe that clusters associated with the two candidates +of the Democrats, namely Joe Biden and Bernie Sanders, are +closer and more strongly connected to each other than any of +them is with Republican candidate Donald Trump’s cluster. +Through our experiments with CT-TN, we demonstrate that, +while network information alone would not suffice to achieve +top performance on stance detection, it is a valuable feature +when used in combination with text, indeed outperforming any +ablated models that solely use text or network data. +VII. CONCLUSION +To tackle the challenging task cross-target stance detection +from social media posts, we have introduced a novel model, +CT-TN, which aggregates multimodal text and network em- +beddings into a model. With a set of experiments across six + +Comparison of few-shot performance (F1-macro +on P-stance dataset +0.8 +RoBERTa +0.75 +Like +0.7 +Friend +Follower +0.65 +Like+RoBERTa +0.6 +Friend+RoBERTa +0.55 +Follower+RoBERTa +Like+Friend+Follower +0.5 +AII (CT-TN) +0.45 +CrossNet +0.4 +.TGA-Net +100 +200 +300 +400BernieSanders +JoeBiden +DonaldTrump9 +different source-destination target pairs, we demonstrate the +overall effectiveness of CT-TN, outperforming state-of-the-art +models such as CrossNet and TGA-Net. While all models +struggle with small numbers of shots used for training, CT- +TN achieves a noticeable performance gain after 300 shots +associated with the destination target are incorporated into the +training data. In addition to showing the effectiveness of the +novel CT-TN model, we also demonstrate the importance of +considering network features for cross-target stance detection, +among which the ‘likes’ feature leads to highest performance +gains. +Our work in turn opens a set of avenues for future re- +search. While we demonstrate that we can achieve competitive +performance with 300+ shots, future work could look into +further improving models that perform competitively when +fewer shots available, which is particularly important where +there are limited resources for labelling new data. Our research +demonstrates the effectiveness of CT-TN for 2-class stance +detection, while future research could further look into ex- +tending it to 3-class stance detection. While datasets enabling +cross-target stance detection are very limited to date, not least +datasets for which network data can be gathered, we hope +to see more suitable datasets in the future, which would also +enable further experiments using a bigger set of target pairs. +ACKNOWLEDGMENTS +Parisa Jamadi Khiabani is funded by the Islamic Devel- +opment Bank (IsDB). We thank the authors of the P-stance +dataset for kindly providing us with the tweet IDs which +enables us to complement the dataset. +REFERENCES +[1] A. Perrin, “Social media usage,” Pew research center, vol. 125, pp. 52– +68, 2015. +[2] L. Tian, X. Zhang, Y. Wang, and H. Liu, “Early detection of rumours +on twitter via stance transfer learning,” in European Conference on +Information Retrieval. +Springer, 2020, pp. 575–588. +[3] B. Liu, “Sentiment analysis and opinion mining,” Synthesis lectures on +human language technologies, vol. 5, no. 1, pp. 1–167, 2012. +[4] D. Biber and E. 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Walker, “Internet argument corpus +2.0: An sql schema for dialogic social media and the corpora to go with +it,” in Proceedings of the Tenth International Conference on Language +Resources and Evaluation (LREC’16), 2016, pp. 4445–4452. + diff --git a/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/load_file.txt b/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40b962795ceb7e829b3aaa4f80834243e11be1ff --- /dev/null +++ b/bdE3T4oBgHgl3EQfdgpw/content/tmp_files/load_file.txt @@ -0,0 +1,800 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf,len=799 +page_content='1 Few-shot Learning for Cross-Target Stance Detection by Aggregating Multimodal Embeddings Parisa Jamadi Khiabani, Arkaitz Zubiaga Queen Mary University of London, UK {p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='jamadikhiabani,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='zubiaga}@qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='uk Abstract—Despite the increasing popularity of the stance detec- tion task, existing approaches are predominantly limited to using the textual content of social media posts for the classification, overlooking the social nature of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The stance detection task becomes particularly challenging in cross-target classifi- cation scenarios, where even in few-shot training settings the model needs to predict the stance towards new targets for which the model has only seen few relevant samples during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To address the cross-target stance detection in social media by leveraging the social nature of the task, we introduce CT-TN, a novel model that aggregates multimodal embeddings derived from both textual and network features of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We conduct experiments in a few-shot cross-target scenario on six different combinations of source-destination target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' By comparing CT-TN with state-of-the-art cross-target stance detection models, we demonstrate the effectiveness of our model by achieving average performance improvements ranging from 11% to 21% across different baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Experiments with different numbers of shots show that CT-TN can outperform other models after seeing 300 instances of the destination target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Further, ablation experiments demonstrate the positive contribution of each of the components of CT-TN towards the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We further analyse the network interactions between social media users, which reveal the potential of using social features for cross- target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Index Terms—stance detection, social media, multimodal, clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' INTRODUCTION I N the information-driven world we now inhabit, a large amount of opinion texts can be found on the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The presence of this content is ever growing as social networking platforms become increasingly popular, which according to recent statistics are used by around 65% of American adults [1], attracting a great deal of public attention [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' However, given the volume of content posted daily, monitoring the opinions expressed in social media platforms remains a time- consuming task which is not manageable without the support of automated means [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Hence, there is a need to develop novel methods for automated classification and processing of these texts to determine the stance expressed in texts with the aim of mining public opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Stance classification is concerned with identifying a per- son’s or a post’s standpoint towards a target [4], which is generally classified as one of in favor of (supporting) or against (opposing) the target in question [5], [6], [7], [4], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Stance classification from social media data is however a challenging task [9], [10], given the diverse and informal nature of social media data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Despite recent progress in stance classification, there is still substantial room for improvement, particularly when it comes to enabling generalisability of classifiers to deal with new targets [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This is the case in cross-target stance classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' where a classifier has seen training data associated with particular targets but needs to predict the stance towards new targets, of which the model has seen no or few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' An ability to deal with new targets is however important, given the evolving nature of the targets intended for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' For example, an interest for mining stances towards US President Donald Trump can eventually shift towards Joe Biden as the new president takes office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Previous research in cross-target stance classification has introduced approaches that leverage the textual content of posts, generally by using transfer learning strategies [12], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Previous research has however been limited to the sole use of the textual content of posts to determine the stance, hence overlooking the potential of other features inherent to social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Following the intuition of the theory of homophily, which suggests that like-minded users will tend to follow each other and like each other’s posts, we therefore propose further digging into network features for cross-target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Our objective here is both to demonstrate that network information can be uniquely valuable to enhance the cross-target stance detection task as well as to propose a novel methodology that effectively does so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We propose a novel method, CT-TN, which encapsulates text and network features through a proposed architecture that aggregates both feature types for improved stance classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Our aim here, in line with much of the previous work, is to use a small number of instances from the destination target in the training phase through few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To the best of our knowledge, CT-TN is the first multimodal architecture for cross-target stance classification which com- bines text and network features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' By experimenting on six dif- ferent combinations of source and destination targets in a few- shot cross-target stance detection scenario, we demonstrate the effectiveness of CT-TN to consistently outperform two state- of-the-art cross-target stance detection models as well as a state-of-the-art pre-trained language model, RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Through our study, we make the following key contributions: We propose CT-TN (Cross-Target Text-Net) model, a model that encapsulates multimodal embeddings by inte- grating state-of-the-art text and graph embedding strate- gies for the cross-target stance classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='04535v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='CL] 11 Jan 2023 2 We investigate the effectiveness of CT-TN in the few-shot cross-target stance detection task by using the P-Stance dataset, one of the largest stance datasets which enables experimenting with combinations of six different source and destination target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We perform ablation experiments to investigate the im- pact of the different components that form CT-TN, as- sessing whether and the extent to which each of them is contributing positively to improved performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In addition to our initial experiments with 400 shots from the destination target used for training, we further investigate scenarios where fewer shots are available, investigating its impact on the CT-TN model, and as- sessing how many shots the model needs to perform competitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We find that our model can consistently outperform state- of-the-art text-based cross-target stance detection models such as TGA-Net and CrossNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Ablation experiments with CT- TN alternatives demonstrates the contribution of its different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While we demonstrate the effectiveness of CT- TN, we also observe that it becomes less effective when we dramatically reduce the number of shots used during training, suggesting that the contribution of carefully integrated network features becomes useful after 300+ shots, but is less reliable when fewer shots (100 or 200) are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Paper structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The remainder of this paper is organised as follows: Section II discusses work related to ours looking at the challenges of cross-target stance detection as well as the use of multimodal embeddings for stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We introduce our proposed method CT-TN in Section III, followed by the experiment settings which we describe in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We present our experiments results in Section V as well as we further discuss and delve into them in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We conclude the paper in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' RELATED WORK We discuss related work in two main directions: research on cross-target stance detection and the use of multimodal embeddings in stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Cross-target Stance Detection Despite a substantial body of research in stance detection in recent years [13], [5], [14], the more challenging task of cross-target stance detection has received less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' One of the first approaches to cross-target stance detection is Bicond [15], which combined multiple layers of LSTM models in different settings encoding the texts from left to right and from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' [16] developed CrossNet, which added an Aspect Attention Layer to the Bicond model, which enabled discovering domain-specific aspects for cross-target stance inference, utilising self-attention to signal the core parts of a stance-bearing sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Their model consists of four main layers: Embedding Layer, Context Encoding Layer, Aspect Attention Layer, and Prediction Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Their model showed to outperform the Bicond model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' A different type of approach focused on transferable topic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This is the case of the VTN model [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This model uses shared latent topics between two targets as transferable knowledge in order to achieve model adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The latent topics are determined by using Neural variational inference [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Another model by [10], called SEKT, proposed to leverage external knowledge to perform stance detection across targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Still limited to processing textual content, they pro- posed to generate a semantic-emotion heterogeneous graph (SE-graph) which is fed to a GCN and a BiLSTM for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' One of the best-known models among those presented recently is Topic-Grouped Attention (TGA), introduced by [18], and is one of our key baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The model consists of four main phases: (i) Contextual Conditional Encoding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' using contextual emebddings like BERT to embed a document and topic together, (ii) Generalized Topic Representations (GTR), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' using Ward hierarchical clustering, (iii) Topic-Grouped Attention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' using learned scaled dot- product attention (compute similarity scores), and (iv) Label Prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' feed-forward neural network to compute the output probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This model proved competitive in com- parison with a range of other baseline models, showing greater generalisability across different targets than other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' However, existing approaches are limited to processing the textual part of the posts only for the cross-target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Here, we further design this research by proposing the first model that leverages network features in addition to text, through our proposed model CT-TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To the best of our knowledge, this is the first study that takes advantage of using network/social information along with text into the cross- target stance detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our experiments, in addition to CT-TN, we also experiment with TGA-Net and CrossNet as competitive baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Multimodal Embeddings and Stance Detection Text and Graph Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' There has been a substantial body of research in recent years in developing embedding approaches that deal with either texts or graphs separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' When it comes to text-based embeddings, non- contextualised representations such as Word2vec [19] and GloVe [20] were soon followed by more sophisticated, contextualised representations such as ELMo [21] and OpenAI’s GPT [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The latter use unidirectional language models in order to learn general language representations, restricting the efficiency of the pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Recently, researchers spend more time on applying transfer learning by fine-tuning large pre-trained language models for downstream NLP/NLU tasks, including a small number of examples which achieves distinguish performance improve- ment regarding these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Despite the fact that pre-trained language models are used for this approach, they suffer from a main limitation which is needing huge corpora for pre-training plus the high computational cost of days needed for training [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Transformer models [23], including the likes of BERT [24] have more recently become the state-of-the-art models for text 3 representation in text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Models under this category include BERT, RoBERTa [25], XLM [26] and XLM-R [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our work, we make use of the RoBERTa model as a component of CT-TN, as well as a baseline model when used on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' When it comes to graph embeddings, different approaches have been proposed which operate at the node, sub-graph or different levels of granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' These types of model include DeepWalk [28], a method based on random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' A more popular method for generating embeddings from graphs is Node2Vec [29], which consists of a flexible biased random walk procedure to explore networks, being one of the first Deep Learning attempts to learn from graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' There are similarities between Deepwalk [28] and Node2Vec [29] in that they both maximise the probability of node co-occurrences in sampled random-walks approaches across the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' However there is a difference based on how the random walks are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The former uses unbiased random walks, whereas the latter biases the random walks using two random walk hyperparameters return parameter (p) and in-out parameter (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The p parameter controls the likelihood of immediately revisiting a vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' and the q parameter q is responsible to controlling the likelihood that the walk revisits a vertex’s one- hop neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Deepwalk can in fact be viewed as a special case of node2vec where p=q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our work, we use PecanPy [30] as the component to implement graph embeddings, which is an optimised imple- mentation of Node2Vec that makes it more efficient thanks to its paralellisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We use PecanPy to extract embeddings from three different types of network information: followers, friends and likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Multimodal Embeddings in Stance Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Combining textual embeddings with graph embeddings has been studied in previous research, however barely in the context of stance de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This is the case of [31], who proposed an approach to enrich a BERT transformer by incorporating knowledge graph embeddings trained from Wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Through experiments on a book classification task, they showed that their method could outperform other text-only baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Despite the inherently social nature of the stance detection task, the vast majority of the research has been limited to textual features and embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The main exception to this is the work by [5], who demonstrated that social signals could also be helpful to predict the stance expressed by a user, suggesting that it could even be possible to predict the stance of a user who has not posted anything, solely based on their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This work is however limited to in-target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To the best of our knowledge, no work has studied the multimodal aggregation of textual and graph embeddings for cross-target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Through the introduction of CT- TN, we aim to propose the first approach that effectively achieves this in the cross-target stance detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Problem Formulation We formulate the stance detection task as that in which each of the posts in a collection P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=', pn} has to be classified into one of the stances S = {favor, against}, where each post pi expresses a stance towards a target t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We have a dataset where each post expresses a stance towards one of the targets in a collection T = {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=', tm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The cross- target stance detection task consists in predicting the stance expressed in posts referring to target ti, where the training data is composed of posts referring to other targets excluding ti, hence requiring a transfer of knowledge from a set of targets to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In the few-shot cross-target stance detection scenario, however, we experiment with a small number of instances referring to ti incorporated into the training data, to relax this cross-target scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our particular case, we experiment with 400 instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' 400 shots) pertaining to ti incorporated into the training data in the few-shot scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' in subsequent experiments, we test with smaller numbers of instances in the few-shot scenario, namely 100, 200 and 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Proposed Method: CT-TN The CT-TN architecture consists of three main types of components: (i) text-based embedding generation and classifi- cation, (ii) three instances of network encoding components for graph-based embedding generation and classification, which are used for feeding followers, friends and likes, and (iii) output aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The first two types of components are executed in parallel to produce isolated stance predictions, after which the final component aggregates the predictions for final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Figure 1 demonstrates general architecture of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In what follows we describe the specifics of these three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Architecture of the proposed model, CT-TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Component #1: Text-based classification: Contextual Conditional Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This component takes in the textual content of the input posts using bidirectional conditional en- coding (conditioning the document representation on the topic) layer followed by a feed-forward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Previous works have shown the advantage of utilising contextual em- beddings [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We embed the user generated text through the RoBERTa language model [25] to embed a document and topic jointly (768 dimension vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' RoBERTa can take as input either one or two sentences, and uses the special token [SEP] Majority Voting Predicted Labels (FAVOR/AGAINST) Cassification Cassification Cassification Cassification Layer Layer Layer Layer FollowerGraph Like Graph Friend Graph TextEmbedding Embedding Embedding Embedding Follower Like Friend Tweet4 to separate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In order to input both the textual content and target information associated with the post, we feed the following to the model: “[CLS] + target + [SEP] + context”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This component produces an output with its own prediction for the stance of a particular post, as either supporting or opposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Component #2: Graph-based classification: Network Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The CT-TN model uses three different instances of the network encoding model for graph-based classification, for representing three types of inputs: followers, friends and likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To generate embeddings using the Node2Vec architecture [29], we use the PecanPy implementation [30] which optimises its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The node embeddings calculated using PecanPy (128 dimension vector) can be used as feature vectors in a downstream task such as node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our case, user IDs are considered as graph nodes and the relation- ships between users (friends/followers/likes) are provided as graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Each of the three components implemented here through network encodings produce their own predictions on a particular post (supporting or opposing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Component #3: Output aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The final component takes as input the predictions made by all the above compo- nents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' the text-based and three network-based components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To aggregate the predictions of all these four components, the “output aggregation” component implements a voting ensemble (or a “majority voting“ strategy) which combines the different predictions, ultimately choose the class with a larger number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Model Hyperparameters We use the RoBERTa base model (roberta-base-cased) as our pre-trained language model, due to its improved perfor- mance over similar transformer models such as BERT [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' It consists of 12 transformer layers, each of which adopts a hidden state size of 768 and 12 attention headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Training for RoBERTa text embedding is performed with batch size b = 128, dropout probability d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='2, learning rate= 3e-5 (AdamW optimiser) and 40 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While we trained graph embedding models as follow: batch size b = 128, dropout probability d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='2, learning rate= 1e-2 (SGD optimiser) and 100 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' For model training, we use multi-class cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While previous research has addressed the stance detection task in both 3-class [33], [34] and 2-class [35], [36] settings, our focus here is on the latter, while the extension of our proposed model to 3-class is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' EXPERIMENTS Next, we provide the details of the dataset we use in our research, as well as the baseline methods we compare our method against, followed by experiment settings and evaluation metrics used in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Dataset We chose to use the P-Stance dataset [11], given that it is an order of magnitude larger than other publicly available datasets and because it provides more than one target, as required for our research in cross-target stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The P-Stance dataset is originally composed of tweets pertaining to three different political figures as targets: “Donald Trump,” “Joe Biden,” and “Bernie Sanders.” A sample of the P-Stance dataset is provided as Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The original P-Stance dataset, as published by the authors, only contains the tweet texts associated with their stance labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This original dataset lacked the network information that we needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Upon request, the authors kindly shared 9,307 tweet IDs, which we use to reconstruct and expand the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This includes retrieving full tweet metadata, from which we can extract the user IDs, which would then allow us to retrieve the user network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Given the focus of our research in 2-class classification (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' favor or against), we retrieve metadata for the tweets associated with these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This led to 4,212 tweets with available metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The resulting dataset has a distribution of labels as shown in Table II, and a distribution in the number of tweets per target as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While the number of tweets across targets is very similar, we observe some differences in the number of favor and against tweets, with Donald Trump having the largest ratio of against tweets, and Bernie Sanders having the largest ratio of favor tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' For these available tweets, we further complement the data retrieval as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Distribution of targets in the P-Stance dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Having the collection of tweet IDs, tweet metadata and user IDs, we proceeded with the retrieval of additional data including networks of the users (followers and friends) and likes (tweets they liked from other users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We detail each of these additional data collection steps next: Retrieval of followers: Followers include the set of users who follow a particular user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' For the user IDs in the dataset, we retrieve the complete list of followers for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This leads to a list of user IDs followed by each user, which allows us to build a network of followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Retrieval of friends: Friends constitute the set of users followed by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Similar to the list of followers, this provides a list of user IDs per user, with which we can build a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Retrieval of likes: For each user, we retrieve the tweets they liked from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Given that we are interested in building networks of users, we obtain the user IDs pertaining to the tweets liked by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This again allows us to build a network, very similar to the friend / 1600 1400 1200 1000 count 800 600 400 200 0 DonaldTrump JoeBiden BernieSanders Target5 TABLE I A SAMPLE OF THE P-STANCE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Tweet Target Stance How Joe Biden would make community college free and fix student loans via @politico Joe Biden FAVOR Glad our GREAT President called out the so called whistleblower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' If there is a Senate trail, they may call the whistleblower to testify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' BTW Trump is not impeached until crazy Nancy send the articles over to the Senate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Trump will not be convicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Vote Trump 2020 Donald Trump FAVOR #Bernie Sanders says he’s ’one of the poorer members of the #UnitedStatesSenate’ #BetOil is A Multimillionaire,#Warren has A 5 Million dollar Home,#Hillary HAS several Mansions plus A Super Millionaire!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Whats Your Point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Bernie Sanders AGAINST TABLE II THE STATISTICS OF THE RESULTING DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Target Favor Against Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' length Donald Trump 519 947 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 Joe Biden 702 716 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 Bernie Sanders 776 553 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='5 follower networks above, in this case based on the user IDs whose tweets have been liked by each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Hence, for each user, we have four features: (i) the textual content of their tweet, (ii) the network of followers, (iii) the network of friends, and (iv) the network of likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Each of these is associated with a favor or against label, which we aim to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' After aggregating all these four features, we end up with a dataset of 4,144 tweets (posted by 3,871 distinct users) for which we have all features available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Baseline Methods We evaluate and compare our model with several strong baselines, including two of the main state-of-the-art cross- target stance detection models as well as the widely-used Transformer model RoBERTa: CrossNet [16]: This model is a variant of BiCond, which leverages a self-attention layer to capture important words in the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' TGA-NET [18]: A new model has been proposed for (few-shot) cross-target stance detection that implicitly captures relationships between topics using generalized topic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' RoBERTa [25]: The method fine-tunes a pre-trained BERT model to perform cross-target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Specif- ically, we convert the given context and target to “[CLS] + target + [SEP] + context” structure for source and target domain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Note that all of the above baselines make use of the textual content of the posts, as opposed to our proposed CT-TN also incorporating network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Experiment Settings Experiments for the proposed few-shot cross-target ap- proach are conducted in 100-shot, 200-shot, 300-shot, and 400- shot settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' injecting N samples of destination target into (source-target based) train data and then predicting the stance on the test data consist of only destination target) with 5 different random seeds: 24, 524, 1024, 1524, and 2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Then we average the 5 seeds’ results per shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Evaluation Metrics In line with previous research in stance detection [15], [16], [18], we also adopt the macro-averaged F1 score (MacFavg) as the main metric to evaluate the performance in our ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our case with binary classification involving the support and oppose classes, the resulting metric is the arithmetic mean of the F1 scores for each class, as follows: MacFavg = F1support + F1oppose 2 (1) where each of F1support and F1oppose is defined as follows: F1c = 2 ∗ precisionc ∗ recallc precisionc + recallc (2) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' RESULTS We next present results of our proposed CT-TN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We first discuss results of the model compared to a set of competitive baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We then delve into the results by analysing the performance of ablated versions of the model, and by looking at the impact of the number of shots used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' CT-TN vs Baselines Table III shows the results of CT-TN for the six com- binations of source-destination targets under consideration, compared with the baseline models RoBERTa, CrossNet and TGA-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In addition to the results for each of the target pairs, we also show the average performance of each model across all pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We observe that CT-TN consistently outperforms both cross- target stance detection models, CrossNet and TGA-Net, when we look at each target pair independently as well as at 6 TABLE III MACRO-AVERAGED F1 SCORES FOR CT-TN VS BASELINE MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Source Trump Sanders Sanders Biden Trump Biden Average Test Sanders Trump Biden Sanders Biden Trump RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='66 CrossNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='56 TGA-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='62 CT-TN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 TABLE IV MACRO-AVERAGED F1 SCORES FOR THE FAVOR AND AGAINST CLASSES WITH CT-TN VS BASELINE MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Source Trump Sanders Sanders Biden Trump Biden Average Test Sanders Trump Biden Sanders Biden Trump ‘Favor’ class RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='64 CrossNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='53 TGA-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 CT-TN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='76 ‘Against’ class RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='67 CrossNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 TGA-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='64 CT-TN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='79 the overall average absolute improvements of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='21 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' CT-TN also performs remarkably better for than RoBERTa for a number of target pairs, not least Trump-Sanders, Sanders- Trump and Biden-Trump, with absolute improvements of 19%, 21% and 20% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This improvement is more modest for Biden-Sanders (7%), with similar performances for the Sanders-Biden and Trump-Biden (0%) target-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' On average, CT-TN still outperforms RoBERTa by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='11, showing that it is more consistent across targets and overall more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We believe that the strongest improvements of CT-TN with respect to the baselines come particularly for targets with significantly different ideology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' those combining Trump and Sanders, and Trump and Biden);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' this suggests that for more distant targets, textual models may be more limited in capturing these substantial linguistic differences, whereas a network-based model generalises better in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We next delve into the performance scores of the models broken down by category: favor and against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Table IV shows the results for the favor and against categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We see that the improvement of the CT-TN model with respect to the baselines is consistent across both classes, hence showing that CT-TN provides a positive boost that impacts both classes positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The extent of the improvement across classes is also consistent with the overall results shown above, as we see that the set of target pairs where CT-TN achieves the highest improvement matches those with the highest improvement in the overall results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Trump-Sanders, Sanders-Trump and Biden-Trump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Overall, CT-TN achieves improvements of 12% in both the favor and against class over the second best model, RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Ablated versions of CT-TN To evaluate the effectiveness of each of the text and network components of CT-TN, we perform a set of ablation of experiments with different sets of these components removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Table V shows the performance scores of the full CT-TN model compared with ablated versions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We can see that the full CT-TN model achieves top perfor- mance in five of the six source-destination target pairs, with the exception of the Sanders-Trump pair where the use of likes only outperforms the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' For the rest of the target pairs, the full CT-TN either outperforms all ablated models or achieves the same performance as one of the ablated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Interestingly, however, even if some of the ablated models perform at the same level as the full model on some occasions, there is no consistency on the best ablated model across target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Given the uncertainty on the selection of the best ablated model in each case, it is more reliable to use the full CT- TN model instead, which is more consistent across all target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Indeed, this consistency is also demonstrated in the highest average performance across targets, with an average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Among the ablated models, those using the likes feature show competitive performance, with the model using only likes and the model combining likes, friends and followers both achieving second-best performance with an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' This in turn suggests that, among the network features, the likes are the most useful ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Reducing the number of shots Experiments so far have relied on the use of 400 shots associated with the destination target, showing competitive 7 TABLE V MACRO-AVERAGED F1 ON FULL CT-TN VS ABLATED VERSIONS OF CT-TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' LI: LIKE, FR: FRIENDS, FL: FOLLOWERS, RB: ROBERTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Source Trump Sanders Sanders Biden Trump Biden Average Test Sanders Trump Biden 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 TABLE VI MACRO-AVERAGED F1 SCORES ON MODELS USING DIFFERENT NUMBERS OF TRAINING SHOTS (100-400) FROM THE DESTINATION TARGET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Trump → Sanders Sanders → Trump #100 #200 #300 #400 #100 #200 #300 #400 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 CrossNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='46 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='59 CT-TN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='82 performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We are further interested in investigating how CT-TN performs with fewer shots, as well as to assess the number of shots the model needs to outperforms other baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Table VI shows the results for varying numbers of shots, ranging from 100 to 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' These results show a clear trend where the CT-TN model becomes remarkably effective with 300 shots used for training, after which it starts to outperform baseline models, generally by a margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Conversely, results also show that using 200 or fewer shots is insufficient for CT- TN, where baseline models CrossNet and TGA-Net can per- form better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Hence, CT-TN becomes especially reliable as the number of shots increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' however, performance scores are substantially lower for all models when the number of shots is 200 or fewer, hence suggesting that it is worth labelling some more instances up to 300 to achieve a substantial performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Figure 3 shows the performance of the full CT-TN model, ablated models as well as baseline models with different numbers of shots used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In addition to the re- sults presented in Table VI, this figure enables additional visual comparison by also incorporating ablated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' These results reaffirm our previous observations, showing that it is especially after 300 shots that CT-TN and its ablated models become effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' All CT-TN based models achieve a remarkable gain of performance from 200 to 300 shots, which becomes less pronounced when shots are increased from 300 to 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' DISCUSSION Through our experiments, CT-TN has proven to be a very competitive model achieving state-of-the-art performance when it is given a moderate number of training data associated with the destination target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In our case, we have seen that the 8 TABLE VII SAMPLES WITH CORRECT PREDICTION ONLY BY CC-TN, WHERE BASELINE MODELS MISPREDICTED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' EXAMPLES EXTRACTED FROM EXPERIMENTS FOR THE BIDEN-TRUMP TARGET PAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Tweet Real label CC-TN RoBERTa CrossNet TGA-NET Guess we will have to wait forever!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Were working with heartless #CLOWNS her!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' #TRUMP #Trump FAVOR FAVOR AGAINST AGAINST AGAINST What was Nancy Pelosi doing when realDonaldTrump was putting the #Coronavirus task force together?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Hand- ing out impeachment pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Voting #Trump and red down the entire ticket!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' FAVOR FAVOR AGAINST AGAINST AGAINST People talk about the #GOP being the party of Lincoln and Reagan, well add realDonaldTrump to it because he is a game changing #POTUS with policy like this!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' FAVOR FAVOR AGAINST AGAINST AGAINST Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' The overall performance of 11 few-shot cross-target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' model can outperform all other baselines with 300 instances pertaining to the destination target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We are however interested in further delving into the performance of CT-TN, which we do next by looking at some of its correct predictions as well as further investigating the structure of the network data it uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' To better understand the benefits of CT-TN, we delve into some of the examples where CT-TN made a correct prediction and the baseline models made a wrong prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We show some of these CT-TN’s correct predictions in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We observe that these are indeed difficult to predict solely from text for an automated model, not least because there are no explicitly positive keywords, often requiring more complex understanding of the text which is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In situations like these, information derived from the network through CT- TN can be particularly valuable to correct these otherwise challenging predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Looking at the network data, Figure 4 shows a visualisation of the aggregate of follower, friend and like connections of supporters of each of the political candidates in the dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Bernie Sanders (blue), Joe Biden (green) and Donald Trump (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Interestingly, we can observe three clear clusters of supporters of each candidate, with strong connections within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Network visualisation of followers, friends and likes for users expressing supporting stance towards Bernie Sanders (purple), Joe Biden (green) and Donald Trump (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' clusters and fewer connections across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Further, we can also observe that clusters associated with the two candidates of the Democrats, namely Joe Biden and Bernie Sanders, are closer and more strongly connected to each other than any of them is with Republican candidate Donald Trump’s cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Through our experiments with CT-TN, we demonstrate that, while network information alone would not suffice to achieve top performance on stance detection, it is a valuable feature when used in combination with text, indeed outperforming any ablated models that solely use text or network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' CONCLUSION To tackle the challenging task cross-target stance detection from social media posts, we have introduced a novel model, CT-TN, which aggregates multimodal text and network em- beddings into a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' With a set of experiments across six Comparison of few-shot performance (F1-macro on P-stance dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='8 RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='75 Like 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='7 Friend Follower 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='65 Like+RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='6 Friend+RoBERTa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='55 Follower+RoBERTa Like+Friend+Follower 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='5 AII (CT-TN) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='45 CrossNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content='TGA-Net 100 200 300 400BernieSanders JoeBiden DonaldTrump9 different source-destination target pairs, we demonstrate the overall effectiveness of CT-TN, outperforming state-of-the-art models such as CrossNet and TGA-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While all models struggle with small numbers of shots used for training, CT- TN achieves a noticeable performance gain after 300 shots associated with the destination target are incorporated into the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' In addition to showing the effectiveness of the novel CT-TN model, we also demonstrate the importance of considering network features for cross-target stance detection, among which the ‘likes’ feature leads to highest performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Our work in turn opens a set of avenues for future re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While we demonstrate that we can achieve competitive performance with 300+ shots, future work could look into further improving models that perform competitively when fewer shots available, which is particularly important where there are limited resources for labelling new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Our research demonstrates the effectiveness of CT-TN for 2-class stance detection, while future research could further look into ex- tending it to 3-class stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' While datasets enabling cross-target stance detection are very limited to date, not least datasets for which network data can be gathered, we hope to see more suitable datasets in the future, which would also enable further experiments using a bigger set of target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' ACKNOWLEDGMENTS Parisa Jamadi Khiabani is funded by the Islamic Devel- opment Bank (IsDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' We thank the authors of the P-stance dataset for kindly providing us with the tweet IDs which enables us to complement the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} +page_content=' Perrin, “Social media usage,” Pew research center, vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE3T4oBgHgl3EQfdgpw/content/2301.04535v1.pdf'} diff --git a/ctE2T4oBgHgl3EQfagfL/content/tmp_files/2301.03876v1.pdf.txt b/ctE2T4oBgHgl3EQfagfL/content/tmp_files/2301.03876v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6551958f307470850cc4a4fd32809dd44cee15a --- /dev/null +++ b/ctE2T4oBgHgl3EQfagfL/content/tmp_files/2301.03876v1.pdf.txt @@ -0,0 +1,685 @@ +Depolarization Induced III-V Triatomic Layers with Tristable +Polarization States +Changming Ke,1, 2 Yihao Hu,1, 2 and Shi Liu1, 2, ∗ +1Key Laboratory for Quantum Materials of Zhejiang Province, +Department of Physics, School of Science, +Westlake University, Hangzhou, Zhejiang 310030, China +2Institute of Natural Sciences, Westlake Institute for Advanced Study, +Hangzhou, Zhejiang 310024, China +Abstract +The integration of ferroelectrics that exhibit high dielectric, piezoelectric, and thermal suscepti- +bilities with the mainstream semiconductor industry will enable novel device types for widespread +applications, and yet there are few silicon-compatible ferroelectrics suitable for device downscal- +ing. We demonstrate with first-principles calculations that the enhanced depolarization field at +the nanoscale can be utilized to soften unswitchable wurtzite III-V semiconductors, resulting in +ultrathin two-dimensional (2D) sheets possessing reversible polarization states. A 2D sheet of AlSb +consisting of three atomic planes is identified to host both ferroelectricity and antiferroelectricity, +and the tristate switching is accompanied by a metal-semiconductor transition. The thermody- +namics stability and potential synthesizability of the triatomic layer are corroborated with phonon +spectrum calculations, ab initio molecular dynamics, and variable-composition evolutionary struc- +ture search. We propose a 2D AlSb-based homojunction field effect transistor that supports three +distinct and nonvolatile resistance states. This new class of III-V semiconductor-derived 2D ma- +terials with dual ferroelectricity and antiferroelectricity opens up the possibility for nonvolatile +multibit-based integrated nanoelectronics. +∗ liushi@westlake.edu.cn +1 +arXiv:2301.03876v1 [cond-mat.mtrl-sci] 10 Jan 2023 + +Ferroelectricity, as an extensively studied dipolar ordering state of insulators, is charac- +terized by electrically switchable polarization. The strong coupling between polarization, +strain, and electronic degrees of freedom of ferroelectrics have made them critical compo- +nents in numerous devices such as sensors, actuators, and nonvolatile memories [1, 2]. The +continuing demand for miniaturized electronics has imposed stringent requirements on fer- +roelectrics. In particular, to incorporate ferroelectric functionalities into integrated circuits +via the current semiconductor manufacturing process, materials with nanoscale switchable +dipoles and silicon compatibility are essential [3]. +Two-dimensional (2D) ferroelectrics with long-range dipolar ordering in atom-thick crys- +talline layers are promising materials for ferroelectric-based nanoelectronics because of their +various merits such as the uniform atomic thickness for high-density integration and the +easy preparation of high-quality interface in van der Waals heterostructures [4]. However, +similar to perovskite ferroelectrics, most 2D ferroelectrics also suffer from the depolariza- +tion effect such that they often have the polarization developed in-plane [5, 6], a feature +that is inconvenient for lateral downscaling. Atomically thin monolayers with out-of-plane +polarization (POP) remains rare, and few notable examples confirmed experimentally are +CuInP2S6 [7], α-In2Se3 [8–13], MoTe2 [14], and WTe2 [15]. Additionally, it remains unclear +how to integrate these 2D ferroelectrics with the mainstream semiconductor technology. +A strategy to obtain new ferroelectrics suitable for integrated systems is to “soften” +silicon-compatible piezoelectrics to make them switchable by applying appropriate “stres- +sors” [16]. For example, by substituting Sc into a well-known nitride piezoelectric, AlN, +Fichtner et al. realized a giant switchable polarization (80–110 µC/cm2) in Al1−xScxN [17]. +More recently, starting with another widely used piezoelectric, ZnO, Ferri et al. synthesized +thin films of Zn1−xMgxO and reported even larger switchable polarization of > 100 µC/cm2 +and coercive fields below 3 MV/cm at room temperatures [16]. In both cases, the essence is +to destabilize an unswitchble piezoelectric by applying a chemical stressor. +We propose to “physically soften” silicon-compatible piezoelectrics represented by III-V +wurtzite piezoelectrics via dimension reduction. Products based on III-V semiconductors +have been widely employed in mobile devices, wireless networks, satellite communications, +and optoelectronics [18–20]. For example, the 4th-generation (4G) wireless networks depend +on thin-film bulk acoustic resonators consisting of piezoelectric wurtzite AlN. At present, +the industry of III-V semiconductor manufacturing is well established. Several approaches +2 + +such as direct growth of III-V on Si, III-V on lattice engineered substrate, and III-V on +Ge-Si template have been developed to integrate III-V compounds with the cutting-edge +modern complementary metal oxide semiconductor (CMOS) technology [21, 22]. There- +fore, III-V semiconductor-based 2D ferroelectrics, if available, will reduce the barrier of +integrating ferroelectric functionalities with silicon-based technology and lower the cost of +commercialization. +The physical stressor we employ is the enhanced depolarization field at the nanosale. The +depolarization field (Ed) arising from the incomplete screening of surface polarization bound +charges scales inversely with the film thickness (Ed ∝ Ps/d with Ps the remnant polarization +and d the film thickness) [23]. In thin films of conventional perovskite ferroelectrics such +as PbTiO3, the intrinsic double-well energy landscape of a ferroelectric will eventually be +flattened out by the pronounced depolarization field in thin films below a critical thickness, +leading to a nonpolar paraelectric ground state (Fig. 1a top panel). In contrast, some piezo- +electrics such as wurtzite AlN are unswitchable in bulk because the barrier (∆U) separating +two polar states is prohibitively large such that the switching field exceeds the dielectric +breakdown limit. Utilizing the increased depolarization energy (fd) with reduced dimension +(fd ∝ P 2 +s /d) to compensate ∆U, we suggest it is feasible to soften piezoelectrics to 2D ferro- +electrics with switchable POP. Another competing phase that could emerge in thin films is +an antipolar phase with neighboring antiparallel dipoles that has zero depolarization energy. +It is expected that the neighboring dipoles in bulk wurtzite piezoelectrics strongly favor the +parallel alignment with a coupling strength characterized by J; forming an antipolar phase +thus comes with an energy cost, fc ∝ zJ, with z the coordination number of a local dipole. +Heuristically, the competition between ∆U, fd, and fc determines the ground state (polar, +antipolar, or paraelectric) in free-standing thin films. Moreover, a triple potential well may +emerge by engineering the relative magnitudes of competing energy terms (Fig. 1a bottom +panel). +We explore our design principle with first-principles density functional theory (DFT) cal- +culations, focusing on ultrathin 2D sheets of wurtzite III-V compounds (III=Al, Ga, In; +V=N, P, As). We discover a nonpolar diatomic layer (2L) and a triatomic layer (3L) with +spontaneous local inversion symmetry breaking. Specifically for 3L sheets, it can adopt a +high-energy polar state with POP and a low-energy antipolar state with neighboring an- +tiparallel dipoles. +Interestingly, the polar and antipolar states in 3L AlSb are both dy- +3 + +namically stable, as confirmed by phonon spectrum calculations and ab initio molecular +dynamics (AIMD), and these two states are comparable in energy, making 3L AlSb an un- +usual tristable system that supports both ferroelectricity and antiferroelectricity. Moreover, +the electronic degree of freedom is directly coupled to the polar ordering in 3L AlSb, and +the tristate switching is accompanied with a metal-semiconductor transition. We propose +a 2D homojunction field effect transistor (FET) consisting of 2L and 3L AlSb. The carrier +type and density in the semiconducting channel of 2L AlSb can be effectively regulated by +the polarization state of 3L AlSb, leading to three distinct and nonvolatile resistance states. +The deterministic ferroelectric domain engineering at the nanoscale could be used to pattern +the 2L-3L homojunction as high-density periodic arrays of p-n junctions and p-i-n junctions. +The proposed 2D sheets of III-V compounds supporting tristable polarization states offer +promise for experimental investigation and for the development and design of nonvolatile +multistate functional applications such as high-density memory and synaptic electronics. +DFT calculations are performed using Vienna ab initio Simulation Package (VASP) [24, +25]. +The interaction between the core ion and electrons is described by the projector +augmented wave (PAW) method [26]. The PBEsol functional is chosen as the exchange- +correlation functional [27]. The vacuum layer along the c axis is thicker than 15 ˚A in the +slab model, and the dipole correction is employed to remove the spurious interaction between +different periodic images. We use an energy cutoff of 700 eV, a 8×8×1 Monkhorst–Pack +k-point mesh, and an energy convergence threshold of 10−8 eV for electronic self-consistent +calculations. The convergence criterion for an optimized structure is 10−7 eV in energy. The +structural stability at finite temperatures is studied by NV T AIMD simulations using the Γ- +point sampling with the temperature controlled using the Nos´e-Hoover thermostat [28, 29]. +The phonon spectrum is computed using the frozen phonon approach as implemented in +Phonopy [30] in conjunction with VASP. +The 2D sheet is constructed by cutting the bulk along the c plane, and the thickness of +the film is defined as the number (nL) of atomic planes (Fig. 1b). In the case of monolayers +(nL=1), we find that all nitrides favor the planar structure [31] whereas monolayers of +other III-V compounds are buckled honeycomb structures characterized by the presence of +POP and small values of ∆U (< 0.2 eV, Fig. 1c). We note that III–V buckled honeycomb +monolayers have been studied previously with DFT [32–34], though the 2D ferroelectricity +was not appreciated. The formation energy per formula unit (f.u.) of an isolated 2D sheet +4 + +with respect to the bulk counterpart is defined as Ef +vac = E2D/nL − E3D/N3D, where E2D +is the energy of the 2D sheet consisting of nL atomic planes and E3D is the energy of a +cell in bulk containing N3D atomic planes. Though several III-V monolayers are potential +2D ferroelectrics featuring small switching barriers, their formation energies are rather large +(> 0.8 eV/f.u., Fig. 1d), hinting at the difficulty of synthesis in experiments. +When the thickness increases to nL=2, for III-V compounds (III=Al, Ga, In, V=P, As, +Sb), the initial wurtzite-like configuration is no longer stable, and the optimized diatomic +layer denoted as 2L acquires the inversion symmetry thus being nonpolar (Fig. 1b). Further +increasing the thickness to nL=3 surprisingly revives POP. The triatomic layer labeled as +3L has a buckled central layer that breaks the out-of-plane inversion symmetry (Fig. 1b). +Structurally, both 2L and 3L have group-V anions being the outermost surface layers. We +note that 3L sheets have much lower formation energies than monolayers albeit with higher +∆U (Fig. 1c-d). +This seems to suggest III-V compounds in the 3L form are easier to +synthesize but remain unswitchable. +Following the design principle, we investigate possible competing antipolar phases in 3L +sheets. +We identify an antipolar phase that has the energy consistently lower than the +polar phase (Fig. 1e and Fig. S1 in Supporting Information). Based on Shirane’s energetic +criterion on antiferroelectricity, an antiferroelectric is an antipolar crystal with free energy +comparable to that of the reference polar crystal that has aligned sublattice local dipoles [35]. +Therefore, we suggest 3L sheets of AlSb and GaSb likely host antiferroelectricity as their +polar and antipolar phases are close in energy. In below, we demonstrate that 3L AlSb is +an unusual tristable system that supports both ferroelectricity and antiferroelectricity. +Figure 2a presents the phonon spectra of 3L AlSb in polar and antipolar phase, respec- +tively. Since the phonon spectra have no imaginary vibrational frequencies over the whole +Brillouin zone, polar and nonpolar phases are dynamically stable and each locates at a lo- +cal minimum of the potential energy surface. We perform AIMD simulations at elevated +temperatures to check the structural stability against larger atomic distortions due to ther- +mal fluctuations. The evolution of the total energy at 600 K during AIMD simulations +is shown for both phases in Fig. 2b, revealing no sign of structural destruction or recon- +struction. This serves as a strong evidence to corroborate the room-temperature stability +of 3L AlSb. A defining feature of (anti)ferroelectricity is the polarization reversibility. As +depicted in Fig. 2c, the barrier separating the polar and antipolar phase obtained with the +5 + +nudged elastic band (NEB) method is 0.1 eV that is lower than the barrier in conventional +perovskite ferroelectrics such as PbTiO3 (0.17 eV) [36], indicating a switchable polarization +by an external electric field. Therefore, 3L AlSb is a rare 2D material characterized by +tristable and electrically switchable polarization states and thus hosts both ferroelectricity +and antiferroelectricity. +In addition, we perform a structural search using the variable-composition evolutionary +algorithm as implemented in USPEX [37–39] with a 6-atom slab model confined within 9 ˚A. +Figure. 2d compiles the DFT formation enthalpies of all identified 2D crystals for Al1−xSbx. +We find that ferroelectric 3L AlSb has a convex hall distance of zero, further supporting its +thermodynamic stability and synthesizability. +The emergence of POP in ferroelectric 3L AlSb can be understood by determining the +electric free energy (F) of wurtzite AlSb under an open-circuit boundary condition (OCBC) +that has D = 0 where D is the electric displacement. For an intermediate configuration λ +obtained by linear interpolating the ground-state polar configuration (λ = 1) and the high- +symmetry nonpolar configuration (λ = 0, space group P63/mmc), the free energy F(λ) +under D = 0 can be estimated as [40, 41] +F(λ) = U(λ) + Ω(λ) 1 + 1 +2χ∞(λ) +ϵ0[1 + χ∞(λ)]2P 2(λ) +(1) +where U(λ), P(λ), χ∞(λ), and Ω(λ) are the DFT total (internal) energy per unit cell, electric +polarization, high-frequency dielectric permittivity along the polar direction, and the unit +cell volume of AlSb at configuration λ, respectively, and ϵ0 is the vacuum permittivity. The +internal energy U(λ) becomes the electric free energy under short-circuit boundary condition +(SCBC, E = 0) and the second term is the depolarization energy fd associated with the +depolarization field under OCBC. All quantities required to evaluate F(λ) are bulk values +easily accessible via conventional DFT calculations. This analytical formation of F(λ) has +been used to understand the origin of hyperferroelectricity [40] in thin films under OCBC. +As shown in Fig. 3a, the potential well of U(λ) is rather deep under SCBC. After intro- +ducing the depolarization effect under OCBC, the well becomes shallower and the ground +state remains polar as F(λ) reaches the minimum at λ =0.7. It is noted that Eq. 1 does not +consider the impact of surface reconstruction or the change in χ∞(λ) with reduced dimen- +sion. Nevertheless, the simple analytical model of Eq. 1 predicts that AlSb has a low-energy +polar state under OCBC, resembling a hyperferroelectric [42]. We also plot the DFT energy +6 + +profile for the ferroelectric-antiferroelectric transition in 3L AlSb in Fig. 3a. By compar- +ing the analytical and DFT results, we suggest the surface reconstruction of 3L AlSb that +has group-V anions becoming the outmost surface layers strongly stabilize the polar phase +(λ = 1.1), while the emergence of a low-energy antiferroelectric phase (not captured by the +analytical model) is critical for the polarization switchability. +We now consider the electronic properties of 2L and 3L AlSb. Semilocal density func- +tionals such as PBE often underestimate the band gap due to the remnant self-interaction +error. To obtain accurate electronic structures of 3L AlSb, we employ a newly developed +pseudohybrid Hubbard density functional, extend Agapito–Cuetarolo–Buongiorno Nardelli +(eACBN0) [43–45]. The eACBN0 function is a DFT+U+V method with self-consistently +computed Hubbard U (V ) parameters that account for the onsite (intersite) Coulomb inter- +actions, thus capable of capturing the local variations of Coulomb screening. Particularly for +low-dimensional materials, eACBN0 yields better descriptions of the electronic structures +than hybrid density functionals such as HSE06 [46] that assumes fixed dielectric screen- +ing [44, 47]. Figure 3b-c presents the eACBN0 band structures for 3L AlSb in ferroelectric +and antiferroelectric phases (see the comparison of eACBN0 and HSE06 band structures +in Supporting Information). We find that the ferroelectric phase is a semimetal while the +antiferroelectric phase is a semiconductor with a band gap of 0.7 eV. The semimetal nature +of the ferroelectric phase is due to the built-in depolarization field that induces a band bend- +ing [48, 49] such that the valence band maximum (VBM) and the conduction band minimum +(CBM) are dominated by the states of P − and P + surfaces, respectively (Fig. 3b). Moreover, +we compute the field-induced forces (Fig. 3b inset) and find that nearly all atoms are affected +by an applied field. This indciates the (semi)metallic ferroelectric 3AlSb remains electrically +switchable, similar to 2D metallic WTe2 [15]. In contrast, the antiferroelectric phase has null +depolarization field and the band gap is mostly determined by the hybridization of Al-3p +and Sb-5p states. The strong coupling between the polarization state and the band gap +in 3L AlSb enables intrinsically voltage-switchable metal-semiconductor transition [50], a +useful feature to realize on-off states for device applications. +The ferroelectric field effect transistor (FeFET) comprising a semiconductor as the chan- +nel material and a ferroelectric as the gate insulator is an attractive architecture to realize +low-power, high-speed, and high-density nonvolatile memory. Our DFT calculations show +that 2L AlSb is a nonpolar semiconductor with a band gap of 0.5 eV. Taking advantage +7 + +of the semiconducting property of 2L AlSb and the tristable polarization states affored by +3L AlSb, we propose a 2D homojunction FET using 3L AlSb as the gate insulator and 2L +AlSb as the channel material (Fig. 4a). The design based on a homojunction could simply +the fabrication process and improve the device performance over the heterojucntion-based +device by reducing interfacial defects. +We compute the eACBN0 band structures of a 2L-3L homojunction with 3L AlSb adopt- +ing different polarization states. The contributions from states of 2L AlSb are highlighted in +the band structures to reveal the electrical properties of the channel. As shown in Fig. 4d-f, +the conductivity of 2L AlSb is readily regulated by the polarization state of 3L AlSb. Specif- +ically, when the 3L AlSb adopts the antiferroelectric state, the channel consisting of 2L AlSb +is a semicondcutor with a band gap of ≈0.6 eV (Fig. 4d). When the polarization of 3L AlSb +is switched toward 2L AlSb, the band structure of the homojunction reveals a n-type doped +2L AlSb (Fig. 4e). This can be understood from the band diagram (right before the charge +transfer) illustrated in Fig. 4b. Because the VBM of 3L AlSb is higher in energy than the +CBM of 2L AlSb, high-energy electrons in 3L AlSb naturally relax to the conduction bands +of 2L AlSb, effectively n-type doping the channel. Finally, in the case where 3L AlSb has the +polarization pointing away from 2L AlSb, the channel becomes hole doped as the electrons +in 2L AlSb relax to the CBM of 3L AlSb that is lower in energy (Fig. 4c). Therefore, the +tristable polarization states of 3L AlSb create three resistance states of the channel, suitable +for nonvolatile multistate functional applications. +In addition, the nanoscale deterministic ferroelectric domain engineering can be employed +to configure the 2L-3L homojunction into high-density p-n junction arrays as well as p-i- +n junction arrays where the tristable polarization states of 3L AlSb control the carrier +type and density in 2L AlSb, as shown in Fig. 4g. The voltage-configurable multidomain +pattern offers a platform to design energy-efficient, high-density synaptic electronics and +neuromorphic systems. +In summary, we propose a strategy to obtain switchable 2D polar materials with promis- +ing compatibility with the main stream semiconductor industry. The depolarization field +that is often considered detrimental to ferroelectric properties is used as a physical stressor to +convert unswitchable bulk III-V semiconductors to 2D materials with reversible polarization. +The delicate competition between the local polarization energy, the global depolarization en- +ergy, and the neighboring dipolar coupling in 2D gives rise to a thickness-sensitive phase +8 + +competition. The triatomic layer of AlSb is demonstrated to exhibit tristable polarization +states thus hosting both ferroelectricity and antiferroelectricity. We have explored the func- +tionalities of AlSb-based 2D homojunctions consisting of diatomic and triatomic layers and +predicted the emergence of three distinct and nonvolatile resistance states characterized by +different carrier type and density. 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(b) Construction +of ultrathin 2D sheets by cutting wurtzite III-V piezoelectrics along the c plane. The thickness of +the sheet is defined as the number (nL) of atomic planes. The right panel shows the optimized +structures of monolayer (1L), diatomic layer (2L), and triatomic layer (3L). (c) Energy barrier +(∆U) in bulk and 1L and 3L sheets. (d) Formation energy (Ef +vac) of 1L, 2L, and 3L sheets. (e) +Energy of antipolar (EAP) and polar (EP) phases in 3L sheets relative to the paraelectric (EPE) +phase. +14 + +C +a +1L +△U (Ferroelectrics) +△U+fa +0.8 +Energy +GaSb +AlSb +2L +InSb +0.6 +AlAs +GaAs +e +0.4 +InAs +GaP +U (Pizoelectrics) +AIP +0.2 +Inp +Energy +0.0 +10n + Bulk +anion +Bulk +n. = 1 +Unswitchable +Polarization +d +e +n =3 +1.6 +Polar +0.0 +u +f +(Cn +1.2 +(eV/ +-0.4 +(eV/ +InP +EpE +0.8 +InAs +GaAs GaSb +GaP +InSb +Antipolar +-0.8 +0.4 +E +AlIAs AISb +AIP +-1.2 +Polar +Antipolar +0.0 +nL =2 +AlP AlAs AlSb GaP GaAs GaSb +InP +InAsFIG. 2. (a) Phonon dispersion relationships of 3L AlSb in the polar phase (left) and antipolar +phase (right). (b) AIMD simulations. The top pannel shows the energy evolution as a function of +time at 600 K. The bottom pannel shows the distribution of out-of-plane local displacements (∆z) +of Al atoms in the central layer. (c) Minimum energy path obtained with NEB connecting the +polar and antipolar phases. (d) Convex hull of AlxSb1−x from variable-composition evolutionary +structure search. +15 + +C +a +Polar +Antipolar +400 +300 +0.1 +(cm +一 +200 +200 +100 +-0.1 +100 +d +0. +b +M +K +K +0.5 +2 +9 +10 +4 +Enthalpy of formation (eVlatom) +Time (ps) +Polar +-103 +0.4 +Antipolar +104 +0.3 +-108 +0.1 +20% +Percentage +15% +Percentage. +16% +0.0 +12% +%0 +V +8% +-0.1 +Antipolar +Polar +5% +4% +P2/c +Pmmm +-0.2 +Cm (FE phase) +P1 +. +%0 +L +-1.5 +-1.0 +-0.5 +0.5 +1.5 +-1.0 +-0.5 +0.5 +0.0 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +△z (A) +△z (A) +Composition: Al/(Al+Sb)FIG. 3. (a) Electric free energy F(λ) of AlSb under SCBC (E = 0) and OCBC (D = 0). The DFT +minimum energy pathway (blue line) connecting the ferroelectric (FE) and antiferroelectric (AFE) +phases in 3L AlSb is plotted for comparison. Electronic band structures of (b) ferroelectric and +(c) antiferroelectric 3L AlSb computed with eACBN0. The ferroelectric phase is a semimetal and +the projected band structure has atomic orbital contributions from P − and P + surfaces colored +in blue and red, respectively. The inset in (b) shows the atomic forces induced by an electric field +applied against POP, showing that nearly all atoms are affected by the applied field despite that +the ferroelectric phase is a semimetal. +16 + +a +OCBC, D=0 +0 +入= 0.7 +F(2) (eV) +DFT +Ferroelectric +Antiferroelectric +SCBC.E=0 +? +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +b +入 +2 +1 +(eV) +FE +0 +E +-1 +-2 +M +K +M +C +Z +1 +(eV) +AFE +0 +E +-1 +-2 +M +K +MFIG. 4. (a) Sechmatic of a 2D homojunction field effect transistor consisting of semiconducting +2L AlSb and tristable 3L AlSb. (b) Band bending diagrams of 2L-3L homojunction. The depo- +larization field Ed in ferroelectric 3L AlSb creates a potential step ∆Φ across the sheet. When +POP points toward 2L AlSb, the high-energy electrons transfer from 3L to 2L, making 2L n-type +doped. The polarization reversal will lead to a p-type doped 2L AlSb. Projected electronic band +structures and density of states of 2L-3L homojunction with 3L adopting (d) antiferroelectric, (e) +upward polarization, and (f) downward polarization, showing atomic orbital contributions from 2L +AlSb. (g) Schematic of voltage-configurable multidomain-determined high-density p-n and p-i-n +junction arrays. +17 + +e +d +Source +Drain +ev +2L +3L +-2 +e +Gate +b +Pop +PoP +n-type +c +ev +E +Vacuum +AΦ +AΦ +-2 +M +M +Level +f +p-type +doped +e +VBM +1 +CBM +e +n-type +h+ +e +p-type +(eV) +doped +0 +E +-1 +g +Writing +p-n +31 +个个 +h+ +e +h+ +e +p-i-n \ No newline at end of file diff --git a/ctE2T4oBgHgl3EQfagfL/content/tmp_files/load_file.txt b/ctE2T4oBgHgl3EQfagfL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3bc1b6dc1a1ca6f4aecb562715f4bc8dab89b4b --- /dev/null +++ b/ctE2T4oBgHgl3EQfagfL/content/tmp_files/load_file.txt @@ -0,0 +1,775 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf,len=774 +page_content='Depolarization Induced III-V Triatomic Layers with Tristable Polarization States Changming Ke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2 Yihao Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2 and Shi Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' ∗ 1Key Laboratory for Quantum Materials of Zhejiang Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Westlake University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Zhejiang 310030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' China 2Institute of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Westlake Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Zhejiang 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' China Abstract The integration of ferroelectrics that exhibit high dielectric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' piezoelectric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' and thermal suscepti- bilities with the mainstream semiconductor industry will enable novel device types for widespread applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' and yet there are few silicon-compatible ferroelectrics suitable for device downscal- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We demonstrate with first-principles calculations that the enhanced depolarization field at the nanoscale can be utilized to soften unswitchable wurtzite III-V semiconductors, resulting in ultrathin two-dimensional (2D) sheets possessing reversible polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' A 2D sheet of AlSb consisting of three atomic planes is identified to host both ferroelectricity and antiferroelectricity, and the tristate switching is accompanied by a metal-semiconductor transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The thermody- namics stability and potential synthesizability of the triatomic layer are corroborated with phonon spectrum calculations, ab initio molecular dynamics, and variable-composition evolutionary struc- ture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We propose a 2D AlSb-based homojunction field effect transistor that supports three distinct and nonvolatile resistance states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This new class of III-V semiconductor-derived 2D ma- terials with dual ferroelectricity and antiferroelectricity opens up the possibility for nonvolatile multibit-based integrated nanoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' ∗ liushi@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='03876v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='mtrl-sci] 10 Jan 2023 Ferroelectricity, as an extensively studied dipolar ordering state of insulators, is charac- terized by electrically switchable polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The strong coupling between polarization, strain, and electronic degrees of freedom of ferroelectrics have made them critical compo- nents in numerous devices such as sensors, actuators, and nonvolatile memories [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The continuing demand for miniaturized electronics has imposed stringent requirements on fer- roelectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In particular, to incorporate ferroelectric functionalities into integrated circuits via the current semiconductor manufacturing process, materials with nanoscale switchable dipoles and silicon compatibility are essential [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Two-dimensional (2D) ferroelectrics with long-range dipolar ordering in atom-thick crys- talline layers are promising materials for ferroelectric-based nanoelectronics because of their various merits such as the uniform atomic thickness for high-density integration and the easy preparation of high-quality interface in van der Waals heterostructures [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' However, similar to perovskite ferroelectrics, most 2D ferroelectrics also suffer from the depolariza- tion effect such that they often have the polarization developed in-plane [5, 6], a feature that is inconvenient for lateral downscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Atomically thin monolayers with out-of-plane polarization (POP) remains rare, and few notable examples confirmed experimentally are CuInP2S6 [7], α-In2Se3 [8–13], MoTe2 [14], and WTe2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Additionally, it remains unclear how to integrate these 2D ferroelectrics with the mainstream semiconductor technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' A strategy to obtain new ferroelectrics suitable for integrated systems is to “soften” silicon-compatible piezoelectrics to make them switchable by applying appropriate “stres- sors” [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' For example, by substituting Sc into a well-known nitride piezoelectric, AlN, Fichtner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' realized a giant switchable polarization (80–110 µC/cm2) in Al1−xScxN [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' More recently, starting with another widely used piezoelectric, ZnO, Ferri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' synthesized thin films of Zn1−xMgxO and reported even larger switchable polarization of > 100 µC/cm2 and coercive fields below 3 MV/cm at room temperatures [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In both cases, the essence is to destabilize an unswitchble piezoelectric by applying a chemical stressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We propose to “physically soften” silicon-compatible piezoelectrics represented by III-V wurtzite piezoelectrics via dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Products based on III-V semiconductors have been widely employed in mobile devices, wireless networks, satellite communications, and optoelectronics [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' For example, the 4th-generation (4G) wireless networks depend on thin-film bulk acoustic resonators consisting of piezoelectric wurtzite AlN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' At present, the industry of III-V semiconductor manufacturing is well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Several approaches 2 such as direct growth of III-V on Si, III-V on lattice engineered substrate, and III-V on Ge-Si template have been developed to integrate III-V compounds with the cutting-edge modern complementary metal oxide semiconductor (CMOS) technology [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' There- fore, III-V semiconductor-based 2D ferroelectrics, if available, will reduce the barrier of integrating ferroelectric functionalities with silicon-based technology and lower the cost of commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The physical stressor we employ is the enhanced depolarization field at the nanosale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The depolarization field (Ed) arising from the incomplete screening of surface polarization bound charges scales inversely with the film thickness (Ed ∝ Ps/d with Ps the remnant polarization and d the film thickness) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In thin films of conventional perovskite ferroelectrics such as PbTiO3, the intrinsic double-well energy landscape of a ferroelectric will eventually be flattened out by the pronounced depolarization field in thin films below a critical thickness, leading to a nonpolar paraelectric ground state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1a top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In contrast, some piezo- electrics such as wurtzite AlN are unswitchable in bulk because the barrier (∆U) separating two polar states is prohibitively large such that the switching field exceeds the dielectric breakdown limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Utilizing the increased depolarization energy (fd) with reduced dimension (fd ∝ P 2 s /d) to compensate ∆U, we suggest it is feasible to soften piezoelectrics to 2D ferro- electrics with switchable POP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Another competing phase that could emerge in thin films is an antipolar phase with neighboring antiparallel dipoles that has zero depolarization energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' It is expected that the neighboring dipoles in bulk wurtzite piezoelectrics strongly favor the parallel alignment with a coupling strength characterized by J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' forming an antipolar phase thus comes with an energy cost, fc ∝ zJ, with z the coordination number of a local dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Heuristically, the competition between ∆U, fd, and fc determines the ground state (polar, antipolar, or paraelectric) in free-standing thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Moreover, a triple potential well may emerge by engineering the relative magnitudes of competing energy terms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1a bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We explore our design principle with first-principles density functional theory (DFT) cal- culations, focusing on ultrathin 2D sheets of wurtzite III-V compounds (III=Al, Ga, In;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' V=N, P, As).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We discover a nonpolar diatomic layer (2L) and a triatomic layer (3L) with spontaneous local inversion symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Specifically for 3L sheets, it can adopt a high-energy polar state with POP and a low-energy antipolar state with neighboring an- tiparallel dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Interestingly, the polar and antipolar states in 3L AlSb are both dy- 3 namically stable, as confirmed by phonon spectrum calculations and ab initio molecular dynamics (AIMD), and these two states are comparable in energy, making 3L AlSb an un- usual tristable system that supports both ferroelectricity and antiferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Moreover, the electronic degree of freedom is directly coupled to the polar ordering in 3L AlSb, and the tristate switching is accompanied with a metal-semiconductor transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We propose a 2D homojunction field effect transistor (FET) consisting of 2L and 3L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The carrier type and density in the semiconducting channel of 2L AlSb can be effectively regulated by the polarization state of 3L AlSb, leading to three distinct and nonvolatile resistance states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The deterministic ferroelectric domain engineering at the nanoscale could be used to pattern the 2L-3L homojunction as high-density periodic arrays of p-n junctions and p-i-n junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The proposed 2D sheets of III-V compounds supporting tristable polarization states offer promise for experimental investigation and for the development and design of nonvolatile multistate functional applications such as high-density memory and synaptic electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' DFT calculations are performed using Vienna ab initio Simulation Package (VASP) [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The interaction between the core ion and electrons is described by the projector augmented wave (PAW) method [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The PBEsol functional is chosen as the exchange- correlation functional [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The vacuum layer along the c axis is thicker than 15 ˚A in the slab model, and the dipole correction is employed to remove the spurious interaction between different periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We use an energy cutoff of 700 eV, a 8×8×1 Monkhorst–Pack k-point mesh, and an energy convergence threshold of 10−8 eV for electronic self-consistent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The convergence criterion for an optimized structure is 10−7 eV in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The structural stability at finite temperatures is studied by NV T AIMD simulations using the Γ- point sampling with the temperature controlled using the Nos´e-Hoover thermostat [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The phonon spectrum is computed using the frozen phonon approach as implemented in Phonopy [30] in conjunction with VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The 2D sheet is constructed by cutting the bulk along the c plane, and the thickness of the film is defined as the number (nL) of atomic planes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In the case of monolayers (nL=1), we find that all nitrides favor the planar structure [31] whereas monolayers of other III-V compounds are buckled honeycomb structures characterized by the presence of POP and small values of ∆U (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 eV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We note that III–V buckled honeycomb monolayers have been studied previously with DFT [32–34], though the 2D ferroelectricity was not appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The formation energy per formula unit (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=') of an isolated 2D sheet 4 with respect to the bulk counterpart is defined as Ef vac = E2D/nL − E3D/N3D, where E2D is the energy of the 2D sheet consisting of nL atomic planes and E3D is the energy of a cell in bulk containing N3D atomic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Though several III-V monolayers are potential 2D ferroelectrics featuring small switching barriers, their formation energies are rather large (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1d), hinting at the difficulty of synthesis in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' When the thickness increases to nL=2, for III-V compounds (III=Al, Ga, In, V=P, As, Sb), the initial wurtzite-like configuration is no longer stable, and the optimized diatomic layer denoted as 2L acquires the inversion symmetry thus being nonpolar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Further increasing the thickness to nL=3 surprisingly revives POP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The triatomic layer labeled as 3L has a buckled central layer that breaks the out-of-plane inversion symmetry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Structurally, both 2L and 3L have group-V anions being the outermost surface layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We note that 3L sheets have much lower formation energies than monolayers albeit with higher ∆U (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This seems to suggest III-V compounds in the 3L form are easier to synthesize but remain unswitchable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Following the design principle, we investigate possible competing antipolar phases in 3L sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We identify an antipolar phase that has the energy consistently lower than the polar phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1e and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' S1 in Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Based on Shirane’s energetic criterion on antiferroelectricity, an antiferroelectric is an antipolar crystal with free energy comparable to that of the reference polar crystal that has aligned sublattice local dipoles [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Therefore, we suggest 3L sheets of AlSb and GaSb likely host antiferroelectricity as their polar and antipolar phases are close in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In below, we demonstrate that 3L AlSb is an unusual tristable system that supports both ferroelectricity and antiferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Figure 2a presents the phonon spectra of 3L AlSb in polar and antipolar phase, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Since the phonon spectra have no imaginary vibrational frequencies over the whole Brillouin zone, polar and nonpolar phases are dynamically stable and each locates at a lo- cal minimum of the potential energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We perform AIMD simulations at elevated temperatures to check the structural stability against larger atomic distortions due to ther- mal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The evolution of the total energy at 600 K during AIMD simulations is shown for both phases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2b, revealing no sign of structural destruction or recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This serves as a strong evidence to corroborate the room-temperature stability of 3L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' A defining feature of (anti)ferroelectricity is the polarization reversibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2c, the barrier separating the polar and antipolar phase obtained with the 5 nudged elastic band (NEB) method is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1 eV that is lower than the barrier in conventional perovskite ferroelectrics such as PbTiO3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='17 eV) [36], indicating a switchable polarization by an external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Therefore, 3L AlSb is a rare 2D material characterized by tristable and electrically switchable polarization states and thus hosts both ferroelectricity and antiferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In addition, we perform a structural search using the variable-composition evolutionary algorithm as implemented in USPEX [37–39] with a 6-atom slab model confined within 9 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2d compiles the DFT formation enthalpies of all identified 2D crystals for Al1−xSbx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We find that ferroelectric 3L AlSb has a convex hall distance of zero, further supporting its thermodynamic stability and synthesizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The emergence of POP in ferroelectric 3L AlSb can be understood by determining the electric free energy (F) of wurtzite AlSb under an open-circuit boundary condition (OCBC) that has D = 0 where D is the electric displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' For an intermediate configuration λ obtained by linear interpolating the ground-state polar configuration (λ = 1) and the high- symmetry nonpolar configuration (λ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' space group P63/mmc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' the free energy F(λ) under D = 0 can be estimated as [40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 41] F(λ) = U(λ) + Ω(λ) 1 + 1 2χ∞(λ) ϵ0[1 + χ∞(λ)]2P 2(λ) (1) where U(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' P(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' χ∞(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' and Ω(λ) are the DFT total (internal) energy per unit cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' electric polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' high-frequency dielectric permittivity along the polar direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' and the unit cell volume of AlSb at configuration λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' and ϵ0 is the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The internal energy U(λ) becomes the electric free energy under short-circuit boundary condition (SCBC, E = 0) and the second term is the depolarization energy fd associated with the depolarization field under OCBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' All quantities required to evaluate F(λ) are bulk values easily accessible via conventional DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This analytical formation of F(λ) has been used to understand the origin of hyperferroelectricity [40] in thin films under OCBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 3a, the potential well of U(λ) is rather deep under SCBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' After intro- ducing the depolarization effect under OCBC, the well becomes shallower and the ground state remains polar as F(λ) reaches the minimum at λ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' It is noted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1 does not consider the impact of surface reconstruction or the change in χ∞(λ) with reduced dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Nevertheless, the simple analytical model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1 predicts that AlSb has a low-energy polar state under OCBC, resembling a hyperferroelectric [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We also plot the DFT energy 6 profile for the ferroelectric-antiferroelectric transition in 3L AlSb in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' By compar- ing the analytical and DFT results, we suggest the surface reconstruction of 3L AlSb that has group-V anions becoming the outmost surface layers strongly stabilize the polar phase (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1), while the emergence of a low-energy antiferroelectric phase (not captured by the analytical model) is critical for the polarization switchability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We now consider the electronic properties of 2L and 3L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Semilocal density func- tionals such as PBE often underestimate the band gap due to the remnant self-interaction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' To obtain accurate electronic structures of 3L AlSb, we employ a newly developed pseudohybrid Hubbard density functional, extend Agapito–Cuetarolo–Buongiorno Nardelli (eACBN0) [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The eACBN0 function is a DFT+U+V method with self-consistently computed Hubbard U (V ) parameters that account for the onsite (intersite) Coulomb inter- actions, thus capable of capturing the local variations of Coulomb screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Particularly for low-dimensional materials, eACBN0 yields better descriptions of the electronic structures than hybrid density functionals such as HSE06 [46] that assumes fixed dielectric screen- ing [44, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Figure 3b-c presents the eACBN0 band structures for 3L AlSb in ferroelectric and antiferroelectric phases (see the comparison of eACBN0 and HSE06 band structures in Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We find that the ferroelectric phase is a semimetal while the antiferroelectric phase is a semiconductor with a band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The semimetal nature of the ferroelectric phase is due to the built-in depolarization field that induces a band bend- ing [48, 49] such that the valence band maximum (VBM) and the conduction band minimum (CBM) are dominated by the states of P − and P + surfaces, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Moreover, we compute the field-induced forces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 3b inset) and find that nearly all atoms are affected by an applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This indciates the (semi)metallic ferroelectric 3AlSb remains electrically switchable, similar to 2D metallic WTe2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In contrast, the antiferroelectric phase has null depolarization field and the band gap is mostly determined by the hybridization of Al-3p and Sb-5p states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The strong coupling between the polarization state and the band gap in 3L AlSb enables intrinsically voltage-switchable metal-semiconductor transition [50], a useful feature to realize on-off states for device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The ferroelectric field effect transistor (FeFET) comprising a semiconductor as the chan- nel material and a ferroelectric as the gate insulator is an attractive architecture to realize low-power, high-speed, and high-density nonvolatile memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Our DFT calculations show that 2L AlSb is a nonpolar semiconductor with a band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Taking advantage 7 of the semiconducting property of 2L AlSb and the tristable polarization states affored by 3L AlSb, we propose a 2D homojunction FET using 3L AlSb as the gate insulator and 2L AlSb as the channel material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The design based on a homojunction could simply the fabrication process and improve the device performance over the heterojucntion-based device by reducing interfacial defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We compute the eACBN0 band structures of a 2L-3L homojunction with 3L AlSb adopt- ing different polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The contributions from states of 2L AlSb are highlighted in the band structures to reveal the electrical properties of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4d-f, the conductivity of 2L AlSb is readily regulated by the polarization state of 3L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Specif- ically, when the 3L AlSb adopts the antiferroelectric state, the channel consisting of 2L AlSb is a semicondcutor with a band gap of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='6 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' When the polarization of 3L AlSb is switched toward 2L AlSb, the band structure of the homojunction reveals a n-type doped 2L AlSb (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' This can be understood from the band diagram (right before the charge transfer) illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Because the VBM of 3L AlSb is higher in energy than the CBM of 2L AlSb, high-energy electrons in 3L AlSb naturally relax to the conduction bands of 2L AlSb, effectively n-type doping the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Finally, in the case where 3L AlSb has the polarization pointing away from 2L AlSb, the channel becomes hole doped as the electrons in 2L AlSb relax to the CBM of 3L AlSb that is lower in energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Therefore, the tristable polarization states of 3L AlSb create three resistance states of the channel, suitable for nonvolatile multistate functional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In addition, the nanoscale deterministic ferroelectric domain engineering can be employed to configure the 2L-3L homojunction into high-density p-n junction arrays as well as p-i- n junction arrays where the tristable polarization states of 3L AlSb control the carrier type and density in 2L AlSb, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The voltage-configurable multidomain pattern offers a platform to design energy-efficient, high-density synaptic electronics and neuromorphic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' In summary, we propose a strategy to obtain switchable 2D polar materials with promis- ing compatibility with the main stream semiconductor industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The depolarization field that is often considered detrimental to ferroelectric properties is used as a physical stressor to convert unswitchable bulk III-V semiconductors to 2D materials with reversible polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The delicate competition between the local polarization energy, the global depolarization en- ergy, and the neighboring dipolar coupling in 2D gives rise to a thickness-sensitive phase 8 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The triatomic layer of AlSb is demonstrated to exhibit tristable polarization states thus hosting both ferroelectricity and antiferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' We have explored the func- tionalities of AlSb-based 2D homojunctions consisting of diatomic and triatomic layers and predicted the emergence of three distinct and nonvolatile resistance states characterized by different carrier type and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The readily regulable doping by the tristable polarization states potentially enables facile fabrications of high-density periodic p-n and p-i-n junctions at the nanoscale for nanoelectric and optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' ACKNOWLEDGMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' acknowledge the supports from Westlake Education Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The compu- tational resource is provided by Westlake HPC Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 9 [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Scott, Applications of modern ferroelectrics, Science 315, 954 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' [2] L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Jeon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Liu, On-demand quantum spin hall insulators controlled by two-dimensional ferroelectricity, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Horiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 9, 1440 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' [50] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Duan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Xu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Liu, A two-dimensional multiferroic metal with voltage- tunable magnetization and metallicity, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Horiz 8, 2316 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (a) Utilizing the depolarization energy (fd) to soften unswitchable piezoelectrics with large barrier ∆U separating two polar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The delicate balance between ∆U, fd, and the energy cost (fc) to form antiparallel neighboring dipoles may lead to a triple well in thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (b) Construction of ultrathin 2D sheets by cutting wurtzite III-V piezoelectrics along the c plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The thickness of the sheet is defined as the number (nL) of atomic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The right panel shows the optimized structures of monolayer (1L), diatomic layer (2L), and triatomic layer (3L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (c) Energy barrier (∆U) in bulk and 1L and 3L sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (d) Formation energy (Ef vac) of 1L, 2L, and 3L sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (e) Energy of antipolar (EAP) and polar (EP) phases in 3L sheets relative to the paraelectric (EPE) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 14 C a 1L △U (Ferroelectrics) △U+fa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 Energy GaSb AlSb 2L InSb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='6 AlAs GaAs e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 InAs GaP U (Pizoelectrics) AIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 Inp Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 10n Bulk anion Bulk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' = 1 Unswitchable Polarization d e n =3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='6 Polar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 u f (Cn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 (eV/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 (eV/ InP EpE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 InAs GaAs GaSb GaP InSb Antipolar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 E AlIAs AISb AIP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 Polar Antipolar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 nL =2 AlP AlAs AlSb GaP GaAs GaSb InP InAsFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (a) Phonon dispersion relationships of 3L AlSb in the polar phase (left) and antipolar phase (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (b) AIMD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The top pannel shows the energy evolution as a function of time at 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The bottom pannel shows the distribution of out-of-plane local displacements (∆z) of Al atoms in the central layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (c) Minimum energy path obtained with NEB connecting the polar and antipolar phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (d) Convex hull of AlxSb1−x from variable-composition evolutionary structure search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 15 C a Polar Antipolar 400 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1 (cm 一 200 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1 100 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' b M K K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='5 2 9 10 4 Enthalpy of formation (eVlatom) Time (ps) Polar 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 Antipolar 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='3 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='1 20% Percentage 15% Percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 16% 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 △z (A) △z (A) Composition: Al/(Al+Sb)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (a) Electric free energy F(λ) of AlSb under SCBC (E = 0) and OCBC (D = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The DFT minimum energy pathway (blue line) connecting the ferroelectric (FE) and antiferroelectric (AFE) phases in 3L AlSb is plotted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Electronic band structures of (b) ferroelectric and (c) antiferroelectric 3L AlSb computed with eACBN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The ferroelectric phase is a semimetal and the projected band structure has atomic orbital contributions from P − and P + surfaces colored in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The inset in (b) shows the atomic forces induced by an electric field applied against POP, showing that nearly all atoms are affected by the applied field despite that the ferroelectric phase is a semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 16 a OCBC, D=0 0 入= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='7 F(2) (eV) DFT Ferroelectric Antiferroelectric SCBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='E=0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content='2 b 入 2 1 (eV) FE 0 E 1 2 M K M C Z 1 (eV) AFE 0 E 1 2 M K MFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (a) Sechmatic of a 2D homojunction field effect transistor consisting of semiconducting 2L AlSb and tristable 3L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (b) Band bending diagrams of 2L-3L homojunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The depo- larization field Ed in ferroelectric 3L AlSb creates a potential step ∆Φ across the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' When POP points toward 2L AlSb, the high-energy electrons transfer from 3L to 2L, making 2L n-type doped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' The polarization reversal will lead to a p-type doped 2L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' Projected electronic band structures and density of states of 2L-3L homojunction with 3L adopting (d) antiferroelectric, (e) upward polarization, and (f) downward polarization, showing atomic orbital contributions from 2L AlSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' (g) Schematic of voltage-configurable multidomain-determined high-density p-n and p-i-n junction arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} +page_content=' 17 e d Source Drain ev 2L 3L 2 e Gate b Pop PoP n-type c ev E Vacuum AΦ AΦ 2 M M Level f p-type doped e VBM 1 CBM e n-type h+ e p-type (eV) doped 0 E 1 g Writing p-n 31 个个 h+ e h+ e p-i-n' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfagfL/content/2301.03876v1.pdf'} diff --git a/fNE3T4oBgHgl3EQfHglt/content/tmp_files/2301.04324v1.pdf.txt b/fNE3T4oBgHgl3EQfHglt/content/tmp_files/2301.04324v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6292a9dab1daf083af61d9ee5f280e1742eb773 --- /dev/null +++ b/fNE3T4oBgHgl3EQfHglt/content/tmp_files/2301.04324v1.pdf.txt @@ -0,0 +1,1372 @@ +MNRAS 000, 1–11 (2015) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Investigating the impact of reactions of C and CH with molecular +hydrogen on a glycine gas-grain network +Johannes Heyl,1★ Thanja Lamberts,2,3 Serena Viti3,1 and Jonathan Holdship3,1 +1Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT, London, UK +2Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, PO Box 9502, 2300 RA Leiden, The Netherlands +3Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The impact of including the reactions of C and CH with molecular hydrogen in a gas-grain network is assessed via a sensitivity +analysis. To this end, we vary 3 parameters, namely, the efficiency for the reaction C+H2 −−−→ CH2, and the cosmic ray ionisation +rate, with the third parameter being the final density of the collapsing dark cloud. A grid of 12 models is run to investigate +the effect of all parameters on the final molecular abundances of the chemical network. We find that including reactions with +molecular hydrogen alters the hydrogen economy of the network; since some species are hydrogenated by molecular hydrogen, +atomic hydrogen is freed up. The abundances of simple molecules produced from hydrogenation, such as CH4, CH3OH and +NH3, increase, and at the same time, more complex species such as glycine and its precursors see a significant decrease in their +final abundances. We find that the precursors of glycine are being preferentially hydrogenated, and therefore glycine itself is +produced less efficiently. +Key words: astrochemistry – ISM: abundances – ISM: molecules +1 INTRODUCTION +Interstellar dust plays a significant role in the rich chemistry that +takes place in the interstellar medium. It is widely believed that +complex-organic molecules (COMs) form on interstellar dust (Herbst +& van Dishoeck 2009; Caselli & Ceccarelli 2012) since for certain +molecules, grain-surface reactions are more efficient than gas-phase +reactions. This is particularly important in cold astronomical envi- +ronments where some gas-phase reactions may be highly inefficient, +because a "third body" is needed to take up the excess heat of an +exothermic reaction. Dust grains thus act as an energy sink allowing +the chemistry to thrive and this can lead to the formation of more +complex organic molecules. +Both experimental work and modelling has shown that one such +molecules, namely the amino acid glycine can be formed through +energetic processing of the ices during the warm-up phase of star +formation (Bernstein et al. 2002; Woon 2002; Lee et al. 2009; Bossa +et al. 2009; Ciesla & Sandford 2012; Garrod 2013; Sato et al. 2018), +although there is evidence to suggest that glycine would undergo +destruction under increased irradiation (Pernet et al. 2013; Maté +et al. 2015). In addition, in a joint experimental and modeling effort, +Ioppolo et al. (2021) suggested that non-energetic mechanisms such +as atom-addition reactions might be a promising route for glycine +formation. +A new grain-surface reaction, inserting C atoms in H2 to form +CH2 via C + H2 −−−→ CH2, was recently proposed to be barrierless +by Simončič et al. (2020), based on earlier lab work by Krasnokutski +★ E-mail: johannes.heyl.19@ucl.ac.uk +et al. (2016). They included this reaction in their network and found a +far more rapid conversion of C to CH4. Subsequently, Lamberts et al. +(2022) performed a combined experimental and computational work +to investigate the importance of reactions with molecular hydrogen +for the formation of methane. It was found that while the former +reaction might not be fully barrierless, and the barrier likely depends +on the binding site, the reaction CH + H2 −−−→ CH3 does in fact pro- +ceed without a barrier. The reason these ‘dihydrogenation’ reactions +might be of interest is that they make H2 more chemically active, the +importance of which was recognized already by Hasegawa & Herbst +(1993) and by Meisner et al. (2017) in the context of water formation. +Typically, H2 has one of the lowest binding energies of grain-surface +species, lower than even atomic H (Al-Halabi & van Dishoeck 2007; +Wakelam et al. 2017; Molpeceres & Kästner 2020), which allows +the molecule to diffuse readily on the surface. Moreover, the molec- +ular hydrogen abundance in molecular clouds and pre-stellar cores +is much higher than that of atomic hydrogen (van Dishoeck & Black +1988; Goldsmith & Li 2005). +By including these reactions in chemical models, one might first of +all expect changes in the CH4 abundance, but it is equally interesting +to consider the effect on downstream species such as complex organic +molecules, whose typical abundances are far lower. Their sensitiv- +ity to new reactions should be considered, as their more abundant +precursors might see changes in their abundances. +In this work, we look to build on the work by Simončič et al. +(2020) and Lamberts et al. (2022) to investigate the impact of the +dihydrogenation reactions of C and CH on our gas-grain chemical +network. In particular, we are interested in observing the effect these +reactions have on the production of glycine and its precursors. Our +© 2015 The Authors +arXiv:2301.04324v1 [astro-ph.GA] 11 Jan 2023 + +2 +J.Heyl et al. +glycine network is based on the kinetic Monte Carlo network used +in Ioppolo et al. (2021), using in part updated rate constants from +recent literature, as indicated in Table 2. +We start by describing the astrochemical model, our choice of +parameters and how we evaluate the network sensitivity in Section 2. +We then discuss the results as well as the astrochemical implications +in Section 3 and summarize our conclusions in 4. +2 METHODOLOGY +2.1 The Astrochemical Model +In this work, the gas-grain chemical code UCLCHEM was used +(Holdship et al. 2017)1. UCLCHEM makes use of a rate equation +approach to modelling the gas and grain-surface and bulk abun- +dances. The gas-phase reaction network is taken from the UMIST +database (McElroy et al. 2013). The grain-surface network used was +the default one as available on GitHub. +Various reaction mechanisms are implemented in UCLCHEM. +The grain-surface reaction mechanisms that exist in UCLCHEM +include the Eley-Rideal mechanism as well as the Langmuir- +Hinshelwood diffusion mechanism, which were implemented in Qué- +nard et al. (2018), as was the competition formula from Chang et al. +(2007) and Garrod & Pauly (2011). The binding energies that are +used to calculate the diffusion reaction rate are taken from Wake- +lam et al. (2017). We also included an updated version of the glycine +grain-surface network from Ioppolo et al. (2021), also including both +the reactions C + H2 −−−→ CH2 and CH + H2 −−−→ CH3 as sum- +marized in Table 2. Note that the reaction OH + H2 −−−→ H2O + H +had been already included, based on previous work by, e.g., Meisner +et al. (2017). The code also includes thermal and non-thermal des- +orption, such as due to H2 formation, cosmic ray ionisation as well +as UV-induced desorption. Note that the astrochemical model used +in Ioppolo et al. (2021) makes use of the non-diffusive grain-surface +chemistry that is described in Garrod & Pauly (2011) and Jin & Gar- +rod (2020). This is not used in UCLCHEM. The implications of this +will be discussed later in this work. +UCLCHEM is used to model two distinct phases of the star forma- +tion process. Phase 1 is the free-fall collapse phase of a dark cloud +for a default value of 5 million years, whereas Phase 2 models the +warm-up phase immediately following Phase 1, with the initial den- +sity of Phase 2 equal to the final density of Phase 1. Phase 2 runs for +1 million years. Further details of the code can be found in Holdship +et al. (2017). +2.2 Parameter Selection +To assess the importance of the two proposed reactions to the net- +work under various interstellar conditions, three parameters were +varied, as listed in Table 2. The standard cosmic ray ionisation rate in +UCLCHEM is 𝜁 = 1.3×10−17 s−1. This is in line with typical values +that are of the order 10−17 s−1 in diffuse ISM conditions (O’Donnell +& Watson 1974; Black et al. 1978; Hartquist et al. 1978; Indriolo & +McCall 2013). However, there exist observations of higher cosmic +ray ionisation rates (Indriolo et al. 2007; Indriolo & McCall 2012), +which is why we also include analysis of a region with cosmic ray +ionisation rate of 10𝜁. Cosmic ray ionisation is typically expected +to break larger molecules into smaller radicals. We did not consider +1 https://uclchem.github.io/ +lower values of the cosmic ray ionisation rate, as these are typi- +cally not observed. The cosmic ray dependency on column density +in O’Donoghue et al. (2022) covered a range of values that were, +however, already covered by the factor of 10 we consider here. While +they found differences for lower densities during the collapse phase, +these were ironed out once the collapse reached larger final densities, +which is why here we do not include this dependency on column +density. +Three different astronomical regions were modelled: +(i) a dark cloud with a final density of 105 cm−3 +(ii) a low-mass protostar with a final density of 106 cm−3 +(iii) a high-mass protostar, with a final density of 107 cm−3 +The heating profiles during Phase 2 for the last two cases are based +on Viti et al. (2004) and differ for each astronomical object. The dark +cloud simulation was only run for Phase 1, but was allowed to run +for a further million years to allow the chemistry to settle. +Another parameter that was varied was the efficiency, 𝛼, of the +extent to which the reaction C + H2 −−−→ CH2 is barrierless. While +Simončič et al. (2020) considered the reaction to be fully barrierless, +Lamberts et al. (2022) found that the reaction barrier likely depends +on the binding site. As such, our grid of models considers efficiencies +for the reaction of 0 (the reaction is not included), 0.05 (5% of binding +sites lead to a barrierless reaction and 95% of the binding sites have +an infinitely high barrier) and 1 (the reaction is fully barrierless). +What this means practically is that the reaction rate is multiplied by +the efficiency. The reaction CH + H2 −−−→ CH3 was included as only +being barrierless, based on Lamberts et al. (2022). +2.3 Evaluating the network sensitivity +We quantify the effect of the new reactions on the model by con- +sidering the change in abundances of the species that are the most +affected when taking the ratio of the abundances of the modified +and original models. The modified model is the chemical network +which has 𝛼 = 1, whereas the original model was taken to be the +network which had neither of the dihydrogenation reactions. These +two scenarios were taken to be the extremes of the parameter range +in terms of including these reactions. The ratio is most sensitive +to strong deviations in the molecular abundances as a result of the +dihydrogenation reactions. +This ratio is defined for each species 𝑖 as: +𝛿𝑖(𝑡) = +𝑥𝑀 +𝑖 (𝑡) +𝑥𝑂 +𝑖 (𝑡) +, +(1) +where 𝑥𝑀 +𝑖 (𝑡) is the abundance of species 𝑖 in the modified model at +time 𝑡 and 𝑥𝑂 +𝑖 (𝑡) is the abundance of the same species in the original +model at time 𝑡. +We only considered species which had a value above a “threshold +of detectability". This was to ensure that we did not look at species +whose original and changed abundances were below what can be ob- +served from an astronomical point-of-view. For grain-surface species +this threshold was set to 10−8 with respect to hydrogen whereas for +gas-phase species this threshold equalled 10−12 with respect to hy- +drogen. We took 10−8 as a lower-limit threshold for grain-surface +species, as this was the order of magnitude of the lowest reported +abundances in Boogert et al. (2015). Similarly, the gas-phase thresh- +old was taken based on the abundances of COMs typically observed +in the gas-phase, such as in Jiménez-Serra et al. (2016, 2021). +MNRAS 000, 1–11 (2015) + +Impact of C and CH reacting with H2 +3 +Reaction No. +Reaction +Reference +1 +CO + OH −−−→ HOCO +Arasa et al. (2013) +2 +HOCO + H −−−→ H2 + CO2 +Goumans et al. (2008) +3 +HOCO + H −−−→ HCOOH +Goumans et al. (2008); Ioppolo et al. (2011) +4 +CH4 + OH −−−→ CH3 + H2O +Lamberts et al. (2017) +5 +NH2 + CH3 −−−→ NH2CH3 +Ioppolo et al. (2021) +6 +NH3 + CH −−−→ NH2CH2 +Balucani et al. (2009) +7 +NH2CH2 + H −−−→ NH2CH3 +Ioppolo et al. (2021) +8 +NH2CH3 + H −−−→ NH2CH2 + H2 +Oba et al. (2014) +9 +NH2CH3 + OH −−−→ NH2CH2 + H2O +Ioppolo et al. (2021) +10 +NH2CH2 + HOCO −−−→ NH2CH2COOH +Woon (2002) +11 +H2 + OH −−−→ H2O + H +Meisner et al. (2017) +12 +O2 + H −−−→ HO2 +Lamberts et al. (2013) +13 +HO2 + H −−−→ OH + OH +Lamberts et al. (2013) +14 +HO2 + H −−−→ H2 + O2 +Lamberts et al. (2013) +15 +HO2 + H −−−→ H2O + O +Lamberts et al. (2013) +16 +OH + OH −−−→ H2O2 +Lamberts et al. (2013) +17 +OH + OH −−−→ H2O + O +Lamberts et al. (2013) +18 +H2O2 + H −−−→ H2O + OH +Lamberts & Kästner (2017) +19 +N + O −−−→ NO +Ioppolo et al. (2021) +20 +NO + H −−−→ HNO +Fedoseev et al. (2012) +21 +HNO + H −−−→ H2NO +Fedoseev et al. (2012) +22 +HNO + H −−−→ NO + H2 +Fedoseev et al. (2012) +23 +HNO + O −−−→ NO + OH +Ioppolo et al. (2021) +24 +HN + O −−−→ HNO +Ioppolo et al. (2021) +25 +N + NH −−−→ N2 +Ioppolo et al. (2021) +26 +NH + NH −−−→ N2 + H2 +Ioppolo et al. (2021) +27 +C + O −−−→ CO +Ioppolo et al. (2021) +28 +CH3 + OH −−−→ CH3OH +Qasim et al. (2018) +29 +C + H2 −−−→ CH2 +Simončič et al. (2020); Lamberts et al. (2022) +30 +CH + H2 −−−→ CH3 +Lamberts et al. (2022) +Table 1. Table of the reactions added to the standard UCLCHEM network. +Parameter +Values +Comment +Final Density of Phase 1 and Initial Density of Phase 2 +105 cm−3, 106 cm−3, 107 cm−3 +Final density of Phase 1 same as initial density of Phase 2 +Efficiency for barrierless C + H2 −−−→ CH2 +0, 0.05, 1 +Efficiency of 0 is equivalent to reaction being excluded. +Cosmic Ray Ionisation Rate +𝜁 , 10𝜁 +𝜁 is the standard cosmic ray ionisation rate of 1.3 × 10−17 s−1 +Table 2. The parameters that were varied in this work to assess the effect of the two reactions. +MNRAS 000, 1–11 (2015) + +4 +J.Heyl et al. +We can also define a quantity that tracks the absolute change in +the abundance of species: +Δ𝑖(𝑡) = 𝑥𝑀 +𝑖 (𝑡) − 𝑥𝑂 +𝑖 (𝑡) = 𝑥𝑂 +𝑖 (𝑡)[𝛿𝑖(𝑡) − 1], +(2) +This value indicates how species with relatively large abundances, +such as elemental species or their hydrogenation products, are re- +distributed. +3 RESULTS AND ASTROCHEMICAL IMPLICATIONS +We find that even though the amounts by which various species are +affected differs for each stage of star formation, the general trends are +broadly similar. As such, we group our analysis per phase. Tables 3 +and 4 summarise the changes in terms of 𝛿. The effect of the enhanced +cosmic ray ionisation rate is discussed in Section 3.1.3. +Our results differ from Ioppolo et al. (2021) in that, while glycine +does form on the grains, it does not do so in Phase 1, as UCLCHEM +does not utilise non-diffusive grain-surface mechanisms. Instead, +glycine forms on the grains as the temperature increases in Phase 2. +3.1 Impact of the Parameters +In this sub-section we consider the role that the physical and chemical +parameters play. Tables 3 and 4 show the changes in abundance +when we compare the original network without the dihydrogenation +reactions with the 𝛼 = 1 case. Figures 1 and 2 show the time series +of the abundances for glycine and its precursors. +3.1.1 Final Density +The final density of the collapsing cloud had a minor effect on the +final abundances of the species in Phase 1. For all three astronomical +objects modelled in Phase 1, we observe a significant decrease of +grain-surface CH and C when the reactions are included and see +an enhancement of grain-surface CH2, CH3 and CH4. However, +the values of 𝛿 as well as their original abundances seem to be +independent of the density, suggesting a saturation effect. +In Phase 2, we observe that the final density of the collapsing +cloud does affect the extent to which the added reactions influence +the final abundances. We notice that several hydrogenation-based +species have greater abundances at lower densities, including species +such as HOCO, H2O2, CH3CCH and H2CO. +3.1.2 Efficiency +For more abundant species, such as H2O and CH3OH, we find that +the results obtained from using a branching fraction of 0.05 for the +barrierless dihydrogenation of C are essentially the same as using a +efficiency of 1 (the reaction is fully barrierless). +We do find that the efficiency parameter plays a role in the final +abundances of glycine and its precursors during the warm-up phase +of low and high-mass stars. This can be seen in Figures 1 and 2. +For Phase 1, the species are not detectable except for the original +configuration. However, we still observe that for the other three con- +figurations an increasing value of 𝛼 corresponds to an increased level +of depletion. In Phase 2, the configurations are all detectable and this +same hierarchy remains in the gas-phase. +3.1.3 Cosmic Ray Ionisation Rate +The degree of cosmic ray ionisation is found to play an important role +in enhancing or counteracting the role of the dihydrogenation reac- +tions. The cosmic ray destruction routes we include in our standard +network are from Garrod et al. (2008). These consist of hydrogen +abstraction reactions and reactions that produce radical-radical pairs +of products. An enhanced cosmic ray ionisation rate leads to the +destruction of many of hydrogenated species, such as CH4, NH3, +H2O and CH3OH, as well as their precursors. This leads to further +hydrogen reservoirs being released and radicals being formed which +can go on to form glycine and its precursors. Because no cosmic +ray destruction mechanisms for these complex, larger, species are +included, we find that these are more abundantly produced. +This is important to consider in the context of glycine. In Figures +1 and 2, we plot the time dependence of the abundance of glycine +precursors for eight different parameter sets, including the enhanced +cosmic ray ionisation rate. In Phase 1, we find that on the grains, the +enhanced cosmic ray ionisation rate depletes the species. In Phase 2, +the effect varies by configuration and species. The original config- +uration consistently leads to a decrease of all plotted species in the +presence of enhanced cosmic ray ionisation. The 𝛼 = 0 configuration +is depleted for the methylamine radical and glycine, but enhanced for +methylamine. The 𝛼 = 0.05 and 𝛼 = 1 configurations are depleted for +methylamine and glycine, but enhanced for the methylamine radical. +3.2 General Implications +As can be seen in Tables 3 and 4, the inclusion of reactions with +molecular hydrogen affects the hydrogen economy of the reaction +network. Previously, the reaction network had a significant amount +of H2 being adsorbed or produced on the surface with no chemical de- +struction mechanisms. The H2 molecules are a previously untapped +hydrogen reservoir that is now being utilised (Hasegawa & Herbst +1993). Because one H2 frees up two H atoms on the surface, other +atomic hydrogenation reactions can take place more easily. There- +fore, we observe the increase in the abundances of species in Phases +1 and 2 that are the products of hydrogenation. While for many of +the more common species, the relative increase, i.e., 𝛿 is small, the +abundance increases in absolute terms. There are large relative and +absolute changes in the network of less abundant species, such as +NH2CH2, NH2CH3 and NH2CH2COOH and there are fairly large +absolute changes in the network of highly abundant species, such as +C and its hydrogenation products. +We can also comment on the carbon budget. The previously defined +Δ parameter allows us to consider how carbon is redistributed as a +result of the new reactions being included. For instance, for the dark +cloud during Phase 1, the total Δ for the main carbon-based grain- +surface species that increase +Δtotal(#CH2 + #CH3 + #CH4 + #H2CS + #CH3OH) = 2.9 × 10−6 . +is nearly equal to that of the total decrease Δ of main grain-surface +species: +Δtotal(#C + #CH + #NCH4 + #NH2CH3 = 2.5 × 10−6 . +From this we can see that the dihydrogenation reactions redis- +tribute the carbon between the aforementioned species. The re- +maining carbon is redistributed to other species in the network in +smaller amounts. We also observe that besides the methyl radi- +cal, also species that contain the CH3 group, such as CH3OH and +CH3CN see increases in their abundances, via the reactions CH3 + +OH −−−→ CH3OH and CH3 + CN −−−→ CH3CN. +MNRAS 000, 1–11 (2015) + +Impact of C and CH reacting with H2 +5 +Dark Cloud +Low-Mass Star +High-Mass Star +Species +𝛿 +Original Abundances +Species +𝛿 +Original Abundances +Species +𝛿 +Original Abundances +#CH2 +2.8 +4.1 ×10−7 +#CH2 +2.8 +4.1 ×10−7 +#CH2 +2.8 +4.1 ×10−7 +#CH3 +2.3 +2.6 ×10−7 +#CH3 +2.3 +2.6×10−7 +#CH3 +2.3 +2.6 ×10−7 +#CH4 +1.3 +4.0 ×10−6 +#CH4 +1.3 +3.8 ×10−6 +#CH4 +1.3 +3.8 ×10−6 +#NH3 +1.1 +3.8 ×10−6 +#NH3 +1.1 +3.7 ×10−6 +#NH3 +1.1 +3.7 ×10−6 +#H2CS +1.1 +2.4 ×10−8 +#H2CS +1.1 +2.4 ×10−8 +#H2CS +1.1 +2.4 ×10−8 +#CH3OH +1.04 +1.5 ×10−5 +#CH3OH +1.04 +1.3 ×10−5 +#CH3OH +1.04 +1.3 ×10−5 +#HNC +1.03 +2.3 ×10−8 +#HNC +1.04 +2.3 ×10−8 +#HNC +1.04 +2.3 ×10−8 +#H2SiO +1.03 +3.3 ×10−7 +#H2SiO +1.03 +1.1 ×10−8 +#H2SiO +1.03 +3.4 ×10−7 +#HCN +1.02 +1.7 ×10−7 +#HO2 +1.03 +2.3 ×10−7 +#HO2 +1.03 +2.3 ×10−7 +#O2 +1.02 +1.8 ×10−6 +NO +1.03 +1.0 ×10−10 +#HCN +1.02 +1.6 ×10−7 +#CH +1.1 ×10−15 +7.2 ×10−7 +#CH +2.0 ×10−15 +7.2 ×10−7 +#CH +2.1 ×10−15 +7.2 ×10−7 +#C +2.4 ×10−13 +1.4 ×10−6 +#C +2.5 ×10−13 +1.4 ×10−6 +#C +2.5 ×10−13 +1.4 ×10−6 +#NCH4 +3.7 ×10−13 +1.5 ×10−7 +#NCH4 +3.4 ×10−13 +1.5 ×10−7 +#NCH4 +3.4 ×10−13 +1.5 ×10−7 +#NH2CH3 +8.0 ×10−13 +1.9 ×10−7 +#NH2CH3 +8.3 ×10−13 +2.0 ×10−7 +#NH2CH3 +8.3 ×10−13 +2.0 ×10−7 +NH2CH3 +1.5 ×10−12 +8.7 ×10−10 +#Si +0.98 +5.6 ×10−8 +#Si +0.98 +5.6 ×10−8 +CH +0.96 +9.3 ×10−10 +#SiH +0.99 +2.5 ×10−8 +#SiH +0.99 +2.5 ×10−8 +CH3 +0.98 +1.5 ×10−9 +#SiH2 +0.99 +1.3 ×10−8 +#SiH2 +0.99 +1.3 ×10−8 +#Si +0.98 +5.7 ×10−8 +#O +0.99 +7.8 ×10−5 +#SI +0.99 +6.7 ×10−5 +#SiH +0.99 +2.6 ×10−8 +#H3CO +0.99 +1.7 ×10−6 +#H3CO +0.99 +1.7 ×10−6 +#SiH2 +0.99 +1.4 ×10−8 +#HNO +0.99 +1.2 ×10−5 +#HNO +0.99 +1.2 ×10−5 +Table 3. Summary of the species that experienced the greatest increases (top section) and decreases (bottom section) for each of the three astronomical objects +in Phase 1. Species with a "#" are grain-surface species. All other species are gas-phase. +Low-Mass Star +High-Mass Star +Species +𝛿 +Original Abundances +Species +𝛿 +Original Abundances +HOCO +3.7 +9.3 ×10−10 +HOCO +2.1 +4.3 ×10−8 +H2O2 +2.6 +4.3 ×10−9 +CH3OH +2.0 +1.8 ×10−9 +CH3CHO +2.2 +1.0×10−7 +CH3CHO +2.0 +1.5 ×10−7 +CH3OH +2.1 +3.7 ×10−9 +C2H4 +2.0 +2.5 ×10−9 +CH3CN +1.7 +1.0 ×10−9 +CH2CO +1.9 +1.8 ×10−10 +C4H +1.6 +3.2 ×10−10 +H2CO +1.7 +9.3 ×10−9 +C3H2 +1.5 +5.6 ×10−9 +CH3 +1.7 +1.1×10−10 +CH3CCH +1.5 +2.4×10−8 +NH3 +1.6 +1.3 ×10−8 +NH3 +1.5 +2.7 ×10−7 +CH3CN +1.5 +7.1 ×10−10 +NH2CHO +1.4 +2.7 ×10−7 +C2H2 +1.5 +1.1 ×10−8 +NCH4 +3.8 ×10−5 +9.2 ×10−7 +NCH4 +4.7 ×10−5 +8.3 × 10−7 +NH2CH3 +2.4 ×10−3 +1.6 ×10−7 +NH2CH3 +2.5 ×10−3 +1.7 ×10−7 +NH2CH2COOH +6.0 ×10−2 +6.3 ×10−9 +NH2CH2COOH +6.3 ×10−3 +7.2 ×10−8 +H2S +0.88 +2.0 ×10−9 +NO +0.82 +4.0 ×10−6 +SO2 +0.92 +4.4 ×10−8 +NCCN +0.96 +3.9 × 10−7 +MG+ +0.93 +8.0 ×10−8 +O2 +0.96 +7.1 ×10−6 +O +0.95 +1.3 × 10−5 +HCOO +0.96 +1.9 × 10−10 +CH2OH +0.95 +6.4 ×10−8 +C2N +0.97 +3.5 ×10−8 +O2 +0.96 +4.2 ×10−5 +O +0.97 +3.6 × 10−8 +SO +0.97 +1.9 ×10−6 +CO2 +0.97 +7.6 × 10−6 +Table 4. Summary of the species that experienced the greatest increases (top section) and decreases (bottom section) for each of the three astronomical objects +in Phase 2. All species listed are gas-phase. +In a similar fashion, nitrogen is redistributed throughout the net- +work. The grain-surface ammonia abundance increases by 10%, +i.e., 3.8×10−7. The decrease in #NCH4 and #NH2CH3 accounts +for 3.4×10−7 or ∼ 90%. +3.3 Implications for Simple Grain-Surface Species +In the light of the recent ice observations with the James Webb +Space Telescope, both published (Yang et al. 2022) and upcoming +(McClure et al. 2017), it is important to consider the effect on the +main ice constituents. Figure 3 shows the time-evolution of the abun- +dances of grain-surface H2O, CO, CO2, CH3OH, H2CO, NH3 and +MNRAS 000, 1–11 (2015) + +6 +J.Heyl et al. +CH4 in Phase 1 of a dark cloud. These are species that have been se- +curely or likely identified in the ices (Boogert et al. 2015). The shaded +areas in the plots indicate the 68% confidence interval for the mea- +sured abundances, taken from Boogert et al. (2015). In Boogert et al. +(2015), the abundances were given in terms of the median value as +well as the upper and lower quartiles. It was assumed that the spread +in the measurements was Gaussian, which meant that the interquar- +tile range represented 1.36𝜎. This spread in measurements is due to +both observational error and source-to-source variation. We observe +that we recover the measured abundances for most of the species +within the uncertainty, with the exception of grain-surface CO2. The +inclusion of the dihydrogenation reactions does not change how well +the models agree with the abundance measurements, however, for +all hydrogenation products we observe that the inclusion of reac- +tions with molecular hydrogen increases their abundance, as a result +of the additional atomic hydrogen on the surface. In short, despite +uncertainties surrounding activation energies, networks and binding +energies, we are able to recover observational abundances reasonably +well when we include the reactions with molecular hydrogen and this +gives us confidence that the predictions we make for glycine and its +precursors are accurate. +3.4 Implications for Glycine and its Precursors +In Tables 3 and 4, we observe that the abundances of glycine and +its precursors decreases if molecular hydrogen is part of the reaction +network. We can also explain why the abundance of precursors of +glycine, gas and grain NH2CH3 and NH2CH2 decrease. The former +is formed through the reaction NH2 + CH3, but since more atomic +H is present on the grains, both radical species are preferentially +hydrogenated. The inclusion of H2 as a reacting species, not just in +the context of the two reactions we consider in this work, introduces +greater competition for radicals that are needed for the formation of +complex organic molecules. This results in the lower abundances of +NH2CH3 and NH2CH2. +We can also use this to justify the impact of the efficiency. Figures +1 and 2 plot the time series for the various efficiencies as well as +with enhanced cosmic ray ionisation in Phase 1 and 2, respectively. +We previously remarked that the original configuration produced the +most of glycine and its precursors. For the other configurations, the +greater the value of 𝛼, the greater the depletion of these species. This +makes sense when one considers that an increasing value of 𝛼 results +in more H2 being consumed and therefore more atomic H becoming +available to hydrogenate precursors. +We now look to compare our results with observations. We do +this separately for glycine and its precursors. We also discuss the +implications of not using non-diffusive grain-surface mechanisms in +our code, such as the ones discussed in Garrod & Pauly (2011) and +Jin & Garrod (2020). +3.4.1 Methylamine and the methylamine radical +Methylamine (NH2CH3) and the methylamine radical (NH2CH2) +are important precursors of glycine. The hydrogen abstraction of +methylamine to form the methylamine radical is crucial, as there is +growing evidence to suggest that the reaction NH2CH2 + HOCO +–> NH2CH2COOH is a feasible glycine formation route (Ramesh +& Yuan-Pern 2022). Confirmed detections of methylamine in high- +mass star forming regions are summarised in Table 5. We observe +improved level of agreement between our model outputs and obser- +vations when the reactions are included with 𝛼 = 1. We observed +significant enhancement when the cosmic ray ionisation rate was +increased. This suggests that if dihydrogen is chemically active on +the grains, one would need to consider regions of high cosmic ray +ionisation rate to detect these precursors of glycine, as these reactions +reduce the abundance of methylamine. In the case of the Bøgelund +et al. (2019) observation, we have confidence in the value of our +ratio, as the chemical network for methanol is well-established. +However, the entirety of the above discussion regarding the agree- +ment of our results with observations is incomplete without dis- +cussing the effect of the nondiffusive reaction mechanisms being +absent in our modelling. These mechanisms are of particular use +when considering reactions between reactants which are likely to re- +act very slowly via the Langmuir-Hinshelwood diffusion mechanism, +such as the reaction between CO and OH to form CO2. Methylamine +and the methylamine radical are formed via reactions 6 and 7, which +involve species with high binding energies, thereby making their for- +mation at 10K inefficient via diffusion. As a result, the fact that we do +not include the non-diffusive mechanisms means that methylamine +and its radical are under-produced. +3.4.2 Glycine +While there may be no confirmed detection for glycine in the liter- +ature, various estimates exist. In Gibb et al. (2004), an upper limit +of 0.3% with respect to water was determined, whereas in Jiménez- +Serra et al. (2014), this was estimated to be around 0.1%. In this +work, we find that when the dihydrogenation reactions are not in- +cluded this value is 0.07% and when we include both reactions then +it is 2×10−4%. We should note that in the absence of experimentally- +motivated gas-phase glycine destruction reactions the values derived +in this work are only upper limits, if one neglects non-diffusive mech- +anisms. In the previous sub-section, we discussed that methylamine +and its radical are underproduced. This will result in glycine being +underproduced as well, not just due to the underproduction of its +precursors, but also because reaction 10 is less efficient if assumed +to be diffusion-only. +4 CONCLUSION +In this work, we considered the effect of including the reactions of +H2 with C and CH in our grain-surface network. We ran a grid of +12 models that vary the final density of the collapsing cloud, the +efficiency for the ‘barrier’ of C + H2 −−−→ CH2 as well as the cosmic +ray ionisation rate. +Making molecular hydrogen chemically active unlocks a previ- +ously untapped reservoir of hydrogen, and therefore freeing up the +use of atomic hydrogen for hydrogenation reactions. A particularly +interesting consequence of this is that making H2 more chemically +active decreased the abundances of glycine and its precursors. This +may aid in explaining why methylamine, the methylamine radical as +well as glycine have remained undetected so far. +We note that we do not have a comprehensive gas-phase network +for glycine and its precursors. That is likely to be a limitation. While +it is still likely that glycine and its precursors form on the grains +and then evaporate into the gas-phase, it is possible that there would +be gas-phase destruction routes as well. Additionally, cosmic-ray +ionisation destruction routes on the grains and in the gas-phase are +likely also needed, as these typically break large molecules down +into smaller radicals which are then recycled for further gas-phase +reactions. As such, the abundances we obtain for glycine and its +precursors are likely to only be upper limits. +MNRAS 000, 1–11 (2015) + +Impact of C and CH reacting with H2 +7 +0 +1 +2 +3 +4 +5 +6 +Time (Years) +1e6 +10 +28 +10 +25 +10 +22 +10 +19 +10 +16 +10 +13 +10 +10 +10 +7 +Abundance +#NCH4 +0 +1 +2 +3 +4 +5 +6 +Time (Years) +1e6 +10 +29 +10 +26 +10 +23 +10 +20 +10 +17 +10 +14 +10 +11 +10 +8 +Abundance +NCH4 +0 +1 +2 +3 +4 +5 +6 +Time (Years) +1e6 +10 +28 +10 +25 +10 +22 +10 +19 +10 +16 +10 +13 +10 +10 +10 +7 +Abundance +#NH2CH3 +0 +1 +2 +3 +4 +5 +6 +Time (Years) +1e6 +10 +29 +10 +26 +10 +23 +10 +20 +10 +17 +10 +14 +10 +11 +10 +8 +Abundance +NH2CH3 +Original += 0 += 0.05 += 1 +Original + CR += 0 + CR += 0.05 + CR += 1 + CR +Figure 1. Time series of the abundances of grain-surface and gas-phase NH2CH2 and NH2CH3 in Phase 1 of a dark cloud. Furthermore, we observe that the +inclusion of the dihydrogenation reactions, regardless of efficiency 𝛼 severely depletes the abundances of the glycine precursors in both phases relative to the +original model which did not include either of the dihydrogenation reactions. Also plotted are the limits of detectability we have used for gas and grain-surface +species. We do not plot glycine, as it is not formed at all in Phase 1. We observe that only the original model is capable of producing ’detectable’ levels of +methylamine and the methylamine radical. For the other configurations, an increase in 𝛼 results in increased depletion of the species relative to the original +model. We also observe that enhanced cosmic ray ionisation depletes the abundances on the grains but not in the gas. +MNRAS 000, 1–11 (2015) + +8 +J.Heyl et al. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time (Years) +1e6 +10 +11 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +Abundance +NCH4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time (Years) +1e6 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +Abundance +NH2CH3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time (Years) +1e6 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +Abundance +NH2CH2COOH +Original += 0 += 0.05 += 1 +Original + CR += 0 + CR += 0.05 + CR += 1 + CR +Figure 2. Time series of the abundances of gas-phase NH2CH2, NH2CH3 and NH2CH2COOH in Phase 2 of a high-mass star. We observe that glycine is +produced in the warm-up phase. The enhanced cosmic ray ionisation rate is found to significantly deplete all three species in the gas-phase for the original model. +For NH2CH2 and NH2CH3, when 𝛼 = 0, 𝛼 = 0.05 or 𝛼 = 1, the enhanced cosmic ray ionisation rate results in an increase of their abundances. For glycine, +the enhanced cosmic ray ionisation rate seems to decrease its gas-phase abundance. +MNRAS 000, 1–11 (2015) + +Impact of C and CH reacting with H2 +9 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#H2O +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#CO +Original += 0 += 0.05 += 1 +Original + CR += 0 + CR += 0.05 + CR += 1 + CR +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#CO2 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#CH3OH +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#H2CO +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#NH3 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#CH4 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +Time (Years) +1e6 +10 +8 +10 +6 +10 +4 +Abundance +#H2S +Figure 3. Time series of the abundances of grain-surface H2O, CO, CO2, CH3OH, H2CO, NH3, CH4 and H2S in Phase 1 of a dark cloud. We include the +species that have securely identified or likely identified. The abundances were adapted from Boogert et al. (2015). The shaded areas include the 1𝜎 region of +abundances. In the case of H2CO, no uncertainty was provided in the original source, so there is no shaded area. Grain-surface H2S only has an upper limit on +its abundance. For both normal and enhanced cosmic ray ionisation rates, the time-series differ very little, which is why it is difficult to distinguish them visually. +MNRAS 000, 1–11 (2015) + +10 +J.Heyl et al. +Reference Molecule +Reference +Abundance Measurements (Relative to Reference Molecule) +Original Model Ratio +New Model Ratio +CH3OH +Bøgelund et al. (2019) +8 × 10−3 − 0.1 +37 +0.02 +H2 +Ohishi et al. (2019) +1.5 ± 1.1 × 10−8 +3.5 × 10−7 +3.9 × 10−10 +Table 5. Table of methylamine abundance measurements relative to reference molecules for high-mass stars. Also included are the corresponding ratios obtained +in this work for high-mass stars with the standard cosmic ray ionisation rate. +An additional limitation is the absence of the non-diffusive reaction +mechanisms discussed in Garrod & Pauly (2011) and Jin & Garrod +(2020). The consequence is that glycine and its precursors do not +form efficiently on the grains at 10 K, which is different to what was +found in Ioppolo et al. (2021). As such, they are under-produced in +our models, whereas diffusion-efficient reactions overproduce certain +species. However, without implementing this formalism in the code, +it is difficult to assess the relative impacts of these mechanisms on +the final abundances. +ACKNOWLEDGEMENTS +We thank the anonymous referee for their constructive comments that +improved the quality of the manuscript. J. Heyl is funded by an STFC +studentship in Data-Intensive Science (grant number ST/P006736/1). +T. Lamberts is grateful for support from NWO via a VENI fellow- +ship (722.017.008). This work was also supported by European Re- +search Council (ERC) Advanced Grant MOPPEX 833460. S. Viti +acknowledges support from the European Union’s Horizon 2020 +research and innovation programme under the Marie Skłodowska- +Curie grant agreement No 811312 for the project “Astro-Chemical +Origins” (ACO). +DATA AVAILABILITY +The data underlying this article are available in the article and in its +online supplementary material. +REFERENCES +Al-Halabi A., van Dishoeck E. F., 2007, MNRAS, 382, 1648 +Arasa C., van Hemert M. C., van Dishoeck E. F., Kroes G. J., 2013, The +Journal of Physical Chemistry A, 117, 7064 +Balucani N., Bergeat A., Cartechini L., Volpi G. G., Casavecchia P., Skouteris +D., Rosi M., 2009, The Journal of Physical Chemistry A, 113, 11138 +Bernstein M. P., Dworkin J. P., Sandford S. 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H., 1988, ApJ, 334, 771 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2015) + diff --git a/fNE3T4oBgHgl3EQfHglt/content/tmp_files/load_file.txt b/fNE3T4oBgHgl3EQfHglt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fc7a66a7ccd9357eb846631e6fa1cc57b59e62a --- /dev/null +++ b/fNE3T4oBgHgl3EQfHglt/content/tmp_files/load_file.txt @@ -0,0 +1,1109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf,len=1108 +page_content='MNRAS 000, 1–11 (2015) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 Investigating the impact of reactions of C and CH with molecular hydrogen on a glycine gas-grain network Johannes Heyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1★ Thanja Lamberts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 Serena Viti3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 and Jonathan Holdship3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 1Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Gower Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' WC1E 6BT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' UK 2Leiden Institute of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Gorlaeus Laboratories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Leiden University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' PO Box 9502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2300 RA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The Netherlands 3Leiden Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Leiden University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' PO Box 9513,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2300 RA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The Netherlands Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The impact of including the reactions of C and CH with molecular hydrogen in a gas-grain network is assessed via a sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' To this end, we vary 3 parameters, namely, the efficiency for the reaction C+H2 −−−→ CH2, and the cosmic ray ionisation rate, with the third parameter being the final density of the collapsing dark cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' A grid of 12 models is run to investigate the effect of all parameters on the final molecular abundances of the chemical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We find that including reactions with molecular hydrogen alters the hydrogen economy of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' since some species are hydrogenated by molecular hydrogen, atomic hydrogen is freed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The abundances of simple molecules produced from hydrogenation, such as CH4, CH3OH and NH3, increase, and at the same time, more complex species such as glycine and its precursors see a significant decrease in their final abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We find that the precursors of glycine are being preferentially hydrogenated, and therefore glycine itself is produced less efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Key words: astrochemistry – ISM: abundances – ISM: molecules 1 INTRODUCTION Interstellar dust plays a significant role in the rich chemistry that takes place in the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' It is widely believed that complex-organic molecules (COMs) form on interstellar dust (Herbst & van Dishoeck 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Caselli & Ceccarelli 2012) since for certain molecules, grain-surface reactions are more efficient than gas-phase reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This is particularly important in cold astronomical envi- ronments where some gas-phase reactions may be highly inefficient, because a "third body" is needed to take up the excess heat of an exothermic reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Dust grains thus act as an energy sink allowing the chemistry to thrive and this can lead to the formation of more complex organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Both experimental work and modelling has shown that one such molecules, namely the amino acid glycine can be formed through energetic processing of the ices during the warm-up phase of star formation (Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Woon 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Bossa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Ciesla & Sandford 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Garrod 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2018), although there is evidence to suggest that glycine would undergo destruction under increased irradiation (Pernet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Maté et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In addition, in a joint experimental and modeling effort, Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) suggested that non-energetic mechanisms such as atom-addition reactions might be a promising route for glycine formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' A new grain-surface reaction, inserting C atoms in H2 to form CH2 via C + H2 −−−→ CH2, was recently proposed to be barrierless by Simončič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2020), based on earlier lab work by Krasnokutski ★ E-mail: johannes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='heyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='19@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='uk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' They included this reaction in their network and found a far more rapid conversion of C to CH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Subsequently, Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) performed a combined experimental and computational work to investigate the importance of reactions with molecular hydrogen for the formation of methane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' It was found that while the former reaction might not be fully barrierless, and the barrier likely depends on the binding site, the reaction CH + H2 −−−→ CH3 does in fact pro- ceed without a barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The reason these ‘dihydrogenation’ reactions might be of interest is that they make H2 more chemically active, the importance of which was recognized already by Hasegawa & Herbst (1993) and by Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017) in the context of water formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Typically, H2 has one of the lowest binding energies of grain-surface species, lower than even atomic H (Al-Halabi & van Dishoeck 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Molpeceres & Kästner 2020), which allows the molecule to diffuse readily on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Moreover, the molec- ular hydrogen abundance in molecular clouds and pre-stellar cores is much higher than that of atomic hydrogen (van Dishoeck & Black 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Goldsmith & Li 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' By including these reactions in chemical models, one might first of all expect changes in the CH4 abundance, but it is equally interesting to consider the effect on downstream species such as complex organic molecules, whose typical abundances are far lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Their sensitiv- ity to new reactions should be considered, as their more abundant precursors might see changes in their abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In this work, we look to build on the work by Simončič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2020) and Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) to investigate the impact of the dihydrogenation reactions of C and CH on our gas-grain chemical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In particular, we are interested in observing the effect these reactions have on the production of glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Our © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04324v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='GA] 11 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' glycine network is based on the kinetic Monte Carlo network used in Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021), using in part updated rate constants from recent literature, as indicated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We start by describing the astrochemical model, our choice of parameters and how we evaluate the network sensitivity in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We then discuss the results as well as the astrochemical implications in Section 3 and summarize our conclusions in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2 METHODOLOGY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 The Astrochemical Model In this work, the gas-grain chemical code UCLCHEM was used (Holdship et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2017)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' UCLCHEM makes use of a rate equation approach to modelling the gas and grain-surface and bulk abun- dances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The gas-phase reaction network is taken from the UMIST database (McElroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The grain-surface network used was the default one as available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Various reaction mechanisms are implemented in UCLCHEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The grain-surface reaction mechanisms that exist in UCLCHEM include the Eley-Rideal mechanism as well as the Langmuir- Hinshelwood diffusion mechanism, which were implemented in Qué- nard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2018), as was the competition formula from Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2007) and Garrod & Pauly (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The binding energies that are used to calculate the diffusion reaction rate are taken from Wake- lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We also included an updated version of the glycine grain-surface network from Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021), also including both the reactions C + H2 −−−→ CH2 and CH + H2 −−−→ CH3 as sum- marized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Note that the reaction OH + H2 −−−→ H2O + H had been already included, based on previous work by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=', Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The code also includes thermal and non-thermal des- orption, such as due to H2 formation, cosmic ray ionisation as well as UV-induced desorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Note that the astrochemical model used in Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) makes use of the non-diffusive grain-surface chemistry that is described in Garrod & Pauly (2011) and Jin & Gar- rod (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This is not used in UCLCHEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The implications of this will be discussed later in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' UCLCHEM is used to model two distinct phases of the star forma- tion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Phase 1 is the free-fall collapse phase of a dark cloud for a default value of 5 million years, whereas Phase 2 models the warm-up phase immediately following Phase 1, with the initial den- sity of Phase 2 equal to the final density of Phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Phase 2 runs for 1 million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Further details of the code can be found in Holdship et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Parameter Selection To assess the importance of the two proposed reactions to the net- work under various interstellar conditions, three parameters were varied, as listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The standard cosmic ray ionisation rate in UCLCHEM is 𝜁 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3×10−17 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This is in line with typical values that are of the order 10−17 s−1 in diffuse ISM conditions (O’Donnell & Watson 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Hartquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Indriolo & McCall 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' However, there exist observations of higher cosmic ray ionisation rates (Indriolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Indriolo & McCall 2012), which is why we also include analysis of a region with cosmic ray ionisation rate of 10𝜁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Cosmic ray ionisation is typically expected to break larger molecules into smaller radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We did not consider 1 https://uclchem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='io/ lower values of the cosmic ray ionisation rate, as these are typi- cally not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The cosmic ray dependency on column density in O’Donoghue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) covered a range of values that were, however, already covered by the factor of 10 we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' While they found differences for lower densities during the collapse phase, these were ironed out once the collapse reached larger final densities, which is why here we do not include this dependency on column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Three different astronomical regions were modelled: (i) a dark cloud with a final density of 105 cm−3 (ii) a low-mass protostar with a final density of 106 cm−3 (iii) a high-mass protostar, with a final density of 107 cm−3 The heating profiles during Phase 2 for the last two cases are based on Viti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2004) and differ for each astronomical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The dark cloud simulation was only run for Phase 1, but was allowed to run for a further million years to allow the chemistry to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Another parameter that was varied was the efficiency, 𝛼, of the extent to which the reaction C + H2 −−−→ CH2 is barrierless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' While Simončič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2020) considered the reaction to be fully barrierless, Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) found that the reaction barrier likely depends on the binding site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' As such, our grid of models considers efficiencies for the reaction of 0 (the reaction is not included), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 (5% of binding sites lead to a barrierless reaction and 95% of the binding sites have an infinitely high barrier) and 1 (the reaction is fully barrierless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' What this means practically is that the reaction rate is multiplied by the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The reaction CH + H2 −−−→ CH3 was included as only being barrierless, based on Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 Evaluating the network sensitivity We quantify the effect of the new reactions on the model by con- sidering the change in abundances of the species that are the most affected when taking the ratio of the abundances of the modified and original models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The modified model is the chemical network which has 𝛼 = 1, whereas the original model was taken to be the network which had neither of the dihydrogenation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' These two scenarios were taken to be the extremes of the parameter range in terms of including these reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The ratio is most sensitive to strong deviations in the molecular abundances as a result of the dihydrogenation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This ratio is defined for each species 𝑖 as: 𝛿𝑖(𝑡) = 𝑥𝑀 𝑖 (𝑡) 𝑥𝑂 𝑖 (𝑡) , (1) where 𝑥𝑀 𝑖 (𝑡) is the abundance of species 𝑖 in the modified model at time 𝑡 and 𝑥𝑂 𝑖 (𝑡) is the abundance of the same species in the original model at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We only considered species which had a value above a “threshold of detectability".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This was to ensure that we did not look at species whose original and changed abundances were below what can be ob- served from an astronomical point-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For grain-surface species this threshold was set to 10−8 with respect to hydrogen whereas for gas-phase species this threshold equalled 10−12 with respect to hy- drogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We took 10−8 as a lower-limit threshold for grain-surface species, as this was the order of magnitude of the lowest reported abundances in Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Similarly, the gas-phase thresh- old was taken based on the abundances of COMs typically observed in the gas-phase, such as in Jiménez-Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2016, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) Impact of C and CH reacting with H2 3 Reaction No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Reaction Reference 1 CO + OH −−−→ HOCO Arasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 2 HOCO + H −−−→ H2 + CO2 Goumans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2008) 3 HOCO + H −−−→ HCOOH Goumans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2011) 4 CH4 + OH −−−→ CH3 + H2O Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017) 5 NH2 + CH3 −−−→ NH2CH3 Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 6 NH3 + CH −−−→ NH2CH2 Balucani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2009) 7 NH2CH2 + H −−−→ NH2CH3 Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 8 NH2CH3 + H −−−→ NH2CH2 + H2 Oba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2014) 9 NH2CH3 + OH −−−→ NH2CH2 + H2O Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 10 NH2CH2 + HOCO −−−→ NH2CH2COOH Woon (2002) 11 H2 + OH −−−→ H2O + H Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2017) 12 O2 + H −−−→ HO2 Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 13 HO2 + H −−−→ OH + OH Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 14 HO2 + H −−−→ H2 + O2 Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 15 HO2 + H −−−→ H2O + O Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 16 OH + OH −−−→ H2O2 Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 17 OH + OH −−−→ H2O + O Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2013) 18 H2O2 + H −−−→ H2O + OH Lamberts & Kästner (2017) 19 N + O −−−→ NO Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 20 NO + H −−−→ HNO Fedoseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2012) 21 HNO + H −−−→ H2NO Fedoseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2012) 22 HNO + H −−−→ NO + H2 Fedoseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2012) 23 HNO + O −−−→ NO + OH Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 24 HN + O −−−→ HNO Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 25 N + NH −−−→ N2 Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 26 NH + NH −−−→ N2 + H2 Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 27 C + O −−−→ CO Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) 28 CH3 + OH −−−→ CH3OH Qasim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2018) 29 C + H2 −−−→ CH2 Simončič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) 30 CH + H2 −−−→ CH3 Lamberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2022) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Table of the reactions added to the standard UCLCHEM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Parameter Values Comment Final Density of Phase 1 and Initial Density of Phase 2 105 cm−3, 106 cm−3, 107 cm−3 Final density of Phase 1 same as initial density of Phase 2 Efficiency for barrierless C + H2 −−−→ CH2 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05, 1 Efficiency of 0 is equivalent to reaction being excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Cosmic Ray Ionisation Rate 𝜁 , 10𝜁 𝜁 is the standard cosmic ray ionisation rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 × 10−17 s−1 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The parameters that were varied in this work to assess the effect of the two reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We can also define a quantity that tracks the absolute change in the abundance of species: Δ𝑖(𝑡) = 𝑥𝑀 𝑖 (𝑡) − 𝑥𝑂 𝑖 (𝑡) = 𝑥𝑂 𝑖 (𝑡)[𝛿𝑖(𝑡) − 1], (2) This value indicates how species with relatively large abundances, such as elemental species or their hydrogenation products, are re- distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3 RESULTS AND ASTROCHEMICAL IMPLICATIONS We find that even though the amounts by which various species are affected differs for each stage of star formation, the general trends are broadly similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' As such, we group our analysis per phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Tables 3 and 4 summarise the changes in terms of 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The effect of the enhanced cosmic ray ionisation rate is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Our results differ from Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021) in that, while glycine does form on the grains, it does not do so in Phase 1, as UCLCHEM does not utilise non-diffusive grain-surface mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Instead, glycine forms on the grains as the temperature increases in Phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 Impact of the Parameters In this sub-section we consider the role that the physical and chemical parameters play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Tables 3 and 4 show the changes in abundance when we compare the original network without the dihydrogenation reactions with the 𝛼 = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Figures 1 and 2 show the time series of the abundances for glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 Final Density The final density of the collapsing cloud had a minor effect on the final abundances of the species in Phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For all three astronomical objects modelled in Phase 1, we observe a significant decrease of grain-surface CH and C when the reactions are included and see an enhancement of grain-surface CH2, CH3 and CH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' However, the values of 𝛿 as well as their original abundances seem to be independent of the density, suggesting a saturation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Phase 2, we observe that the final density of the collapsing cloud does affect the extent to which the added reactions influence the final abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We notice that several hydrogenation-based species have greater abundances at lower densities, including species such as HOCO, H2O2, CH3CCH and H2CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Efficiency For more abundant species, such as H2O and CH3OH, we find that the results obtained from using a branching fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 for the barrierless dihydrogenation of C are essentially the same as using a efficiency of 1 (the reaction is fully barrierless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We do find that the efficiency parameter plays a role in the final abundances of glycine and its precursors during the warm-up phase of low and high-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This can be seen in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For Phase 1, the species are not detectable except for the original configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' However, we still observe that for the other three con- figurations an increasing value of 𝛼 corresponds to an increased level of depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Phase 2, the configurations are all detectable and this same hierarchy remains in the gas-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 Cosmic Ray Ionisation Rate The degree of cosmic ray ionisation is found to play an important role in enhancing or counteracting the role of the dihydrogenation reac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The cosmic ray destruction routes we include in our standard network are from Garrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' These consist of hydrogen abstraction reactions and reactions that produce radical-radical pairs of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' An enhanced cosmic ray ionisation rate leads to the destruction of many of hydrogenated species, such as CH4, NH3, H2O and CH3OH, as well as their precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This leads to further hydrogen reservoirs being released and radicals being formed which can go on to form glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Because no cosmic ray destruction mechanisms for these complex, larger, species are included, we find that these are more abundantly produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This is important to consider in the context of glycine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Figures 1 and 2, we plot the time dependence of the abundance of glycine precursors for eight different parameter sets, including the enhanced cosmic ray ionisation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Phase 1, we find that on the grains, the enhanced cosmic ray ionisation rate depletes the species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Phase 2, the effect varies by configuration and species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The original config- uration consistently leads to a decrease of all plotted species in the presence of enhanced cosmic ray ionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The 𝛼 = 0 configuration is depleted for the methylamine radical and glycine, but enhanced for methylamine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 and 𝛼 = 1 configurations are depleted for methylamine and glycine, but enhanced for the methylamine radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 General Implications As can be seen in Tables 3 and 4, the inclusion of reactions with molecular hydrogen affects the hydrogen economy of the reaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Previously, the reaction network had a significant amount of H2 being adsorbed or produced on the surface with no chemical de- struction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The H2 molecules are a previously untapped hydrogen reservoir that is now being utilised (Hasegawa & Herbst 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Because one H2 frees up two H atoms on the surface, other atomic hydrogenation reactions can take place more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' There- fore, we observe the increase in the abundances of species in Phases 1 and 2 that are the products of hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' While for many of the more common species, the relative increase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=', 𝛿 is small, the abundance increases in absolute terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' There are large relative and absolute changes in the network of less abundant species, such as NH2CH2, NH2CH3 and NH2CH2COOH and there are fairly large absolute changes in the network of highly abundant species, such as C and its hydrogenation products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We can also comment on the carbon budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The previously defined Δ parameter allows us to consider how carbon is redistributed as a result of the new reactions being included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For instance, for the dark cloud during Phase 1, the total Δ for the main carbon-based grain- surface species that increase Δtotal(#CH2 + #CH3 + #CH4 + #H2CS + #CH3OH) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 × 10−6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' is nearly equal to that of the total decrease Δ of main grain-surface species: Δtotal(#C + #CH + #NCH4 + #NH2CH3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 × 10−6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' From this we can see that the dihydrogenation reactions redis- tribute the carbon between the aforementioned species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The re- maining carbon is redistributed to other species in the network in smaller amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We also observe that besides the methyl radi- cal, also species that contain the CH3 group, such as CH3OH and CH3CN see increases in their abundances, via the reactions CH3 + OH −−−→ CH3OH and CH3 + CN −−−→ CH3CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) Impact of C and CH reacting with H2 5 Dark Cloud Low-Mass Star High-Mass Star Species 𝛿 Original Abundances Species 𝛿 Original Abundances Species 𝛿 Original Abundances #CH2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−7 #CH2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−7 #CH2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−7 #CH3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−7 #CH3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6×10−7 #CH3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−7 #CH4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−6 #CH4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−6 #CH4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−6 #NH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−6 #NH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−6 #NH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−6 #H2CS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 #H2CS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 #H2CS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 #CH3OH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−5 #CH3OH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−5 #CH3OH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−5 #HNC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 #HNC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 #HNC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 #H2SiO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−7 #H2SiO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−8 #H2SiO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−7 #HCN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−7 #HO2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−7 #HO2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−7 #O2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−6 NO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−10 #HCN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−7 #CH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−7 #CH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−7 #CH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−7 #C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−6 #C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−6 #C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−6 #NCH4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−7 #NCH4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−7 #NCH4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−7 #NH2CH3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 ×10−7 #NH2CH3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−7 #NH2CH3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−7 NH2CH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−10 #Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−8 #Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−8 CH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='96 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−10 #SiH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−8 #SiH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−8 CH3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−9 #SiH2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 #SiH2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 #Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−8 #O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−5 #SI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−5 #SiH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−8 #H3CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−6 #H3CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−6 #SiH2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 #HNO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−5 #HNO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−5 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Summary of the species that experienced the greatest increases (top section) and decreases (bottom section) for each of the three astronomical objects in Phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Species with a "#" are grain-surface species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' All other species are gas-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Low-Mass Star High-Mass Star Species 𝛿 Original Abundances Species 𝛿 Original Abundances HOCO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−10 HOCO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 H2O2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−9 CH3OH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−9 CH3CHO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0×10−7 CH3CHO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−7 CH3OH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−9 C2H4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−9 CH3CN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−9 CH2CO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−10 C4H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−10 H2CO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−9 C3H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−9 CH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1×10−10 CH3CCH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4×10−8 NH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−8 NH3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−7 CH3CN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−10 NH2CHO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−7 C2H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−8 NCH4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ×10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−7 NCH4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 × 10−7 NH2CH3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 ×10−7 NH2CH3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='7 ×10−7 NH2CH2COOH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−9 NH2CH2COOH 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 ×10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−8 H2S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−9 NO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−6 SO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 NCCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 × 10−7 MG+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 ×10−8 O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='96 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 ×10−6 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 × 10−5 HCOO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 × 10−10 CH2OH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='95 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 ×10−8 C2N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ×10−8 O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 ×10−5 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 × 10−8 SO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 ×10−6 CO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 × 10−6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Summary of the species that experienced the greatest increases (top section) and decreases (bottom section) for each of the three astronomical objects in Phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' All species listed are gas-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In a similar fashion, nitrogen is redistributed throughout the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The grain-surface ammonia abundance increases by 10%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8×10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The decrease in #NCH4 and #NH2CH3 accounts for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4×10−7 or ∼ 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3 Implications for Simple Grain-Surface Species In the light of the recent ice observations with the James Webb Space Telescope, both published (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2022) and upcoming (McClure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2017), it is important to consider the effect on the main ice constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Figure 3 shows the time-evolution of the abun- dances of grain-surface H2O, CO, CO2, CH3OH, H2CO, NH3 and MNRAS 000, 1–11 (2015) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' CH4 in Phase 1 of a dark cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' These are species that have been se- curely or likely identified in the ices (Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The shaded areas in the plots indicate the 68% confidence interval for the mea- sured abundances, taken from Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2015), the abundances were given in terms of the median value as well as the upper and lower quartiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' It was assumed that the spread in the measurements was Gaussian, which meant that the interquar- tile range represented 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='36𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This spread in measurements is due to both observational error and source-to-source variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We observe that we recover the measured abundances for most of the species within the uncertainty, with the exception of grain-surface CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The inclusion of the dihydrogenation reactions does not change how well the models agree with the abundance measurements, however, for all hydrogenation products we observe that the inclusion of reac- tions with molecular hydrogen increases their abundance, as a result of the additional atomic hydrogen on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In short, despite uncertainties surrounding activation energies, networks and binding energies, we are able to recover observational abundances reasonably well when we include the reactions with molecular hydrogen and this gives us confidence that the predictions we make for glycine and its precursors are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 Implications for Glycine and its Precursors In Tables 3 and 4, we observe that the abundances of glycine and its precursors decreases if molecular hydrogen is part of the reaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We can also explain why the abundance of precursors of glycine, gas and grain NH2CH3 and NH2CH2 decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The former is formed through the reaction NH2 + CH3, but since more atomic H is present on the grains, both radical species are preferentially hydrogenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The inclusion of H2 as a reacting species, not just in the context of the two reactions we consider in this work, introduces greater competition for radicals that are needed for the formation of complex organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This results in the lower abundances of NH2CH3 and NH2CH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We can also use this to justify the impact of the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Figures 1 and 2 plot the time series for the various efficiencies as well as with enhanced cosmic ray ionisation in Phase 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We previously remarked that the original configuration produced the most of glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For the other configurations, the greater the value of 𝛼, the greater the depletion of these species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This makes sense when one considers that an increasing value of 𝛼 results in more H2 being consumed and therefore more atomic H becoming available to hydrogenate precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We now look to compare our results with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We do this separately for glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We also discuss the implications of not using non-diffusive grain-surface mechanisms in our code, such as the ones discussed in Garrod & Pauly (2011) and Jin & Garrod (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 Methylamine and the methylamine radical Methylamine (NH2CH3) and the methylamine radical (NH2CH2) are important precursors of glycine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The hydrogen abstraction of methylamine to form the methylamine radical is crucial, as there is growing evidence to suggest that the reaction NH2CH2 + HOCO –> NH2CH2COOH is a feasible glycine formation route (Ramesh & Yuan-Pern 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Confirmed detections of methylamine in high- mass star forming regions are summarised in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We observe improved level of agreement between our model outputs and obser- vations when the reactions are included with 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We observed significant enhancement when the cosmic ray ionisation rate was increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This suggests that if dihydrogen is chemically active on the grains, one would need to consider regions of high cosmic ray ionisation rate to detect these precursors of glycine, as these reactions reduce the abundance of methylamine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In the case of the Bøgelund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2019) observation, we have confidence in the value of our ratio, as the chemical network for methanol is well-established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' However, the entirety of the above discussion regarding the agree- ment of our results with observations is incomplete without dis- cussing the effect of the nondiffusive reaction mechanisms being absent in our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' These mechanisms are of particular use when considering reactions between reactants which are likely to re- act very slowly via the Langmuir-Hinshelwood diffusion mechanism, such as the reaction between CO and OH to form CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Methylamine and the methylamine radical are formed via reactions 6 and 7, which involve species with high binding energies, thereby making their for- mation at 10K inefficient via diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' As a result, the fact that we do not include the non-diffusive mechanisms means that methylamine and its radical are under-produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Glycine While there may be no confirmed detection for glycine in the liter- ature, various estimates exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In Gibb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2004), an upper limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='3% with respect to water was determined, whereas in Jiménez- Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2014), this was estimated to be around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In this work, we find that when the dihydrogenation reactions are not in- cluded this value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='07% and when we include both reactions then it is 2×10−4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We should note that in the absence of experimentally- motivated gas-phase glycine destruction reactions the values derived in this work are only upper limits, if one neglects non-diffusive mech- anisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In the previous sub-section, we discussed that methylamine and its radical are underproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This will result in glycine being underproduced as well, not just due to the underproduction of its precursors, but also because reaction 10 is less efficient if assumed to be diffusion-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 4 CONCLUSION In this work, we considered the effect of including the reactions of H2 with C and CH in our grain-surface network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We ran a grid of 12 models that vary the final density of the collapsing cloud, the efficiency for the ‘barrier’ of C + H2 −−−→ CH2 as well as the cosmic ray ionisation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Making molecular hydrogen chemically active unlocks a previ- ously untapped reservoir of hydrogen, and therefore freeing up the use of atomic hydrogen for hydrogenation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' A particularly interesting consequence of this is that making H2 more chemically active decreased the abundances of glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This may aid in explaining why methylamine, the methylamine radical as well as glycine have remained undetected so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We note that we do not have a comprehensive gas-phase network for glycine and its precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' That is likely to be a limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' While it is still likely that glycine and its precursors form on the grains and then evaporate into the gas-phase, it is possible that there would be gas-phase destruction routes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Additionally, cosmic-ray ionisation destruction routes on the grains and in the gas-phase are likely also needed, as these typically break large molecules down into smaller radicals which are then recycled for further gas-phase reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' As such, the abundances we obtain for glycine and its precursors are likely to only be upper limits.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Abundance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='NH2CH3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 = 1 Original + CR = 0 + CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 + CR = 1 + CR Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Time series of the abundances of grain-surface and gas-phase NH2CH2 and NH2CH3 in Phase 1 of a dark cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Furthermore, we observe that the inclusion of the dihydrogenation reactions, regardless of efficiency 𝛼 severely depletes the abundances of the glycine precursors in both phases relative to the original model which did not include either of the dihydrogenation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Also plotted are the limits of detectability we have used for gas and grain-surface species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We do not plot glycine, as it is not formed at all in Phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We observe that only the original model is capable of producing ’detectable’ levels of methylamine and the methylamine radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For the other configurations, an increase in 𝛼 results in increased depletion of the species relative to the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We also observe that enhanced cosmic ray ionisation depletes the abundances on the grains but not in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 Time (Years) 1e6 10 11 10 10 10 9 10 8 10 7 10 6 Abundance NCH4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 Time (Years) 1e6 10 10 10 9 10 8 10 7 10 6 Abundance NH2CH3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 Time (Years) 1e6 10 10 10 9 10 8 10 7 10 6 Abundance NH2CH2COOH Original = 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 = 1 Original + CR = 0 + CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 + CR = 1 + CR Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Time series of the abundances of gas-phase NH2CH2, NH2CH3 and NH2CH2COOH in Phase 2 of a high-mass star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We observe that glycine is produced in the warm-up phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The enhanced cosmic ray ionisation rate is found to significantly deplete all three species in the gas-phase for the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For NH2CH2 and NH2CH3, when 𝛼 = 0, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 or 𝛼 = 1, the enhanced cosmic ray ionisation rate results in an increase of their abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For glycine, the enhanced cosmic ray ionisation rate seems to decrease its gas-phase abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) Impact of C and CH reacting with H2 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #H2O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #CO Original = 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 = 1 Original + CR = 0 + CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='05 + CR = 1 + CR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #CO2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #CH3OH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #H2CO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #NH3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #CH4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='2 Time (Years) 1e6 10 8 10 6 10 4 Abundance #H2S Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Time series of the abundances of grain-surface H2O, CO, CO2, CH3OH, H2CO, NH3, CH4 and H2S in Phase 1 of a dark cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' We include the species that have securely identified or likely identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The abundances were adapted from Boogert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The shaded areas include the 1𝜎 region of abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' In the case of H2CO, no uncertainty was provided in the original source, so there is no shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Grain-surface H2S only has an upper limit on its abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' For both normal and enhanced cosmic ray ionisation rates, the time-series differ very little, which is why it is difficult to distinguish them visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015) 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='Heyl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Reference Molecule Reference Abundance Measurements (Relative to Reference Molecule) Original Model Ratio New Model Ratio CH3OH Bøgelund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2019) 8 × 10−3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='02 H2 Ohishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2019) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='1 × 10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='5 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='9 × 10−10 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Table of methylamine abundance measurements relative to reference molecules for high-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Also included are the corresponding ratios obtained in this work for high-mass stars with the standard cosmic ray ionisation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' An additional limitation is the absence of the non-diffusive reaction mechanisms discussed in Garrod & Pauly (2011) and Jin & Garrod (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' The consequence is that glycine and its precursors do not form efficiently on the grains at 10 K, which is different to what was found in Ioppolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' As such, they are under-produced in our models, whereas diffusion-efficient reactions overproduce certain species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' However, without implementing this formalism in the code, it is difficult to assess the relative impacts of these mechanisms on the final abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for their constructive comments that improved the quality of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Heyl is funded by an STFC studentship in Data-Intensive Science (grant number ST/P006736/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Lamberts is grateful for support from NWO via a VENI fellow- ship (722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content='008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' This work was also supported by European Re- search Council (ERC) Advanced Grant MOPPEX 833460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQfHglt/content/2301.04324v1.pdf'} +page_content=' Viti acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska- Curie grant agreement No 811312 for the project “Astro-Chemical Origins” (ACO).' metadata={'source': 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a/gdE1T4oBgHgl3EQffAR-/content/tmp_files/2301.03213v1.pdf.txt b/gdE1T4oBgHgl3EQffAR-/content/tmp_files/2301.03213v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84280681dd36344c5664704aefdb34840d9e4f4f --- /dev/null +++ b/gdE1T4oBgHgl3EQffAR-/content/tmp_files/2301.03213v1.pdf.txt @@ -0,0 +1,1578 @@ +EgoTracks: A Long-term Egocentric Visual Object Tracking Dataset +Hao Tang +Kevin Liang +Kristen Grauman +Matt Feiszli +Weiyao Wang +Meta Platforms Inc. +haotang, kevinjliang, grauman, mdf, weiyaowang@meta.com +Figure 1. A sample video from the proposed EgoTracks dataset, with yellow segments of the clip marking when the object (blowtorch) +is visible. Note the frequent disappearances and reappearances of the object over an 8 minute video, with lengthy absences, necessitating +re-detection to track accurately without false positives. The egocentric nature of the video includes the camera-wearer interacting with the +object (Occurrence 2), resulting in significant hand occlusions and dramatic changes in pose. +Abstract +Visual object tracking is a key component to many ego- +centric vision problems. +However, the full spectrum of +challenges of egocentric tracking faced by an embodied AI +is underrepresented in many existing datasets; these tend +to focus on relatively short, third-person videos. Egocen- +tric video has several distinguishing characteristics from +those commonly found in past datasets: frequent large cam- +era motions and hand interactions with objects commonly +lead to occlusions or objects exiting the frame, and ob- +ject appearance can change rapidly due to widely differ- +ent points of view, scale, or object states. Embodied track- +ing is also naturally long-term, and being able to consis- +tently (re-)associate objects to their appearances and dis- +appearances over as long as a lifetime is critical. +Pre- +vious datasets under-emphasize this re-detection problem, +and their “framed” nature has led to adoption of various +spatiotemporal priors that we find do not necessarily gen- +eralize to egocentric video. We thus introduce EgoTracks, a +new dataset for long-term egocentric visual object track- +ing. +Sourced from the Ego4D dataset, this new dataset +presents a significant challenge to recent state-of-the-art +single-object tracking models, which we find score poorly +on traditional tracking metrics for our new dataset, com- +pared to popular benchmarks. We further show improve- +ments that can be made to a STARK tracker to significantly +increase its performance on egocentric data, resulting in +a baseline model we call EgoSTARK. We publicly release +our annotations and benchmark, hoping our dataset leads +to further advancements in tracking. +1. Introduction +First-person or “egocentric” computer vision aims to +capture the real-world perceptual problems faced by an em- +arXiv:2301.03213v1 [cs.CV] 9 Jan 2023 + +Occurrence 2 +Occurrence 17 +Negative Framesbodied AI; it has drawn strong recent interest as an under- +served but highly relevant domain of vision, with important +applications ranging from robotics [15, 57] to augmented +and mixed reality [2,23,59]. Visual object tracking (VOT), +long a fundamental problem in vision, is a core component +to many egocentric tasks, including tracking the progress +of an action or activity, (re-)association of objects in one’s +surroundings, and predicting future states of the environ- +ment. Yet, while the VOT field has made many significant +advancements over the past decade, tracking in egocentric +video remains underexplored. This lack of attention is in +large part due to the absence of a large-scale egocentric +tracking dataset for training and evaluation. While the com- +munity has proposed a number of popular tracking datasets +in recent years, including OTB [69], TrackingNet [51], Got- +10k [26], and LaSOT [16], we find that the strong perfor- +mance that state-of-the-art trackers achieve on these bench- +marks does not translate well to egocentric video, thus es- +tablishing a strong need for such a tracking dataset. +We attribute this performance gap to the many unique +aspects of egocentric views compared to the more tradi- +tional third-person views of previous datasets. In contrast +to intentionally “framed” video, egocentric videos are of- +ten uncurated, meaning they tend to capture many attention +shifts between activities, objects, or locations. Due to the +first-person perspective, large head motions from the cam- +era wearer often result in objects repeatedly leaving and re- +entering the field of view; similarly, hand manipulations of +objects [58] leads to frequent occlusions, rapid variations in +scale and pose, and potential changes in state or appearance. +Furthermore, egocentric video tends to be long (sometimes +representing the entire life of an agent or individual), mean- +ing the volume of the aforementioned occlusions and trans- +formations scales similarly. These characteristics all make +tracking objects in egocentric views dramatically more dif- +ficult than scenarios commonly considered in prior datasets, +and their absence represents an evaluation blindspot. +The aforementioned head motions, locomotion, hand oc- +clusions, and temporal length lead to several challenges. +First, frequent object disappearances and reappearances +causes the problem of re-detection within egocentric track- +ing to become especially critical. +Many previous track- +ing datasets primarily focus on short-term tracking in third- +person videos, providing limited ability to evaluate many of +the challenges of long-term egocentric tracking due to low +quantity and length of target object disappearances. As a +result, competent re-detection is not required for strong per- +formance, leading many recent short-term trackers to ne- +glect it, instead predicting a bounding box for every frame, +which may lead to rampant false positives or tracking the +wrong object. +Additionally, the characteristics of short- +term third-person video have also induced designs relying +on gradual changes in motion and appearance. As we later +Table 1. Statistics of long-term object tracking datasets. An +occurrence is defined as a contiguous set of frames where the ob- +ject is visible, before disappearing. The presented EgoTracks has +the highest number of tracks and target disappearances and reap- +pearances, making it the largest dataset for training and evaluating +long-term trackers. We summarize the training and validation data +from each dataset here, as we do not have the ground truth for the +other datasets’ hold-out test sets. +OxUvA [62] +LaSOT [16] +VOT-LT [32] +Ours +# tracks +200 +1400 +50 +17598 +# occurrences / track +1.5 +4.5 +10.5 +17.7 +Seconds / occurrence +37.8 +21.52 +8 +5.2 +Seconds btw. occurrences +3.5 +0.8 +9.3 +16.1 +show (Section 5.2), many of the motion, context, and scale +priors made by previous short-term tracking algorithms fail +to transfer to egocentric video. Thus, a large-scale long- +term tracking dataset is needed to train and understand the +long-term tracking capability of modern trackers. +To address this gap, we present EgoTracks: a large- +scale long-term egocentric visual object tracking dataset for +training and evaluating long-term trackers. Seeking a real- +istic challenge, we source videos from Ego4D [23], a large- +scale dataset consisting of unscripted, in-the-wild egocen- +tric videos of daily-life activities. The result is a large-scale +dataset to evaluate the tracking and re-detection ability of +SOT models, with more than 20,000 tracks from around +6000 6-minute videos. This constitutes the first large-scale +dataset for visual object tracking in egocentric videos in di- +verse settings, providing a new, significant challenge com- +pared with previous datasets. +We perform a thorough analysis of our new dataset, +demonstrating its difficulty and the need for further research +to develop trackers capable of handling long-term egocen- +tric vision. These experiments reveal valuable insights to- +wards promising future directions in egocentric tracking +and re-detection. Leveraging these intuitions, we propose +multiple simple yet effective changes, such as adjustment of +spatiotemporal priors, egocentric data finetuning and com- +bining multiple templates. We study these proposed strate- +gies on the the state-of-the-art STARK tracker [71] and +train a strong tracker dedicated towards long-term egocen- +tric tracking, EgoSTARK. We hope EgoSTARK can serve +as a strong baseline and facilitate future research. +To summarize, we make the following contributions: +1. We present EgoTracks, the first large-scale long-term +object tracking dataset with diverse egocentric scenar- +ios. We analyze its uniqueness in terms of evaluating +the re-detection performance of trackers. +2. We conduct comprehensive experiments to understand +the performance of many state-of-the-art trackers on +the EgoTracks validation set and observe that due to +the biases and evaluation blindspots of existing third- +person datasets, they tend to struggle. + +Figure 2. EgoTracks is a large-scale egocentric dataset of diverse scenarios (left) and objects (right). +3. We propose a combo of several training and inference- +time improvements to the STARK tracker [71] to adapt +it to long-form egocentric video. +We call this re- +fined model Ego-STARK, which achieves significant +improvements (15% tracking F-score) on EgoTracks. +2. Related work +2.1. Visual Object Tracking Datasets +Visual object tracking studies the problem of joint +spatial-temporal localization of objects in videos. +There +are two main categories: multiple object tracking (MOT) +and single object tracking (SOT). In MOT, a model is pro- +vided a video and a predefined taxonomy, and the model +is required to simultaneously detect, recognize and track +multiple objects. For example, MOT [49] tracks human, +KITTI [20, 45] tracks pedestrians and cars, and TAO [12] +tracks a large taxonomy of 833 categories. Different from +MOT, SOT tracks a single object via a provided initial tem- +plate of the object, and no detection or recognition is in- +volved. Thus, SOT is often taxonomy-free and operates on +generic objects. The community has constructed multiple +popular benchmarks to study this important problem, in- +cluding OTB [69], UAV [50], NfS [30], TC-128 [40], NUS- +PRO [35], GOT-10k [26], VOT [32], and TrackingNet [51]. +These SOT datasets mainly consist of short videos (e.g. a +few seconds). Recently, there have been increasing interests +in long-term tracking. Different from the above datasets, +tracking objects in longer videos (several minutes or more) +poses unique challenges, as the object may involve more +significant transforms, displacements, disappearances, and +reappearances. On top of localizing the object when it is +visible, the model needs to know not predict a box when the +object is absent, and then re-localize the same object when +it reappears. OxUvA [62] is one of the first to benchmark +longer videos (average 2 minutes), with 366 evaluation- +only videos. +LaSOT [16] scales this philosophy with a +benchmark of 1400 videos of more frequent object reap- +pearances. Concurrently, VOT-LT [31] is specifically de- +signed to benchmark videos with frequent object disappear- +ances and reappearances in 50 purposefully selected videos. +Our EgoTracks focuses on long-term SOT and presents +multiple critical and unique attributes: +1) significantly +larger scale, with 17k videos of an average 6 minutes (Ta- +ble 1); 2) more frequent disappearances & reappearances +(avg. 17.7 times) happening in natural, unscripted, real- +world scenarios; 3) data sourced from egocentric videos +shot in-the-wild, involving unique challenging situations, +such as large camera motions, diverse perspective changes, +hand-object interactions, and frequent occlusions. +2.2. Single Object Tracking Methodologies +Many modern approaches use convolutional neural net- +works (CNNs), either with a Siamese network architec- +ture [36, 37, 65], or a correlation-filter based architec- +ture [3,4,7,10,48]. With the Transformer’s recent successes +in computer vision tasks like classification [14] and detec- +tion [5], a line of works leveraging the Transformer archi- +tecture [63] to perform tracking have also emerged. For ex- +ample, inspired by Transformers, TransT [6] uses attention- +based feature fusion to combine features of the object tem- +plate and search image. More recently, several works uti- +lize Transformers as direct predictors to achieve a new state +of the art, such as STARK [71], ToMP [47] and SBT [70]. +These models tokenize frame features from a ResNet [25] +encoder, and use a Transformer to predict the bounding box +and object presence score with the feature tokens. These +methods are often developed on short-term SOT datasets +and assume that the target object stays in the field of view +with minimum occlusions. On the other hand, long-term +trackers [8,27,64] are designed to cope with the problem of +re-detecting objects in their reappearances. Intended to be +aware of potential object disappearances, these approaches +search the whole image for its reappearance. +2.3. Tracking in Egocentric Videos +Multiple egocentric video datasets have been introduced +in the past decades [9, 18, 23, 33, 53, 60]. They offer a host +of interesting challenges, many of which require associating +objects across frames: activity recognition [21, 29, 38, 68, +73], anticipation [17, 19, 22], video summarization [13, 33, +34, 44], human-object interaction [11, 42], episodic mem- + +Doing yardwork/shoveling snow +Indoor Navigation +On a screen +Playing board games +Talking to Colleaguescycling/jogging +Baseball +Farmer +Working in outdoor store +Playing with pets +Watching tv +nGardening +Listening to music +Bike +Working at desk +Walking onstreet +011 +Working in milktea shop +Eating +Playwith cellphone +biology experiments +Readingbooks +Kupune +Going to the park +Talking with family member +Practicing a musical +instrument +Waikingthedog/pet +carr +mechanicCarpenter +mechani +Daily +Camp setup/pack-up/chores +Crafting/knitting/sewing/drawing/painting +Scooter mechanic +Commuting, +road +tri +Grocery +shopping +Going to the gy +ndoors +Making +Potting plantsMaking coffee +Clothes +other +shopping +Fixing something in the home +Taiking onthe phone +Talkingwithfriends/ +housematbag +lawn.mower +televlslon +ball +paper +Naterfanbotte +wi cockbasket +book +phone +.measuring tape +chopping=boar +scIssors +rille +-towel +knife +ricecooker +cup +tissue +flower pot +mobile-phone +pot + trolley +spanner +coolbox..helmet +throw pillow +blende +lam +contalnel +electric jug +slippers +trash-bin +dust pan +kett. +WOO +sandals +plieri +he +dustbin +olasticcontainer +'dustpan +ottl +mouse +brush +weighing--scal +es +o +Jug +pen +pillow +astic + wallet +laundry basket +soap bottle. +tray +1anp shac +paint can +crowal +carton +spade." +'sprayei +hoe +nsinsun +remote +shoes +guitar +Scar +coffee.maker +inergloves +T +paper bag +belt +polythenelbag +tape_measure +oaperitowe +hand drill +sprav-bottle +-plier +Aammera +impact wrencl +oucketchair +shopping list +paint bucket +Ca +Icon'trole +flask +cooking pan- +shopping.basket +...sack +Crewidriver +kitchen=towel +flower-vase. +lidTable 2. Comparison with other object tracking datasets. +Dataset +# Classes +(Eval/Train) +# Videos +(Eval/Train) +Average +Length (s) +Tracks per +Video +Annotation +FPS +Annotation +Type +Egocentric +ImageNet-Vid [56] +30 +1314/4000 +10.6 +2.4 +∼25 +mask +No +YTVIS [72] +40 +645/2238 +4.6 +1.7 +5 +mask +No +DAVIS17 [54] +- +30/60/30/30 +3 +3 +24 +mask +No +GOT-10k [26] +84/480 +360/9335 +12.2 +1 +10 +bbox +No +OxUvA [62] +22/0 +366/0 +141.2 +1.1 +1 +bbox +No +LaSOT [16] +70/70 +280/1120 +82.1 +1 +∼25 +bbox +No +TrackingNet [51] +27/27 +511/30132 +14.7 +1 +∼28 +bbox +No +TAO [12] +785/316 +2407/500 +36.8 +5.9 +1 +mask +No +UVO [67] +- +750/250 +10 +14 +30 +mask +No +EPIC-KITCHENS +VISOR +[11] +242/182 +115/43 +12* +- +0.9** +mask +Yes +EgoTracks (Ours) +- +1.1k/1.2k/3.6k +367.9 +3.8 +5 +bbox +Yes +*: Original video is 720s. **: An alternative dense set of annotations automatically generated by interpolation is also available. +ory [23], visual query [23], and camera-wearer pose infer- +ence [28]. +To tackle these challenges, tracking is lever- +aged in many methodologies [11, 23, 34, 42, 43], yet few +works have been dedicated to this fundamental problem on +its own. EgoTracks provides a unique testbed for devel- +oping tracking methods dedicated to egocentric videos; our +EgoSTARK also serves as a potential plug-and-play module +to solve other tasks where object association is desired. +In egocentric video understanding, two recent works are +closely related: Ego4D [23] and EPIC-KITCHENS VI- +SOR [11]. The seminal Ego4D contains the largest col- +lection of egocentric videos in-the-wild; EgoTracks is an- +notated on a subset of Ego4D. In addition, Ego4D proposes +many novel tasks, such as Episodic Memory, where tracking +is identified as a core component. VISOR was introduced +concurrently this year, annotating short-term (12 sec on av- +erage) videos from on EPIC-KITCHENS [9] with instance +segmentation masks. We believe EgoTracks offers multiple +unique values complementary to EPIC-VISOR: long-term +tracking (6 min vs. 12 sec), significantly larger-scale (6.9k +video clips vs. 158) and more diversified video sources (80+ +scenes vs. kitchen-only) (see Fig. 2). +3. The EgoTracks Dataset +We present EgoTracks: a large-scale long-term egocen- +tric single object tracking dataset, consisting of a total of +22.42k tracks from 5.9k videos. We follow the same data +split as the Ego4D Visual Queries (VQ) 2D benchmark: +3.6k/1.2k/1.1k for train/val/test (Tables 1 and 2). +3.1. Ego4D Visual Queries (VQ) Benchmark +Ego4D [23] is a massive-scale egocentric video dataset, +consisting of 3670 hours of video and hundreds of scenar- +ios, capturing first-person views of daily-life activities in +an unscripted, in-the-wild format. The dataset is accom- +panied by several benchmark tasks, such as episodic mem- +ory, hands and objects, social interaction, and forecasting. +The most relevant task in this benchmark suite to long-term +tracking is episodic memory’s 2D VQ task: Given an ego- +centric video and a static image of an object (a visual crop), +the goal is to localize when and where the object was last +seen in the video. More specifically, the output is a series of +2D bounding boxes in consecutive frames. We observe that +this task is very related to long-term tracking, as the task of +finding an object in a video with a visual template is identi- +cal to the re-detection problem in the long-term tracking lit- +erature. Moreover, Ego4D’s proposed baseline for this task +primarily relies on tracking methods: Siam-RCNN [64] and +KYS [4] for global and local tracking, respectively. +Shortcomings. While highly related, the VQ dataset is not +however immediately suitable for use in long-term tracking. +In particular, the VQ annotation guidelines were roughly +the following: 1) identify three different objects that appear +multiple times in the video; 2) annotate a template for each +object to serve as the query, which should contain the entire +object without any motion blur; 3) annotate an occurrence +of the object that is temporally distant from the template. +Notably, these instructions do not ask for exhaustive anno- +tations over time and are thus quite sparse, limiting their +applicability to tracking. On the other hand, the selection +criteria do result in a strong set of candidate objects to track, +which we leverage to build EgoTracks. +3.2. Annotating VQ for Long-Term Tracking +We thus start with the visual crop and response track +from the VQ annotations, asking annotators to first identify +the object represented by the visual crop, response track, +and object name. Starting from the beginning of the video, +we instruct the annotators to draw a bounding box around +the object each time it appears. Because annotators must go +through each video in its entirety, which contain on average +roughly 1800 frames at 5 frames per second (FPS), this an- +notation task can be labor-intensive, taking roughly 1 to 2 + +Table 3. Attributes of the track in validation set. +Total number +Percentage +All Tracks +4459 +100% +is active +1014 +22.74% +is transformed +241 +5.40% +is recognizable +4450 +99.79% +hours per track to label. An important aspect of this annota- +tion is its exhaustiveness: the entire video is densely anno- +tated for the target object, and any frame without a bounding +box is considered as a negative. Being able to reject nega- +tives examples is an important component to re-detection +in real-world settings, as false positives can impact certain +applications as much as false negatives. +Quality Assurance. All tracks are quality checked by ex- +pert annotators after their initial annotations. To measure +the annotation quality, we employ multi-review on a sub- +set of the validation set. Three independent reviewers are +asked to annotate the same video. +Upon inspection, we +find the overlaps between these independent annotations are +high (> 0.88 IoU). Further, since EgoTracks has a focus on +re-detection, we check the temporal overlap of object pres- +ence and find it to be very consistent across annotators. In +total, the entire annotation effort represented roughly 86k +worker-hours of effort. +3.3. Validation Tracklet Attributes +In addition to the bounding box annotations, we also la- +bel certain relevant attributes to allow for deeper analysis of +tracker performance for the val set. We annotate the follow- +ing three attributes per occurrence in the validation set (see +Figure 3 for examples and Table 3 for statistics): +• is active: Ego4D is a dataset capturing a multitude +of daily activities performed by the camera wearer. +Naturally, this means the camera wearer often inter- +acts with relevant objects with their hands. Objects in +the state of being handled pose a challenge for track- +ing algorithms due to frequent occlusions by hands and +rapid changes in pose. +• is transformed: Certain objects in Ego4D un- +dergo transformations, such as deformation and state +change. Such instances require being able to quickly +adapt to the tracked object having a new appearance. +• is recognizable: Due to occlusions, motion blur, +scale, or other conditions, some objects in long-form, +in-the-wild videos like those in Ego4D can be ex- +tremely difficult to recognize without additional con- +text. We thus annotate if the object is recognizable +solely based on its appearance, without using addi- +tional context information (e.g. other frames). +is_transformed +is_active +not is_recognizable +Figure 3. EgoTracks examples of tracklet attributes. Left: A +micropipette on a bench (top) versus actively used (bottom). Mid- +dle: A paint can (top) is opened (bottom). Right: A hard to recog- +nize blowtorch (bottom) due to distance and motion blur; annota- +tors must rely on context from other frames to identify the object. +4. Analysis of state-of-the-art SOT trackers +We compare the performance of several off-the-shelf +tracking models on EgoTracks’s validation set. Identifying +STARK [71] as the one with the best performance, we con- +duct further ablation studies under different settings using +STARK to further understand its behavior. +4.1. Evaluation protocols and metrics +Evaluation Protocols. +We introduce several evaluation +protocols for EgoTracks, consisting of different combina- +tions of the initial template, evaluated frames, and the tem- +poral direction in which the tracker is run. For the initial +template, we consider two choices: +• Visual Crop Template (VCT): The visual crop im- +ages were specifically chosen to be high-quality views +of the target and served as our annotators’ references +for identifying the object throughout the videos. Thus, +they make ideal candidates for initializing a tracker. +• Occurrence First Frame Template (OFFT): The +tracker is initialized with the first frame of each occur- +rence (see −→ +OO below). While this may result in a lower +quality view of the object, temporal proximity to sub- +sequent frames means it may be closer in appearance. +Note that we exclude the template frame from the calcu- +lation of any evaluation metrics. We also consider several +choices for the evaluated frames and temporal direction: +• Video Start Forward (−→ +VS): The tracker is evaluated +on every frame of the video in causal order, starting +from the first frame. This represents a tracker’s ability +to follow an object through a long video. +• Visual Crop Forward/Backward (←→ +VC): The tracker +is run on the video twice, once starting at the visual + +micropipettemicropipetteblowtorchblowtorch +OGUUHKS +paint- +ca.Deint canFigure 4. Evaluation protocols visualization. +Table 4. EgoTracks performance of various trackers. +Method +AO +F-score +Precision +Recall +KYS [4] +16.09 +13.09 +12.50 +13.74 +DiMP [3] +16.45 +11.84 +10.31 +13.91 +GlobalTrack [27] +23.63 +20.40 +31.28 +15.14 +LTMU [8] +29.33 +27.46 +37.28 +21.74 +ToMP [47] +30.93 +20.95 +19.63 +22.46 +STARK [71] - Res50 +35.99 +30.48 +34.70 +27.17 +STARK [71] - Res101 +35.03 +30.18 +35.30 +26.35 +Tracking by Detection +Mask R-CNN [24]+Oracle +60.00 +- +- +- +GGN [66]+Oracle +75.92 +- +- +- +GGN+InstEmb +15.19 +9.92 +11.75 +8.58 +crop frame and running forward and time, and a sec- +ond time running backwards. This represents an alter- +native way of covering every frame in the video, but +with closer visual similarity between VCT initializa- +tion and the first frames encountered by the tracker. +• Occurrences Only Forward (−−→ +OO): The tracker is +only evaluated on the object occurrences, when the ob- +ject is visible. This simplifies the tracking task and al- +lows us to dis-entangle the challenge of re-detection +from that of simply tracking in an egocentric clip. +We specify protocols by concatenating the appropriate de- +scriptors. We primarily consider VCT-−→ +VS, VCT-←→ +VC, VCT- +−−→ +OO, and OFFT-−−→ +OO (Fig. 4) in our experiments. +Metrics. We adopt common metrics in object tracking. The +most important ones are tracking F-score, precision, and +recall [46]; details on these metrics can be found in [46]. +Trackers are ranked mainly by the F-score. We addition- +ally consider average overlap (AO), success, precision, and +normalized precision as short-term tracking metrics [61]. +4.2. Comparison of SOT trackers +We compare the performance of several CNN-based +tracking algorithms on EgoTracks with the VCT-−→ +VS eval- +uation protocol. Given the large number of existing track- +ing algorithms, we do not aim to be exhaustive but select +high-performing examples representative of different track- +Figure 5. Qualitative results of different trackers. +ing principles, which we briefly describe here. KYS [4] and +DiMP [3] are two typical short-term tracking algorithms +that maintain an online target representation. ToMP [47] +and STARK [71] are two examples of the SOTA short- +term trackers based on Transformers. GlobalTrack [27] is +a global tracker that searches the entire search image for +re-detection. LTMU [8] is a high performance long-term +tracker that combines a global tracker (GlobalTrack) with +a local tracker (DiMP). The performance of these trackers +on EgoTracks are summarized in Table 4. Note, AO in this +table is equivalent to the recall at the probability threshold +of 0. Qualitative results are shown in Figure 5. +We highlight several observations. First, the object pres- +ence scores from most short-term trackers are not very use- +ful, as can be seen from the low precision of KYS (12.5), +DiMP (13.91), and ToMP (22.46), while long-term track- +ers like GlobalTrack and DiMP LTMU achieve higher pre- +cisions at 31.28 and 37.28. This is expected as long-term +trackers are designed to place more emphasis on high re- +detection accuracy, though there clearly is still room for im- +provement. STARK achieves the second highest precision +at 34.70, which is an exception as it has a second train- +ing stage to teach the model to classify whether the ob- +ject is present. Second, more recent works such as ToMP +and STARK achieve better F-score than previous short-term +trackers. This could be partially due to advances in training +strategies, more data, and Transformer-based architectures. +We also include results using the principle of Tracking +by Detection [1, 52]: a detector proposes 100 bounding +boxes, and we select the best using cosine similarity of box +features. We observe that an open-world detector GGN [66] +trained on COCO [41] generalize reasonably well with or- +acle matching, achieving 75.92 AO. However, the associa- +tion problem is very challenging, bringing down the AO to +15.19. Implementation details are in the supplementary. + +Template +VCT-VS +Visual Crop +Evaluation Direction +VCT-VC +VCT-00 +OFFT-00 +Occurrence +First Frame +00empate +emplateTable 5. Comparing tracker initializations. The upper table com- +pares trackers initialized from the first frame in each occurrence +and tracking only that single occurrence (no re-detection or per- +fect re-detection). The lower table compares STARK whole-video +performance, starting from video start frame vs. visual crop frame. +Method +AO +Success +Pre +Prenorm +KYS [4] +33.92 +34.87 +31.22 +34.87 +DiMP [3] +34.80 +35.70 +32.13 +38.98 +ToMP [47] +45.17 +45.93 +41.74 +47.88 +STARK [71] +50.01 +50.64 +45.76 +51.91 +Method +AO +F-score +Precision +Recall +STARK - VCT-−→ +VS +35.99 +30.48 +34.70 +27.17 +STARK - VCT-←→ +VC +40.01 +34.02 +38.31 +30.60 +4.3. Alternative evaluation protocols +We perform additional evaluations according to alterna- +tive evaluation protocols, to gain further insight to tracker +performance. To decouple the re-detection problem from +the other egocentric aspects of EgoTracks, we run experi- +ments with the OFFT-−−→ +OO protocol, which ignores the neg- +ative frames of the video and thus obviates the need for re- +detection, with results in Table 5. Perhaps unsurprisingly, +all trackers do significantly better in this setting, though +there remains much room for improvement, emphasizing +the challenging nature of EgoTracks. We also run exper- +iments with STARK in the VCT-←→ +VC setting (Table 5), in +which case the initial template is temporally adjacent to the +first tracked frames. Here we see a 3-4% improvement to +AO, F-score, precision, and recall compared to the VCT- +−→ +VS protocol, illustrating that trackers like STARK are de- +signed to expect gradual transitions in appearance. Both +these experiments illustrate that the re-detection problem is +a significant challenge for tracking and the need for better +long-term benchmarks requiring more re-detection. +4.4. Effect of attributes on tracking performance +We use the validation set tracklet attribute annotations +described in Section 3.3 to further understand performance +on our evaluation set. For each attribute, we split the track- +lets into two groups, corresponding to the attribute being +true and false. We then use a standard STARK tracker [71] +and report AO for each group of tracklets using the OFFT- +−−→ +OO evaluation protocol in Table 6. As might be expected, +we find that when objects are being actively used by the user +or in the midst of a transformation, AO tends to be lower, +by roughly 6%, likely due to occlusions or changes in ap- +pearance. Additionally, STARK tends to have a harder time +when the object is hard to recognize in the image, whether +due to occlusions, blur, scale, or other conditions. +Table 6. OFFT-−→ +OO AO of standard STARK model [71] for each +attribute, averaged across tracklets. +Attribute +True +False +is active +49.65 +55.73 +is transformed +49.19 +55.31 +is recognizable +55.52 +46.65 +Table 7. Training and test-time hyperparameters comparison. +Method +AO +F-score +Precision +Recall +Data +STARK +35.99 +30.48 +34.70 +27.17 +STARK - ft on VQ +38.94 +33.53 +39.13 +29.33 +STARK - ft on EgoTracks +44.25 +38.20 +42.06 +34.99 +Augmentation +STARK - ft on VQ +38.94 +33.53 +39.13 +29.33 +STARK - ft + multiscale +48.44 +41.92 +42.65 +41.30 +Search window +search size = 320 +35.99 +30.48 +34.70 +27.17 +search size = 480 +48.21 +39.69 +43.95 +36.19 +search size = 640 +52.09 +42.39 +46.23 +39.15 +search size = 800 +54.08 +43.74 +47.60 +40.45 +5. Ego-STARK +Despite not being specifically designed for long-term +tracking, Section 4 suggests STARK [71] to be the most +competitive tracker on EgoTracks. We thus leverage this +tracker for additional analysis, suggesting several improve- +ments to boost performance on EgoTracks. +5.1. Dataset Finetuning +We first demonstrate how STARK trained on third- +person videos significantly benefits from finetuning on ego- +centric data. +We experiment with two versions of Ego- +Tracks: the Ego4D VQ response track dataset (i.e. short- +term subset of EgoTracks) and the full EgoTracks training +set, which contains more point-of-view and scale variations, +as well as hard negatives. As shown in Table 7, finetun- +ing on the VQ response track subset improves the F-score +from 30.48% to 33.53%. Using the full EgoTracks annota- +tion further improves the F-score by 4.67% to 38.2%. This +demonstrates that: 1) finetuning with egocentric data closes +certain domain gaps; 2) training on full EgoTracks provides +additional gains, showing the value of our training set. +5.2. Adjusting Spatiotemporal Priors +Modern trackers often embrace spatiotemporal priors on +object motion, appearance and surroundings, which helped +them on past datasets. However, some of these design deci- +sions translate poorly to long-term egocentric videos. +Search window size. An example is the local search as- +sumption. Many trackers assume the tracked object appears +within a certain range of its previous location. Thus, for ef- +ficiency, these methods often search within a local window +of the next frame. This is reasonable in high FPS, smooth +videos with relatively small motion, commonly in previous +short-term tracking datasets, but in egocentric videos, the +object’s pixel coordinates can change rapidly with frequent +large head motions, and re-detection becomes a key prob- + +Table 8. STARK with different context ratios. Row in bold is the +default STARK setting. CR: context ratio, SRR: search region +ratio, SIS: search image size (in image resolution). +Method +AO +F-score +Precision +Recall +Setting +CR +SRR +SIS +Same SIS +1x +2.5x +320 +28.22 +26.81 +28.68 +25.16 +2x +5x +320 +38.94 +33.53 +39.13 +29.33 +3x +7.5x +320 +44.70 +36.03 +40.28 +32.59 +4x +10x +320 +43.19 +34.32 +37.98 +31.31 +Same SRR +1x +5x +640 +41.50 +31.09 +30.31 +31.91 +3x +5x +208 +39.87 +35.36 +41.54 +30.79 +Same CR +2x +7.5x +480 +48.21 +39.69 +43.95 +36.19 +2x +10x +640 +52.09 +42.39 +46.23 +39.15 +2x +12.5x +800 +54.08 +43.74 +47.60 +40.45 +lem. Therefore, we experiment with expanded search re- +gions beyond what are common in past methods. As we +expand the search size from 320 up to 800, we see dramatic +improvements in AO, F-score, Precision, and Recall (Ta- +ble 7), as STARK is able to correctly locate objects that +were previously outside of its search window due to the +rapid movement of egocentric video. +Multiscale augmentations. The characteristics of egocen- +tric video also affect common SOT assumptions of object +scale. Many trackers are trained with the assumption that +an object’s scale is consistent with the template image and +between adjacent frames. However, large egocentric cam- +era motions, locomotion, and hand interactions with ob- +jects (e.g. bringing an object to one’s face, as in eating) +can translate to objects rapidly undergoing large changes +in scale. We thus propose adding scale augmentations dur- +ing training, randomly resizing the search image by a factor +of s ∈ [0.5, 1.5]. While simple, we find this dramatically +improves performance on EgoTracks, improving STARK’s +AO by nearly 10% and F-score by more than 8% (Table 7). +Context ratio. Past SOT works have found that includ- +ing some of the background can be helpful when extracting +features from the template image, with 2 times the size of +object being common practice. We experiment with differ- +ent context ratios to see if this rule of thumb transfers to +egocentric videos. Because of the local window assump- +tion, the sizes of the template image and search image are +related: +Search Image Size(SIS) +Search Region Ratio(SRR) = +Template Image Size +Context Ratio(CR) += +Object Scale. The template image size is set to a fixed size +128 × 128. When changing the context ratio, we carefully +control the other parameters for a fair comparison. The re- +sults are shown in Table 8. Among all three parameters - +CR, SRR, and SIS, the search region size (determined by +SRR and SIS) has the highest impact on the F-score. This +is expected because there are frequent re-detections, which +require the tracker to search in a larger area for the object, +rather than just within the commonly used local window. +Varying the CR has mixed results so we adhere to the com- +mon practice of using a CR of 2. The best result is achieved +when using SRR 12.5, which covers most of the image and +Table 9. STARK with different numbers of templates. +Method +AO +F-score +Precision +Recall +STARK - 1 template +32.97 +25.42 +25.80 +25.04 +STARK - 3 templates +34.76 +26.84 +28.84 +25.57 +STARK - 5 templates +35.47 +28.03 +29.82 +26.45 +STARK - 7 templates +34.81 +27.83 +30.77 +25.40 +STARK - 9 templates +33.92 +26.89 +30.36 +24.12 +achieves a F-score of 43.74%. +5.3. Multiple templates +Transformer-based architectures can encode arbitrary +length inputs, making it straightforward to consume fea- +tures from an arbitrary number of templates. +The origi- +nal STARK design encodes two templates: the initialization +and a single dynamically updated template. A natural exten- +sion is to include more templates of the target, which may +expose the transformer to different views of the object (par- +ticularly relevent in egocentric video), though low-quality +views may compromise performance [39]. +What’s the right trade-off? We experiment with differ- +ent numbers of templates for a basic STARK model. Moti- +vated by potential applications where a user can take a short +video of an object from different angles [55], we extend the +single visual crop to a visual clip of templates by incorpo- +rating additional frames from the same occurrence where +the visual crop appears as the template. We adopt a simple +template sampling method: uniformly sampling 3, 5, 7, or +9 templates from the visual crop’s occurrence. Uniformly +sampling the videos temporally can be a simple yet effec- +tive heuristic to gather diverse views from an occurrence. +We summarize the results in Table 9. While we observe im- +provements across all metrics using up to 5 templates, per- +formance declines with more. We hypothesize that increas- +ing the number of templates does increase the knowledge +available to STARK for tracking, but after a certain point +it may dilute the information in the templates and make it +difficult for the transformer to synthesize. This highlights +the importance of template selection and multi-view fusion +mechanisms, which inspires promising directions. +6. Conclusion +We present EgoTracks, the first large-scale dataset +for long-term egocentric visual object tracking in diverse +scenes. We conduct extensive experiments to understand +the performance of state-of-the-art trackers on this new +dataset, and find that they struggle considerably, possibly +in part due to overfitting to some of the simpler character- +istics of existing benchmarks. 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In 2015 IEEE International +Conference on Computer Vision (ICCV), pages 4498–4506, +2015. 3 + diff --git a/gdE1T4oBgHgl3EQffAR-/content/tmp_files/load_file.txt b/gdE1T4oBgHgl3EQffAR-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cca4f602abc4e036c4fabd5546fac99fc6d41ead --- /dev/null +++ b/gdE1T4oBgHgl3EQffAR-/content/tmp_files/load_file.txt @@ -0,0 +1,848 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf,len=847 +page_content='EgoTracks: A Long-term Egocentric Visual Object Tracking Dataset Hao Tang Kevin Liang Kristen Grauman Matt Feiszli Weiyao Wang Meta Platforms Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' haotang, kevinjliang, grauman, mdf, weiyaowang@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='com Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' A sample video from the proposed EgoTracks dataset, with yellow segments of the clip marking when the object (blowtorch) is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Note the frequent disappearances and reappearances of the object over an 8 minute video, with lengthy absences, necessitating re-detection to track accurately without false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The egocentric nature of the video includes the camera-wearer interacting with the object (Occurrence 2), resulting in significant hand occlusions and dramatic changes in pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Abstract Visual object tracking is a key component to many ego- centric vision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' However, the full spectrum of challenges of egocentric tracking faced by an embodied AI is underrepresented in many existing datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' these tend to focus on relatively short, third-person videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Egocen- tric video has several distinguishing characteristics from those commonly found in past datasets: frequent large cam- era motions and hand interactions with objects commonly lead to occlusions or objects exiting the frame, and ob- ject appearance can change rapidly due to widely differ- ent points of view, scale, or object states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Embodied track- ing is also naturally long-term, and being able to consis- tently (re-)associate objects to their appearances and dis- appearances over as long as a lifetime is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Pre- vious datasets under-emphasize this re-detection problem, and their “framed” nature has led to adoption of various spatiotemporal priors that we find do not necessarily gen- eralize to egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We thus introduce EgoTracks, a new dataset for long-term egocentric visual object track- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Sourced from the Ego4D dataset, this new dataset presents a significant challenge to recent state-of-the-art single-object tracking models, which we find score poorly on traditional tracking metrics for our new dataset, com- pared to popular benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We further show improve- ments that can be made to a STARK tracker to significantly increase its performance on egocentric data, resulting in a baseline model we call EgoSTARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We publicly release our annotations and benchmark, hoping our dataset leads to further advancements in tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Introduction First-person or “egocentric” computer vision aims to capture the real-world perceptual problems faced by an em- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='03213v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='CV] 9 Jan 2023 Occurrence 2 Occurrence 17 Negative Framesbodied AI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' it has drawn strong recent interest as an under- served but highly relevant domain of vision, with important applications ranging from robotics [15, 57] to augmented and mixed reality [2,23,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Visual object tracking (VOT), long a fundamental problem in vision, is a core component to many egocentric tasks, including tracking the progress of an action or activity, (re-)association of objects in one’s surroundings, and predicting future states of the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Yet, while the VOT field has made many significant advancements over the past decade, tracking in egocentric video remains underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This lack of attention is in large part due to the absence of a large-scale egocentric tracking dataset for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' While the com- munity has proposed a number of popular tracking datasets in recent years, including OTB [69], TrackingNet [51], Got- 10k [26], and LaSOT [16], we find that the strong perfor- mance that state-of-the-art trackers achieve on these bench- marks does not translate well to egocentric video, thus es- tablishing a strong need for such a tracking dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We attribute this performance gap to the many unique aspects of egocentric views compared to the more tradi- tional third-person views of previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In contrast to intentionally “framed” video, egocentric videos are of- ten uncurated, meaning they tend to capture many attention shifts between activities, objects, or locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Due to the first-person perspective, large head motions from the cam- era wearer often result in objects repeatedly leaving and re- entering the field of view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' similarly, hand manipulations of objects [58] leads to frequent occlusions, rapid variations in scale and pose, and potential changes in state or appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Furthermore, egocentric video tends to be long (sometimes representing the entire life of an agent or individual), mean- ing the volume of the aforementioned occlusions and trans- formations scales similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' These characteristics all make tracking objects in egocentric views dramatically more dif- ficult than scenarios commonly considered in prior datasets, and their absence represents an evaluation blindspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The aforementioned head motions, locomotion, hand oc- clusions, and temporal length lead to several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' First, frequent object disappearances and reappearances causes the problem of re-detection within egocentric track- ing to become especially critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Many previous track- ing datasets primarily focus on short-term tracking in third- person videos, providing limited ability to evaluate many of the challenges of long-term egocentric tracking due to low quantity and length of target object disappearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' As a result, competent re-detection is not required for strong per- formance, leading many recent short-term trackers to ne- glect it, instead predicting a bounding box for every frame, which may lead to rampant false positives or tracking the wrong object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Additionally, the characteristics of short- term third-person video have also induced designs relying on gradual changes in motion and appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' As we later Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Statistics of long-term object tracking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' An occurrence is defined as a contiguous set of frames where the ob- ject is visible, before disappearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The presented EgoTracks has the highest number of tracks and target disappearances and reap- pearances, making it the largest dataset for training and evaluating long-term trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We summarize the training and validation data from each dataset here, as we do not have the ground truth for the other datasets’ hold-out test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' OxUvA [62] LaSOT [16] VOT-LT [32] Ours # tracks 200 1400 50 17598 # occurrences / track 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='7 Seconds / occurrence 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='52 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2 Seconds btw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' occurrences 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1 show (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2), many of the motion, context, and scale priors made by previous short-term tracking algorithms fail to transfer to egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Thus, a large-scale long- term tracking dataset is needed to train and understand the long-term tracking capability of modern trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' To address this gap, we present EgoTracks: a large- scale long-term egocentric visual object tracking dataset for training and evaluating long-term trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Seeking a real- istic challenge, we source videos from Ego4D [23], a large- scale dataset consisting of unscripted, in-the-wild egocen- tric videos of daily-life activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The result is a large-scale dataset to evaluate the tracking and re-detection ability of SOT models, with more than 20,000 tracks from around 6000 6-minute videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This constitutes the first large-scale dataset for visual object tracking in egocentric videos in di- verse settings, providing a new, significant challenge com- pared with previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We perform a thorough analysis of our new dataset, demonstrating its difficulty and the need for further research to develop trackers capable of handling long-term egocen- tric vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' These experiments reveal valuable insights to- wards promising future directions in egocentric tracking and re-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Leveraging these intuitions, we propose multiple simple yet effective changes, such as adjustment of spatiotemporal priors, egocentric data finetuning and com- bining multiple templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We study these proposed strate- gies on the the state-of-the-art STARK tracker [71] and train a strong tracker dedicated towards long-term egocen- tric tracking, EgoSTARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We hope EgoSTARK can serve as a strong baseline and facilitate future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' To summarize, we make the following contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We present EgoTracks, the first large-scale long-term object tracking dataset with diverse egocentric scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We analyze its uniqueness in terms of evaluating the re-detection performance of trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We conduct comprehensive experiments to understand the performance of many state-of-the-art trackers on the EgoTracks validation set and observe that due to the biases and evaluation blindspots of existing third- person datasets, they tend to struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' EgoTracks is a large-scale egocentric dataset of diverse scenarios (left) and objects (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We propose a combo of several training and inference- time improvements to the STARK tracker [71] to adapt it to long-form egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We call this re- fined model Ego-STARK, which achieves significant improvements (15% tracking F-score) on EgoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Visual Object Tracking Datasets Visual object tracking studies the problem of joint spatial-temporal localization of objects in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' There are two main categories: multiple object tracking (MOT) and single object tracking (SOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In MOT, a model is pro- vided a video and a predefined taxonomy, and the model is required to simultaneously detect, recognize and track multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' For example, MOT [49] tracks human, KITTI [20, 45] tracks pedestrians and cars, and TAO [12] tracks a large taxonomy of 833 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Different from MOT, SOT tracks a single object via a provided initial tem- plate of the object, and no detection or recognition is in- volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Thus, SOT is often taxonomy-free and operates on generic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The community has constructed multiple popular benchmarks to study this important problem, in- cluding OTB [69], UAV [50], NfS [30], TC-128 [40], NUS- PRO [35], GOT-10k [26], VOT [32], and TrackingNet [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' These SOT datasets mainly consist of short videos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' a few seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Recently, there have been increasing interests in long-term tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Different from the above datasets, tracking objects in longer videos (several minutes or more) poses unique challenges, as the object may involve more significant transforms, displacements, disappearances, and reappearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' On top of localizing the object when it is visible, the model needs to know not predict a box when the object is absent, and then re-localize the same object when it reappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' OxUvA [62] is one of the first to benchmark longer videos (average 2 minutes), with 366 evaluation- only videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' LaSOT [16] scales this philosophy with a benchmark of 1400 videos of more frequent object reap- pearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Concurrently, VOT-LT [31] is specifically de- signed to benchmark videos with frequent object disappear- ances and reappearances in 50 purposefully selected videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Our EgoTracks focuses on long-term SOT and presents multiple critical and unique attributes: 1) significantly larger scale, with 17k videos of an average 6 minutes (Ta- ble 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2) more frequent disappearances & reappearances (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='7 times) happening in natural, unscripted, real- world scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3) data sourced from egocentric videos shot in-the-wild, involving unique challenging situations, such as large camera motions, diverse perspective changes, hand-object interactions, and frequent occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Single Object Tracking Methodologies Many modern approaches use convolutional neural net- works (CNNs), either with a Siamese network architec- ture [36, 37, 65], or a correlation-filter based architec- ture [3,4,7,10,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' With the Transformer’s recent successes in computer vision tasks like classification [14] and detec- tion [5], a line of works leveraging the Transformer archi- tecture [63] to perform tracking have also emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' For ex- ample, inspired by Transformers, TransT [6] uses attention- based feature fusion to combine features of the object tem- plate and search image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' More recently, several works uti- lize Transformers as direct predictors to achieve a new state of the art, such as STARK [71], ToMP [47] and SBT [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' These models tokenize frame features from a ResNet [25] encoder, and use a Transformer to predict the bounding box and object presence score with the feature tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' These methods are often developed on short-term SOT datasets and assume that the target object stays in the field of view with minimum occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' On the other hand, long-term trackers [8,27,64] are designed to cope with the problem of re-detecting objects in their reappearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Intended to be aware of potential object disappearances, these approaches search the whole image for its reappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Tracking in Egocentric Videos Multiple egocentric video datasets have been introduced in the past decades [9, 18, 23, 33, 53, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' They offer a host of interesting challenges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' many of which require associating objects across frames: activity recognition [21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 68,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 73],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' anticipation [17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' video summarization [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' human-object interaction [11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' episodic mem- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Doing yardwork/shoveling snow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Indoor Navigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='On a screen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Playing board games ' metadata={'source': 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+page_content='mechani ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Daily ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Camp setup/pack-up/chores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Crafting/knitting/sewing/drawing/painting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Scooter mechanic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Commuting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' road tri Grocery shopping Going to the gy ndoors Making Potting plantsMaking coffee Clothes other shopping Fixing something in the home Taiking onthe phone Talkingwithfriends/ housematbag lawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='mower televlslon ball paper Naterfanbotte wi cockbasket book phone .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='measuring tape chopping=boar scIssors rille towel knife ricecooker cup tissue flower pot mobile-phone pot trolley spanner coolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='.helmet throw pillow blende lam contalnel electric jug slippers trash-bin dust pan kett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=" WOO sandals plieri he dustbin olasticcontainer 'dustpan ottl mouse brush weighing--scal es o Jug pen pillow astic wallet laundry basket soap bottle." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' tray 1anp shac paint can crowal carton spade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='" \'sprayei hoe nsinsun remote shoes guitar Scar coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content="maker inergloves T paper bag belt polythenelbag tape_measure oaperitowe hand drill sprav-bottle plier Aammera impact wrencl oucketchair shopping list paint bucket Ca Icon'trole flask cooking pan- shopping." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='basket .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='sack Crewidriver kitchen=towel flower-vase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' lidTable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Comparison with other object tracking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Dataset # Classes (Eval/Train) # Videos (Eval/Train) Average Length (s) Tracks per Video Annotation FPS Annotation Type Egocentric ImageNet-Vid [56] 30 1314/4000 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='4 ∼25 mask No YTVIS [72] 40 645/2238 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='7 5 mask No DAVIS17 [54] 30/60/30/30 3 3 24 mask No GOT-10k [26] 84/480 360/9335 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2 1 10 bbox No OxUvA [62] 22/0 366/0 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1 1 bbox No LaSOT [16] 70/70 280/1120 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1 1 ∼25 bbox No TrackingNet [51] 27/27 511/30132 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='7 1 ∼28 bbox No TAO [12] 785/316 2407/500 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='9 1 mask No UVO [67] 750/250 10 14 30 mask No EPIC-KITCHENS VISOR [11] 242/182 115/43 12* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='9** mask Yes EgoTracks (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1k/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2k/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='6k 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='8 5 bbox Yes : Original video is 720s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' **: An alternative dense set of annotations automatically generated by interpolation is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' ory [23], visual query [23], and camera-wearer pose infer- ence [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' To tackle these challenges, tracking is lever- aged in many methodologies [11, 23, 34, 42, 43], yet few works have been dedicated to this fundamental problem on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' EgoTracks provides a unique testbed for devel- oping tracking methods dedicated to egocentric videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' our EgoSTARK also serves as a potential plug-and-play module to solve other tasks where object association is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In egocentric video understanding, two recent works are closely related: Ego4D [23] and EPIC-KITCHENS VI- SOR [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The seminal Ego4D contains the largest col- lection of egocentric videos in-the-wild;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' EgoTracks is an- notated on a subset of Ego4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In addition, Ego4D proposes many novel tasks, such as Episodic Memory, where tracking is identified as a core component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' VISOR was introduced concurrently this year, annotating short-term (12 sec on av- erage) videos from on EPIC-KITCHENS [9] with instance segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We believe EgoTracks offers multiple unique values complementary to EPIC-VISOR: long-term tracking (6 min vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 12 sec), significantly larger-scale (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='9k video clips vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 158) and more diversified video sources (80+ scenes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' kitchen-only) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The EgoTracks Dataset We present EgoTracks: a large-scale long-term egocen- tric single object tracking dataset, consisting of a total of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='42k tracks from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='9k videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We follow the same data split as the Ego4D Visual Queries (VQ) 2D benchmark: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='6k/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2k/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1k for train/val/test (Tables 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Ego4D Visual Queries (VQ) Benchmark Ego4D [23] is a massive-scale egocentric video dataset, consisting of 3670 hours of video and hundreds of scenar- ios, capturing first-person views of daily-life activities in an unscripted, in-the-wild format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The dataset is accom- panied by several benchmark tasks, such as episodic mem- ory, hands and objects, social interaction, and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The most relevant task in this benchmark suite to long-term tracking is episodic memory’s 2D VQ task: Given an ego- centric video and a static image of an object (a visual crop), the goal is to localize when and where the object was last seen in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' More specifically, the output is a series of 2D bounding boxes in consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We observe that this task is very related to long-term tracking, as the task of finding an object in a video with a visual template is identi- cal to the re-detection problem in the long-term tracking lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Moreover, Ego4D’s proposed baseline for this task primarily relies on tracking methods: Siam-RCNN [64] and KYS [4] for global and local tracking, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' While highly related, the VQ dataset is not however immediately suitable for use in long-term tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In particular, the VQ annotation guidelines were roughly the following: 1) identify three different objects that appear multiple times in the video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2) annotate a template for each object to serve as the query, which should contain the entire object without any motion blur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3) annotate an occurrence of the object that is temporally distant from the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Notably, these instructions do not ask for exhaustive anno- tations over time and are thus quite sparse, limiting their applicability to tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' On the other hand, the selection criteria do result in a strong set of candidate objects to track, which we leverage to build EgoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Annotating VQ for Long-Term Tracking We thus start with the visual crop and response track from the VQ annotations, asking annotators to first identify the object represented by the visual crop, response track, and object name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Starting from the beginning of the video, we instruct the annotators to draw a bounding box around the object each time it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Because annotators must go through each video in its entirety, which contain on average roughly 1800 frames at 5 frames per second (FPS), this an- notation task can be labor-intensive, taking roughly 1 to 2 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Attributes of the track in validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Total number Percentage All Tracks 4459 100% is active 1014 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74% is transformed 241 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='40% is recognizable 4450 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='79% hours per track to label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' An important aspect of this annota- tion is its exhaustiveness: the entire video is densely anno- tated for the target object, and any frame without a bounding box is considered as a negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Being able to reject nega- tives examples is an important component to re-detection in real-world settings, as false positives can impact certain applications as much as false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Quality Assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' All tracks are quality checked by ex- pert annotators after their initial annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' To measure the annotation quality, we employ multi-review on a sub- set of the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Three independent reviewers are asked to annotate the same video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Upon inspection, we find the overlaps between these independent annotations are high (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='88 IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Further, since EgoTracks has a focus on re-detection, we check the temporal overlap of object pres- ence and find it to be very consistent across annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In total, the entire annotation effort represented roughly 86k worker-hours of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Validation Tracklet Attributes In addition to the bounding box annotations, we also la- bel certain relevant attributes to allow for deeper analysis of tracker performance for the val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We annotate the follow- ing three attributes per occurrence in the validation set (see Figure 3 for examples and Table 3 for statistics): is active: Ego4D is a dataset capturing a multitude of daily activities performed by the camera wearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Naturally, this means the camera wearer often inter- acts with relevant objects with their hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Objects in the state of being handled pose a challenge for track- ing algorithms due to frequent occlusions by hands and rapid changes in pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' is transformed: Certain objects in Ego4D un- dergo transformations, such as deformation and state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Such instances require being able to quickly adapt to the tracked object having a new appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' is recognizable: Due to occlusions, motion blur, scale, or other conditions, some objects in long-form, in-the-wild videos like those in Ego4D can be ex- tremely difficult to recognize without additional con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We thus annotate if the object is recognizable solely based on its appearance, without using addi- tional context information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' other frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' is_transformed is_active not is_recognizable Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' EgoTracks examples of tracklet attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Left: A micropipette on a bench (top) versus actively used (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Mid- dle: A paint can (top) is opened (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Right: A hard to recog- nize blowtorch (bottom) due to distance and motion blur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' annota- tors must rely on context from other frames to identify the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Analysis of state-of-the-art SOT trackers We compare the performance of several off-the-shelf tracking models on EgoTracks’s validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Identifying STARK [71] as the one with the best performance, we con- duct further ablation studies under different settings using STARK to further understand its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Evaluation protocols and metrics Evaluation Protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We introduce several evaluation protocols for EgoTracks, consisting of different combina- tions of the initial template, evaluated frames, and the tem- poral direction in which the tracker is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' For the initial template, we consider two choices: Visual Crop Template (VCT): The visual crop im- ages were specifically chosen to be high-quality views of the target and served as our annotators’ references for identifying the object throughout the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Thus, they make ideal candidates for initializing a tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Occurrence First Frame Template (OFFT): The tracker is initialized with the first frame of each occur- rence (see −→ OO below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' While this may result in a lower quality view of the object, temporal proximity to sub- sequent frames means it may be closer in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Note that we exclude the template frame from the calcu- lation of any evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We also consider several choices for the evaluated frames and temporal direction: Video Start Forward (−→ VS): The tracker is evaluated on every frame of the video in causal order, starting from the first frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This represents a tracker’s ability to follow an object through a long video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Visual Crop Forward/Backward (←→ VC): The tracker is run on the video twice, once starting at the visual micropipettemicropipetteblowtorchblowtorch OGUUHKS paint- ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='Deint canFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Evaluation protocols visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' EgoTracks performance of various trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Method AO F-score Precision Recall KYS [4] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='09 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='09 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74 DiMP [3] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='45 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='84 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='31 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='91 GlobalTrack [27] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='63 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='40 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='28 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='14 LTMU [8] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='33 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='46 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='28 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74 ToMP [47] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='93 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='95 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='63 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='46 STARK [71] - Res50 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='48 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='17 STARK [71] - Res101 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='03 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='18 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='30 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='35 Tracking by Detection Mask R-CNN [24]+Oracle 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='00 GGN [66]+Oracle 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 GGN+InstEmb 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='75 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='58 crop frame and running forward and time, and a sec- ond time running backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This represents an alter- native way of covering every frame in the video, but with closer visual similarity between VCT initializa- tion and the first frames encountered by the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Occurrences Only Forward (−−→ OO): The tracker is only evaluated on the object occurrences, when the ob- ject is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This simplifies the tracking task and al- lows us to dis-entangle the challenge of re-detection from that of simply tracking in an egocentric clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We specify protocols by concatenating the appropriate de- scriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We primarily consider VCT-−→ VS, VCT-←→ VC, VCT- −−→ OO, and OFFT-−−→ OO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4) in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We adopt common metrics in object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The most important ones are tracking F-score, precision, and recall [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' details on these metrics can be found in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Trackers are ranked mainly by the F-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We addition- ally consider average overlap (AO), success, precision, and normalized precision as short-term tracking metrics [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Comparison of SOT trackers We compare the performance of several CNN-based tracking algorithms on EgoTracks with the VCT-−→ VS eval- uation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Given the large number of existing track- ing algorithms, we do not aim to be exhaustive but select high-performing examples representative of different track- Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Qualitative results of different trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' ing principles, which we briefly describe here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' KYS [4] and DiMP [3] are two typical short-term tracking algorithms that maintain an online target representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' ToMP [47] and STARK [71] are two examples of the SOTA short- term trackers based on Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' GlobalTrack [27] is a global tracker that searches the entire search image for re-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' LTMU [8] is a high performance long-term tracker that combines a global tracker (GlobalTrack) with a local tracker (DiMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The performance of these trackers on EgoTracks are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Note, AO in this table is equivalent to the recall at the probability threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Qualitative results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We highlight several observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' First, the object pres- ence scores from most short-term trackers are not very use- ful, as can be seen from the low precision of KYS (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5), DiMP (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='91), and ToMP (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='46), while long-term track- ers like GlobalTrack and DiMP LTMU achieve higher pre- cisions at 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='28 and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This is expected as long-term trackers are designed to place more emphasis on high re- detection accuracy, though there clearly is still room for im- provement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' STARK achieves the second highest precision at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70, which is an exception as it has a second train- ing stage to teach the model to classify whether the ob- ject is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Second, more recent works such as ToMP and STARK achieve better F-score than previous short-term trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This could be partially due to advances in training strategies, more data, and Transformer-based architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We also include results using the principle of Tracking by Detection [1, 52]: a detector proposes 100 bounding boxes, and we select the best using cosine similarity of box features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We observe that an open-world detector GGN [66] trained on COCO [41] generalize reasonably well with or- acle matching, achieving 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 AO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' However, the associa- tion problem is very challenging, bringing down the AO to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Implementation details are in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Template VCT-VS Visual Crop Evaluation Direction VCT-VC VCT-00 OFFT-00 Occurrence First Frame 00empate emplateTable 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Comparing tracker initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The upper table com- pares trackers initialized from the first frame in each occurrence and tracking only that single occurrence (no re-detection or per- fect re-detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The lower table compares STARK whole-video performance, starting from video start frame vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' visual crop frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Method AO Success Pre Prenorm KYS [4] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='87 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='22 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='87 DiMP [3] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='80 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='13 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='98 ToMP [47] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='17 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='93 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='88 STARK [71] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='01 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='64 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='76 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='91 Method AO F-score Precision Recall STARK - VCT-−→ VS 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='48 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='17 STARK - VCT-←→ VC 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='01 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='02 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='31 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Alternative evaluation protocols We perform additional evaluations according to alterna- tive evaluation protocols, to gain further insight to tracker performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' To decouple the re-detection problem from the other egocentric aspects of EgoTracks, we run experi- ments with the OFFT-−−→ OO protocol, which ignores the neg- ative frames of the video and thus obviates the need for re- detection, with results in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Perhaps unsurprisingly, all trackers do significantly better in this setting, though there remains much room for improvement, emphasizing the challenging nature of EgoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We also run exper- iments with STARK in the VCT-←→ VC setting (Table 5), in which case the initial template is temporally adjacent to the first tracked frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Here we see a 3-4% improvement to AO, F-score, precision, and recall compared to the VCT- −→ VS protocol, illustrating that trackers like STARK are de- signed to expect gradual transitions in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Both these experiments illustrate that the re-detection problem is a significant challenge for tracking and the need for better long-term benchmarks requiring more re-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Effect of attributes on tracking performance We use the validation set tracklet attribute annotations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3 to further understand performance on our evaluation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' For each attribute, we split the track- lets into two groups, corresponding to the attribute being true and false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We then use a standard STARK tracker [71] and report AO for each group of tracklets using the OFFT- −−→ OO evaluation protocol in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' As might be expected, we find that when objects are being actively used by the user or in the midst of a transformation, AO tends to be lower, by roughly 6%, likely due to occlusions or changes in ap- pearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Additionally, STARK tends to have a harder time when the object is hard to recognize in the image, whether due to occlusions, blur, scale, or other conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' OFFT-−→ OO AO of standard STARK model [71] for each attribute, averaged across tracklets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Attribute True False is active 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='65 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='73 is transformed 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='31 is recognizable 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='52 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='65 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Training and test-time hyperparameters comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Method AO F-score Precision Recall Data STARK 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='48 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='17 STARK - ft on VQ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='94 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='53 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='13 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='33 STARK - ft on EgoTracks 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='25 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='20 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='06 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='99 Augmentation STARK - ft on VQ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='94 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='53 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='13 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='33 STARK - ft + multiscale 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='44 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='65 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='30 Search window search size = 320 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='48 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='17 search size = 480 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='21 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='69 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='95 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19 search size = 640 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='09 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='39 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='15 search size = 800 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='08 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='60 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Ego-STARK Despite not being specifically designed for long-term tracking, Section 4 suggests STARK [71] to be the most competitive tracker on EgoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We thus leverage this tracker for additional analysis, suggesting several improve- ments to boost performance on EgoTracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Dataset Finetuning We first demonstrate how STARK trained on third- person videos significantly benefits from finetuning on ego- centric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We experiment with two versions of Ego- Tracks: the Ego4D VQ response track dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' short- term subset of EgoTracks) and the full EgoTracks training set, which contains more point-of-view and scale variations, as well as hard negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' As shown in Table 7, finetun- ing on the VQ response track subset improves the F-score from 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='48% to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='53%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Using the full EgoTracks annota- tion further improves the F-score by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='67% to 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This demonstrates that: 1) finetuning with egocentric data closes certain domain gaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2) training on full EgoTracks provides additional gains, showing the value of our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Adjusting Spatiotemporal Priors Modern trackers often embrace spatiotemporal priors on object motion, appearance and surroundings, which helped them on past datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' However, some of these design deci- sions translate poorly to long-term egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Search window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' An example is the local search as- sumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Many trackers assume the tracked object appears within a certain range of its previous location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Thus, for ef- ficiency, these methods often search within a local window of the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This is reasonable in high FPS, smooth videos with relatively small motion, commonly in previous short-term tracking datasets, but in egocentric videos, the object’s pixel coordinates can change rapidly with frequent large head motions, and re-detection becomes a key prob- Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' STARK with different context ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Row in bold is the default STARK setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' CR: context ratio, SRR: search region ratio, SIS: search image size (in image resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Method AO F-score Precision Recall Setting CR SRR SIS Same SIS 1x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5x 320 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='22 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='16 2x 5x 320 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='94 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='53 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='13 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='33 3x 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5x 320 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='70 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='03 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='28 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='59 4x 10x 320 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='32 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='98 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='31 Same SRR 1x 5x 640 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='50 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='09 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='31 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='91 3x 5x 208 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='87 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='36 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='54 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='79 Same CR 2x 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5x 480 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='21 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='69 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='95 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='19 2x 10x 640 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='09 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='39 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='15 2x 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5x 800 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='08 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='60 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='45 lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Therefore, we experiment with expanded search re- gions beyond what are common in past methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' As we expand the search size from 320 up to 800, we see dramatic improvements in AO, F-score, Precision, and Recall (Ta- ble 7), as STARK is able to correctly locate objects that were previously outside of its search window due to the rapid movement of egocentric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Multiscale augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The characteristics of egocen- tric video also affect common SOT assumptions of object scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Many trackers are trained with the assumption that an object’s scale is consistent with the template image and between adjacent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' However, large egocentric cam- era motions, locomotion, and hand interactions with ob- jects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' bringing an object to one’s face, as in eating) can translate to objects rapidly undergoing large changes in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We thus propose adding scale augmentations dur- ing training, randomly resizing the search image by a factor of s ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' While simple, we find this dramatically improves performance on EgoTracks, improving STARK’s AO by nearly 10% and F-score by more than 8% (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Context ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Past SOT works have found that includ- ing some of the background can be helpful when extracting features from the template image, with 2 times the size of object being common practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We experiment with differ- ent context ratios to see if this rule of thumb transfers to egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Because of the local window assump- tion, the sizes of the template image and search image are related: Search Image Size(SIS) Search Region Ratio(SRR) = Template Image Size Context Ratio(CR) = Object Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The template image size is set to a fixed size 128 × 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' When changing the context ratio, we carefully control the other parameters for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The re- sults are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Among all three parameters - CR, SRR, and SIS, the search region size (determined by SRR and SIS) has the highest impact on the F-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This is expected because there are frequent re-detections, which require the tracker to search in a larger area for the object, rather than just within the commonly used local window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Varying the CR has mixed results so we adhere to the com- mon practice of using a CR of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The best result is achieved when using SRR 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='5, which covers most of the image and Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' STARK with different numbers of templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Method AO F-score Precision Recall STARK - 1 template 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='97 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='42 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='80 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='04 STARK - 3 templates 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='76 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='84 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='84 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='57 STARK - 5 templates 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='47 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='03 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='82 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='45 STARK - 7 templates 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='81 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='83 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='77 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='40 STARK - 9 templates 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='92 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='89 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='36 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='12 achieves a F-score of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='74%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Multiple templates Transformer-based architectures can encode arbitrary length inputs, making it straightforward to consume fea- tures from an arbitrary number of templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' The origi- nal STARK design encodes two templates: the initialization and a single dynamically updated template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' A natural exten- sion is to include more templates of the target, which may expose the transformer to different views of the object (par- ticularly relevent in egocentric video), though low-quality views may compromise performance [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' What’s the right trade-off?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We experiment with differ- ent numbers of templates for a basic STARK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Moti- vated by potential applications where a user can take a short video of an object from different angles [55], we extend the single visual crop to a visual clip of templates by incorpo- rating additional frames from the same occurrence where the visual crop appears as the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We adopt a simple template sampling method: uniformly sampling 3, 5, 7, or 9 templates from the visual crop’s occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Uniformly sampling the videos temporally can be a simple yet effec- tive heuristic to gather diverse views from an occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We summarize the results in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' While we observe im- provements across all metrics using up to 5 templates, per- formance declines with more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We hypothesize that increas- ing the number of templates does increase the knowledge available to STARK for tracking, but after a certain point it may dilute the information in the templates and make it difficult for the transformer to synthesize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' This highlights the importance of template selection and multi-view fusion mechanisms, which inspires promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Conclusion We present EgoTracks, the first large-scale dataset for long-term egocentric visual object tracking in diverse scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We conduct extensive experiments to understand the performance of state-of-the-art trackers on this new dataset, and find that they struggle considerably, possibly in part due to overfitting to some of the simpler character- istics of existing benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' We thus propose several im- provements to the STARK [71] tracker, leading to a strong baseline that we call Ego-STARK, leading to vast improve- ments in performance on egocentric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Lastly, we plan to organize a public benchmark challenge using a held-out test set with a test server as a testbed for new tracking al- gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' By publicly releasing this dataset and organizing the challenge, we hope to encourage advancements in the field of long-term tracking and draw more attention to the challenges of long-term and egocentric videos for this field.' 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Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 10448–10457, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 2, 3, 5, 6, 7, 8 [72] Linjie Yang, Yuchen Fan, and Ning Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Video instance seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5188–5197, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 4 [73] Yipin Zhou and Tamara L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' Temporal perception and prediction in ego-centric video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' In 2015 IEEE International Conference on Computer Vision (ICCV), pages 4498–4506, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} +page_content=' 3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE1T4oBgHgl3EQffAR-/content/2301.03213v1.pdf'} diff --git a/gtE5T4oBgHgl3EQfhw_s/content/tmp_files/2301.05644v1.pdf.txt b/gtE5T4oBgHgl3EQfhw_s/content/tmp_files/2301.05644v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c957b8333706689e12ff4a4891c8bbbd23f80615 --- /dev/null +++ b/gtE5T4oBgHgl3EQfhw_s/content/tmp_files/2301.05644v1.pdf.txt @@ -0,0 +1,1121 @@ +1 + +Excitation-Dependent High-Lying Excitonic Exchange via Interlayer Energy +Transfer from Lower-to-Higher Bandgap 2D Material +Arka Karmakar1*, Tomasz Kazimierczuk1, Igor Antoniazzi1, Mateusz Raczyński1, Takashi +Taniguchi2, Kenji Watanabe3, Adam Babiński1, Abdullah Al-Mahboob4⸸, Maciej R. Molas1# +1 Division of Solid State Physics, Institute of Experimental Physics, Faculty of Physics, University of +Warsaw, Pasteura 5, 02-093 Warsaw, Poland +2 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 +Namiki, Tsukuba, Ibaraki 305-0044, Japan +3 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, +Ibaraki 305-0044, Japan +4 Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA +* arka.karmakar@fuw.edu.pl; ⸸ aalmahboo@bnl.gov; # maciej.molas@fuw.edu.pl +Keywords: 2D material, MoS2, WSe2, heterostructure, excitons, energy transfer, band-nesting + +High light absorption (~15%) and strong photoluminescence (PL) emission in monolayer (1L) transition- +metal dichalcogenide (TMD) makes it an ideal candidate for optoelectronic applications. Competing +interlayer charge (CT) and energy transfer (ET) processes control the photocarrier relaxation pathways in +TMD heterostructures (HSs). In TMDs, long-distance ET can survive up to several tens of nm, unlike the +CT process. Our experiment shows that an efficient ET occurs from the 1L WSe2 to 1L MoS2 with ~9 nm +interlayer hBN, due to the resonant overlapping of the high-lying excitonic states between the two TMDs, +resulting in enhanced HS MoS2 PL emission. This type of ET from the lower-to-higher optical bandgap +material has never been observed. With increasing temperature, the ET process becomes weaker due to the + +2 + +increased electron-phonon scattering, destroying the enhanced MoS2 emission. Our work provides a new +insight into the long-distance ET process and its effect on the photocarrier relaxation pathways. +Introduction: +Group-VI semiconducting transition metal dichalcogenides (TMDs) are formed by stacking of strongly +bonded two-dimensional (2D) X-M-X layers (M = transition metals such as Mo, W etc. and X = chalcogens +such as S, Se, and Te etc.), which are separated by the weakly bond interlayer van der Waals forces1. The +first mechanical exfoliation of the monolayer (1L) molybdenum disulfide (MoS2) film from the bulk crystal +in 2010 led us to observe a strong photoluminescence (PL) emission2,3 due to the indirect-to-direct bandgap +transition from the bulk-to-1L regime4,5. Since then, researchers have been exploring the exciting excitonic +properties6–11 in these 1L TMD materials. Strong light-matter interactions and high light absorption of up +to ~15% in the solar spectrum12 helped researchers to realize the future prospects of 1L TMD-based +optoelectronic device applications13. Heterostructures (HSs) made by the vertical stacking of different +layered materials have shown positive promise for future ultrathin14–16 and flexible17 optoelectronic device +applications. Recent advances in direct and patterned growth of 2D HSs18,19 to obtain a clean large-area +interface have also pushed the effort to make commercially available TMD-based device applications. +However, one of the major challenges in commercializing the promised optoelectronic device applications +is the lack of comprehensive understanding in the competing interlayer processes and their role in the +photocarrier recombination mechanism. +The interlayer charge (CT) and energy transfers (ET) are the two main carrier relaxation pathways in the +semiconductor HSs. Interlayer CT occurs due to the energy band offset in the HS20 and the 'traditional' ET +process happens when nonradiative energy from the excited donor material gets transferred to the acceptor +material via dipole-dipole interactions accompanied by a fluorescence emission from the acceptor +material21,22. ET is observed as a reduction of the donor fluorescence intensity and carrier lifetime followed +by an enhancement of the acceptor fluorescence intensity22. The interlayer CT can be stopped by placing a +thin layer of dielectric material in between the two semiconductors. Britnell et al.23 showed that only four + +3 + +atomic-layer thick hexagonal boron nitride (hBN) is sufficient as a dielectric medium to block the electron +tunneling between the two graphene layers. Unlike the CT process, in TMD HSs the long-distance interlayer +ET process can survive up to several tens of nm24,25. Thus, developing a comprehensive understanding of +the long-distance interlayer ET process is absolute necessity to create practical device applications. +In this work, we study the result of the excitation energy matching with the 1L tungsten diselenide (WSe2) +high-lying excitonic levels and its effect on the interlayer ET process to alter the photocarrier relaxation +pathways in 1L MoS2 with a ~9 nm thick hBN interlayer. Both these TMD materials have overlapping +higher energy B and C (MoS2)/D (WSe2) absorption features26,27. We show that resonant excitations at the +WSe2 B and D absorption regions results in MoS2 PL enhancement in the HS area. We report that this PL +enhancement is due to the interlayer ET process from the WSe2-to-MoS2 layer. This type of long-range ET +process from the lower-to-higher optical bandgap material was never observed before, since ET typically +happens from the higher-to-lower bandgap materials28–36. In this work, we employ multiple optical +spectroscopic techniques at cryogenic temperature (8 K); such as µ-PL, µ-photoluminescence excitation +(PLE) and differential reflectance contrast (RC), complemented by the density functional theory (DFT) +calculation of spin-resolved band structures to study the ET process. Our work reveals a unique interlayer +ET process in the TMD HSs. This will significantly contribute to creating a comprehensive understanding +of the long-range interlayer ET process and its role to influence the photocarrier radiative recombination +processes in these semiconducting HSs. +Results and discussion: +Figure 1a shows the optical micrograph of the fabricated MoS2-hBN-WSe2 HS (see methods for the +fabrication details). The inset of Figure 1a shows the schematic illustration of the side view of the sample. +We introduce ~9 nm thick interlayer hBN (see Supplementary Figure S1) to eliminate any effect related to +the interlayer CT in our system23. The optical absorption of the TMD materials reflects their single-electron +energy band structure. The low temperature RC spectra (see methods for the details) measured at 8 K show +strong overlaps between the B peaks of both materials and the WSe2 D peak with the MoS2 C peak (shaded + +4 + +areas in Figure 1b), which agrees well with the previously published reports26,27. In the later sections, we +discuss how these strong overlaps help us to observe the reported ET from the lower-to-higher bandgap +(WSe2-to-MoS2) material. HS spectrum (Figure 1b) shows similar RC resonance positions as compared to +the individual 1Ls, indicating no major strain-induced effect37 in the HS area. A and B excitonic peaks +occur due to the excitonic transitions at the K+/K- valley in the k-space2,3 and higher energy excitonic +transitions, such as C and D, are the results of the 'band nesting'38,39 in the Brillouin zone. 'Band-nested' +regions occur due to the identical dispersion in valence (VB) and conduction band (CB) over a region in +the Brillouin zone. For 1L MoS2, both the VB maximum and the CB minimum are located at the K+/K- +point in the Brillouin zone. In the case of WSe2, while the VB maximum is located at the K+/K- point, the +CB minimum is situated at the Λ point2,40. The 'band-nesting' region happens in between the Γ and Λ +point38,39. Figure 1c shows the DFT calculated electronic band structures (see Supplementary Information +for the details) along the Γ-K+ direction in the Brillouin zone. For both the band structures, we match the +optical bandgaps with the corresponding PL energies. All types of optical transitions are shown with +different colors of arrow (Figure 1c). PLE maps (see methods for the experimental details) taken at 8 K +show the emission landscapes of the three individual areas (Figures 1d-1f). After saturating the WSe2 +emission in the HS to visualize the MoS2 emission, we observe the significantly enhanced MoS2 PL +emission in the HS area as compared to the 1L region (Figures 2a-2b). The horizontal cuts at the excitation +energies of 2.85 eV and 2.12 eV (black dotted lines in Figures 2a-2b) reveal that the MoS2 PL emission in +the HS is enhanced by a factor of ~1.9 and ~1.7, respectively as compared to the 1L area (Figures 2c-2d). +The PL enhancement factor is defined here as the ratio of PL intensity in the HS area to the 1L area under +the same excitation (and accumulation) conditions. Similarly, PLE (vertical cut along the 1.92 eV emission +energy in Figures 2a-2b) shows an overall increase of the HS MoS2 PL emission throughout the entire +excitation range as compared to the 1L MoS2 region (Figure 2e). We conclude that the MoS2 PL +enhancement in the HS area is a result of an interlayer ET process from the WSe2 layer. It is important to +mention that the total optical absorption in the HS area did not change much as compared to the each 1L +areas (Figure 1b). However, the enhancement in the HS PLE (Figure 2e) suggests that the internal PL + +5 + +efficiency of the HS system was increased due to the ET process. We rule out the possibility of the observed +PL enhancement in the MoS2 emission due to the interference of the backscattering light, because the entire +measured MoS2 area (including the HS) is placed onto the same hBN thickness (inset of Figure 1a). 1L +WSe2 (thickness <1 nm) in the HS area cannot modulate the interference pattern considering the ~9 nm +interlayer and thick substrate hBNs. We also rule out the possible contribution of ET from the hBN defect +states41 in the HS MoS2 PL enhancement process, as the ET from same hBN thickness cannot result in more +HS PL emission as compared to the 1L MoS2 region. +Strong overlaps between the higher energy absorptions in both the investigated materials (Figure 1b) help +us to study the effect of the interlayer ET process under those 'resonant' excitation conditions. PL intensity +map taken at 8 K under the excitation of 2.12 eV (B resonances overlap region) shows an overall enhanced +MoS2 emission in the HS area (Figure 3a). Similarly, an excitation at 2.85 eV energy (WSe2 D and MoS2 +C peaks overlap region) shows an increased MoS2 PL emission throughout the HS area (Figure 3b). Thus, +proving that at both the excitation energies an efficient ET happened from the WSe2-to-MoS2 layer as +discussed in the later section. The PL intensity maps (Figures 3a-3b) also show that the observed +enhancement of the MoS2 PL emission in the HS area is not a localized phenomenon. We note that although +there is some non-uniformity in the HS PL intensity due to the typical inhomogeneous nature of the +exfoliated samples, but the HS PL emission is always higher than the 1L MoS2 area. +In order to study the effect of increasing temperature in our experiments, we performed PLE maps at 25 K, +100 K, and 200 K (Figure 4 and Figure S2). At 25 K, MoS2 emissions in the HS area under both the +excitation energies at ~2.83 eV and 2.2 eV show a similar enhancement factor of ~1.6 (Figure S3). These +values are a slight reduction from the 8 K data. The PLE also shows a similar overall enhancement in the +MoS2 HS emission at 25 K (Figure 4c). Upon further increasing the temperature at 100 K and 200 K, we +observe a complete vanishing of the MoS2 PL enhancement in the HS (Figures 4d-4f). A slight quenching +of the HS MoS2 PLE at 100 K (Figure 4f) could be due to the traditional type-II HS ET28 from the higher- +to-lower bandgap material (MoS2-to-WSe2). + +6 + +For MoS2 and WSe2, the schematics of the A and B transitions based on the VB and CB splitting are shown +in Figure 5a. In these TMD monolayers, VB (VB1 and VB2) and CB (CB1 and CB2) spin splitting occurs +due to the spin-orbit coupling and lack of inversion symmetry10,42, allowing possible absorptions based on +the optical selection rule43,44. In these TMDs, PL emission, which comes from the direct radiative +recombination at the optical bandgap, strongly depends on the spin-state of CB (CB1 or CB2) electron and +VB (VB1 or VB2) hole at the K+/K- point. Based on the allowed electron recombination from the CB1 or +CB2 to the hole situated at the top of VB (VB2), the materials are divided into two categories; 'bright' or +'dark'10, respectively. The calculated momentum-space energy landscape for the allowed optical transitions +from VB2-to-CB1 and VB1-to-CB2 in the MoS2 layer shows a smaller separation of ~150 meV at the K+/K- +point due to the spin splitting (Figures 5b-5c, Figure S4a), which matches well with the previous results45,46. +WSe2 shows a comparatively larger separation of ~500 meV at the K+/K- point47,48 for the VB2-to-CB2 and +VB1-to-CB1 transitions (Figures 5d-5e, Figure S4b). +Optical excitation at the 'band-nested' region (MoS2 C and WSe2 D peak), excites electrons in the valley in +between the Γ-Λ point in MoS2 CB and around the Λ valley in WSe2 CB. These excited photocarriers +(electron and hole) instantly relax to their immediate band extreme points; Λ valley for electron and Γ hill +for hole26. These carriers then further transfer to the band extrema via the extremely fast (<500 fs) +intravalley scattering (kiv)49–51. In our HS, to describe the PL intensity map under the 2.12 eV excitation +(Figure 3a), the only possible mechanism is shown as a schematic illustration in Figure 5f. Upon excitation +with the 2.12 eV photons, photoexcited carries are generated at the WSe2 B excitonic level. Due to the +resonant overlap with the MoS2 B level (Figure 1b), WSe2 B excitonic energy immediately transfer to the +MoS2 B and A band, resulting in more carriers in the MoS2 layer. The extra carriers at the MoS2 B level +transfer to the subsequent band extremum via intervalley transition (kv), followed by a radiative +recombination (kr) process to the ground state (GS). Thus, we obtain an enhanced MoS2 PL emissions in +the HS area with an excitation of 2.12 eV (Figure 3a). However, at an excitation energy of 2.85 eV (MoS2 +C and WSe2 D peak overlap region, Figure 1b), two possible ET channels can play a crucial role. First, ET + +7 + +from the WSe2 D level can directly generate more carriers at the MoS2 C level due to the resonant +overlapping. These extra carriers radiatively recombine at the band extremum via intravalley transition (kiv), +and giving rise to more MoS2 PL emissions in the HS area, as shown in the schematic of Figure 5g (grey +colored ET process). Another possibility is that upon excitation with the 2.85 eV photon carriers generated +at the WSe2 D level scatter to the WSe2 B level via the intravalley transition (kiv) and then transfer to the +MoS2 B and A level via ET process giving rise to the MoS2 PL emission similar as the 2.12 eV excitation +process (black colored ET process in Figure 5g). Interestingly, an excitation at the WSe2 C absorption peak +(2.56 eV) does not result in any MoS2 PL emission (Figure S5), indicating that interlayer coupling between +the suitable levels was not possible at this excitation due to the immediate photoexcited carrier transfer to +the WSe2 A level. Hence, no enhancement in the MoS2 HS PL emission due to the ET process is also +apparent. +Our model to describe the enhanced MoS2 PL emission from the HS area also supports the temperature- +dependent data. Photocarriers go through a series of phonon scattering before relaxing to the ground state. +At low temperature, electron-phonon scattering dominates52. With the increasing temperature, other types +of scattering processes such as anharmonic phonon–phonon scattering and phonon structure scattering53 +start to dominate. Thus, with the increasing temperature, the intravalley transition becomes weaker due to +the multiple-phonon scattering and eventually a minor fraction of the photocarriers generated at the 'band- +nested' region can be transferred to the K+/K- point for radiative recombination. Furthermore, the thermal +activation should make the 'hot' carrier transfer to the band extremum extremely faster (<100 fs)54, +preventing the coupling between the materials' corresponding energy levels. These eventually result in a +complete disappearance of the MoS2 PL enhancement in the HS area at higher temperatures (100 K and +200 K). +Considering the temperature-dependent data we can conclude that at higher excitation energy (~2.85 eV) +ET process via WSe2 B to MoS2 B and A level dominates (black colored ET process in Figure 5h) in our +experiment. Otherwise, with increasing the temperature we should observe an enhanced MoS2 HS PL + +8 + +emission. At cryogenic temperature, the fast intravalley scattering (kiv) in TMDs occur at ~100-500 fs +timescale49–51,54. Whereas, intervalley transitions (kv) occur at a longer timescale of a few ps range55,56. Our +study suggests that the reported ET happened at a faster timescale than the intervalley transition and slower +than the intravalley transition. Otherwise, the ET from the lower optical bandgap WSe2 cannot excite more +carriers in the higher bandgap MoS2, resulting in an enhanced HS MoS2 PL emission. Finding the 'true' ET +timescale in our experiment will require an ultrafast study, which is beyond the scope of this work. It is also +important to mention that with the increasing temperature the effect of band renormalization in the ET +process to alter the radiative recombination pathway of the photocarriers cannot be ignored. A thorough +investigation of the band renormalization effect in the ET process is required in the future work. +In conclusion, our study shows that strong light matter interaction in the 1L MoS2 and WSe2 'band-nested' +region allows us to observe an unusual ET process from the lower-to-higher bandgap (WSe2-to-MoS2) +material. All the previous reports28–36 showed that ET always occurs from the higher-to-lower bandgap (all +types of) low-dimensional materials (such as quantum dots, nanotubes, TMDs, perovskites, etc.) +irrespective of the type of band alignment. This is in a stark contrast to the observed ET process in our +work. The excitation-dependent PL intensity maps prove that the reported HS MoS2 PL enhancement is not +a localized phenomenon due to the materials local property change, the entire HS area shows this enhanced +PL emission. Finally, the temperature-dependent study proves that with the increasing temperature due to +the growing electron-phonon scattering, the carriers transfer to the band extremum become faster, +preventing ET from the WSe2 (smaller gap) to the MoS2 (larger gap) layer. Our findings provide a unique +insight into the interlayer ET process in these layered materials and will help to build a comprehensive +understanding about the competing interlayer processes for developing future TMD-based optoelectronic +device applications. +Methods: +HS fabrication + +9 + +Bottom hBN layer was directly cleaved on the SiO2/Si substrate. MoS2-hBN-WSe2 layers were exfoliated +onto the Gel-Pak (PDMS) films and were stacked layer-by-layer (in reverse order) onto each other using a +home-built semiautomatic transfer stage. MoS2, WSe2 and hBN bulk crystals for exfoliation were obtained +from the Graphene Supermarket, HQ Graphene and National Institute for Materials Science, respectively. +Characterization +We used Bruker Dimension Icon with NanoScope 6 controller in 'ScanAsyst' (peak force tapping) mode to +obtain high resolution AFM image. +The differential RC measurements were performed using a super-continuum light source (without a +monochromator) focused by a Nikon L Plan 100x (N.A. 0.7) objective and directed into a spectrometer. +Sample was loaded in a cryostat and cooled with continuous flow of liquid helium (LHe). The differential +reflectance is defined by (Rs-Rsub)/(Rs+Rsub), where Rs is the reflected light intensity from the TMD sample +areas and Rsub from the hBN/Si substrate. +We performed the µ-PL/PLE experiments by using a super-continuum light source coupled with a +monochromator as an excitation source. The incident light was focused using a Mitutoyo M Plan 50x (N.A. +0.75) objective. The excitation power was constant throughout the measurements and the average power +on the sample was kept ~50 µW (spot diameter ~1 µm) to avoid any high power induced nonlinear effects +from the sample. For PLE experiment sample was loaded in a LHe cryostat to reach the minimum +temperature of ~5 K during the experiments. +Data availability: +All the data necessary to conclude the results are presented in the manuscript and supplementary +information. +Acknowledgements: + +10 + +The work has been supported by the National Science Centre, Poland (grant no. 2017/27/B/ST3/00205 and +2018/31/B/ST3/02111). K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers +19H05790, 20H00354 and 21H05233). Authors acknowledge the help received from the research staffs at +the Center of New Technologies (CeNT) in University of Warsaw. +Author contributions: +A.K. and A.A.M. conceived the project. A.K., A.A.M. and M.R.M. designed the experiments. A.K. did the +sample fabrication. 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Nature Communications 7, 13279 (2016). +55. Song, Y. & Dery, H. Transport Theory of Monolayer Transition-Metal Dichalcogenides through +Symmetry. Phys. Rev. Lett. 111, 026601 (2013). +56. Glazov, M. M. et al. Exciton fine structure and spin decoherence in monolayers of transition metal +dichalcogenides. Phys. Rev. B 89, 201302 (2014). + +15 + + + +Figure 1: Optical characterization of the MoS2-hBN-WSe2 heterostructure (HS). (a) Optical micrograph +of the HS. Inset is the schematic illustration of the sample cross-section. (b) Differential reflectance contrast +(RC) spectra from the three areas on the sample taken at 8 K. Shaded areas indicate the higher energy +excitonic resonances between MoS2 and WSe2. HS shows the characteristics lower energy absorptions from +both the WSe2 (AW) and MoS2 (AM) layer. (c) Single particle band structure of MoS2 and WSe2 along the Γ- +K direction indicating the different optical transitions. Optical bandgaps were matched with the PL +energies. (d)-(f) Photoluminescence excitation (PLE) maps of the three areas taken at 8 K showing the +change of emission intensity as a function of excitation energy. + + +(a) +8 K, WSe2 +52 +3.0 +HS +MoS, +Intensity (a.u.) × 102 +39 +2.8 +2.6 +MoS2 +26 +wu +Interlayer +WSe2 +hBN +hBN +2.4 +9 +13 +hBN +SiO2/Si +10 μm +2.2 +0 +1.6 +1.7 +1.8 +1.9 +2.0 +(b) +0.3 +A↓ +EmissionEnergy(eV) +0 +D +B +8 K, MoS2 +-0.3 +15 +3.0 +-0.6 +WSe2 +8 K +Intensity (a.u.) × 102 +12 +2.8 +0.3 +A +9 +0 +C +2.6 +B +6 +-0.3 +2.4 +-0.6 +3 +MoS2 +2.2 +0.3 +1.6 +1.7 +1.8 +1.9 +2.0 +0 +Emission Energy (eV) +-0.3 +(f) +8 K, HS +-0.6 +(eV) +120 +HS +3.01.0 +2.0 +2.0 +3.0 +Excitation Energy +Intensity (a.u.) × 10 +90 +2.8 +Energy (eV) +(c) +2.6 +60 +K +K +2.4 +30 +K +C +B +D +B +2.2 +K+ +K +0 +1.6 +1.7 +1.8 +1.9 +2.0 +V +K『 +V +K +Emission Energy (eV) +MoS2 +WSe216 + + + +Figure 2: MoS2 PLE intensity comparison between the HS and monolayer area. (a)-(b) PLE maps of the +HS and MoS2 area with the same intensity range taken at 8 K. WSe2 emission intensity in the HS map is +kept saturated to visualize the MoS2 emission. (c)-(d) (MoS2 in) HS and MoS2 PL emission intensities at +2.85 eV and 2.12 eV excitation energies, respectively (along the horizontal dotted lines in Figures 2(a)- +(b)). Under both the excited energies, MoS2 emissions in the HS are significantly enhanced as compared to +the 1L area. (e) Comparison of HS and MoS2 excitation profile at 1.92 eV emission energy (along the +vertical solid lines in Figures 2(a)-(b)). Overall MoS2 shows enhanced PLE intensity in the HS area. + + + + + + + + +(a) +(q) +8 K, HS +8 K, MoS2 +Excitation Energy (eV) +15 +Excitation Energy (eV) +15 +3.0 +3.0 +Intensity (a.u.) × 102 +12 +Intensity (a.u.) × 102 +12 +2.8 +2.8 +9 +9 +2.6 +2.6 +6 +9 +2.4 +2.4 +3 +3 +2.2 +2.2 +0 +1.6 +1.7 +1.8 +1.9 +2.0 +1.6 +1.7 +1.8 +1.9 +2.0 +Emission Energy (eV) +Emission Energy (eV) +(c) +(d) +(e) +Excitation at 2.85 eV +Excitation at 2.12 eV +Emission at 1.92 eV +Intensity (a.u.) × 102 +Intensity (a.u.) × 102 +Intensity (a.u.) x 102 +15 +8 +15 +S +HS +.. MoS2 +MoS2 +6 +MoS2 +10 +10 +4 +5 +5 +2 +0 +0 +0 +1.6 +1.7 +1.8 +1.9 +2.0 +1.6 +1.7 +1.8 +1.9 +2.0 +2.2 2.4 2.6 2.8 3.0 +Emission Energy (eV) +Emission Energy (eV) +Excitation Energy (eV)17 + + +Figure 3: MoS2 PL intensity maps at WSe2 B and D resonant excitations. (a)-(b) MoS2 photoluminescence +(PL) intensity maps at 8 K under 2.12 eV and 2.85 eV excitation energy, respectively. MoS2 emission in the +HS area shows an overall increased PL emission. The scale bars represent 5 µm length. + + + + + + + + + + + + + +(a) +(b) +8 K, MoS2 +8 K, MoS2 +180 +180 +Intensity (a.u.) +Intensity (a.u.) +135 +120 +90 +HS +60 +45 +5 μm +Exc. 2.12 eV +Exc. 2.85 eV +0 +018 + + +Figure 4: MoS2 PLE intensity comparison with increasing temperature. (a)-(b) HS and MoS2 PLE maps +at 25 K. (c) HS and MoS2 PLE comparison along the vertical lines in (a)-(b). HS shows a slightly reduced +MoS2 PLE enhancement as compared to the 8 K map. (d)-(e) HS and MoS2 PLE maps taken at 100 K. (f) +Similar HS and MoS2 PLE comparison at 100 K. MoS2 in the HS area does not show any intensity +enhancement at 100 K as compared to the 1L area. In all the HS maps, WSe2 emission intensities are kept +saturated to visualize the MoS2 emission. + + + + + + + + + + + + + +(a) +(b) +(c) +HS Emission at 1.89 eV +25 K, HS +25 K, MoS2 +140 + (eV) +MoS, Emission at 1.89 eV +140 +3.0 +3.0 +(n'e) +120 +Energy +25 K +Intensity (a.u.) +105 +2.8 +2.8 +Intensity (a.u.) +105 +90 +Intensity +2.6 +70 +2.6 +70 +60 +Excitation +2.4 +2.4 +30 +35 +35 +2.2 +2.2 +0 +1.61.71.81.92.0 +1.61.71.81.92.0 +2.2 2.4 2.6 2.8 3.0 +Emission Energy (eV) +EmissionEnergy (eV) +Excitation Energy (eV) +(d) +(e) +(f) +HS Emission at 1.88 eV +100 K, MoS2 +100 K, HS +(eV) +60 +60 +MoS, Emission at 1.88 ev +3.0 +3.0 +(a.u.) +60 +Excitation Energy +100 K +Intensity (a.u.) +45 +2.8 +Intensity (a.u.) +45 +Intensity +40 +30 +30 +20 +2.4 +15 +15 +2 +2 +PL +0 +1.61.71.81.9 2.0 +1.61.71.81.92.0 +2.2 2.4 2.6 2.8 3.0 +Emission Energy (eV) +Emission Energy (eV) +ExcitationEnergv(eV19 + + +Figure 5: Calculated spin-resolved energy landscape of MoS2 and WSe2. (a) Schematic illustration of the +valence (VB) and conduction band (CB) splitting at the K valley in MoS2 and WSe2, respectively. (b)-(c) +Calculated MoS2 optical transitions along the K--Γ-K+ direction from VB2 to CB1 and VB1 to CB2 (as +shown in (a)), respectively. (d)-(e) Similar calculated WSe2 momentum-space energy landscape along the +K--Γ-K+ direction from VB2 to CB2 and VB1 to CB1 (as shown in (a)), respectively. (f)-(g) Schematic +illustration of the photocarrier relaxation pathways from the higher energy levels to the ground state (GS) +in MoS2 due to the energy transfer (ET) from WSe2 after resonant excitation at (WSe2) B and D excitonic +level, respectively. Different types of transition are shown in the MoS2 layer; such as intravalley scattering +(kiv), intervalley transition (kv), and radiative recombination (kr). + + + + + + + + + + + + +(a) +MoS2 +WSe2 +CB2 +CB2 +(f) +CB1 +CB1 +DCB +BTT +A +B +VB2 +A +VB2 +ET +A +Ex. +VB1 +VB1 +E +K +K +GS +GS +(b) +(c) +MoS2: CB1-VB2 +MoS2: CB2-VB1 +MoS2 +WSe2 +3.4 +3.4 +3.1 +3.1 +2.8 +2.8 +(g) +2.5 +2.5 +C +DCB +K +K+ +2.2 +K +K+ +2.2 +ET +Vkiv +1.9 +1.9 +B +(d) +A +WSe2: CB2-VB2 +WSe2: CB1-VB1 +ET +E +A +3.7 +3.7 +WM +(eV) +3.3 +3.3 +GS +GS +2.9 +2.9 +MoS2 +WSe2 +2.5 +2.5 +K- +K+ +2.1 +K +K+ +2.1 +1.7 +1.720 + +SUPPORTING INFORMATION +Excitation-Dependent High-Lying Excitonic Exchange via Interlayer Energy +Transfer from Lower-to-Higher Bandgap 2D Material +Arka Karmakar1*, Tomasz Kazimierczuk1, Igor Antoniazzi1, Mateusz Raczyński1, Takashi +Taniguchi2, Kenji Watanabe3, Adam Babiński1, Abdullah Al-Mahboob4⸸, Maciej R. Molas1# +1 Division of Solid State Physics, Institute of Experimental Physics, Faculty of Physics, University of +Warsaw, Pasteura 5, 02-093 Warsaw, Poland +2 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 +Namiki, Tsukuba, Ibaraki 305-0044, Japan +3 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, +Ibaraki 305-0044, Japan +4 Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA +* arka.karmakar@fuw.edu.pl +⸸ aalmahboo@bnl.gov +# maciej.molas@fuw.edu.pl + + + + + + + + +21 + +Details of the theoretical calculations: +We computed the ground state band structure of 1Ls MoS2 and WSe2 employing the density functional +theory (DFT) calculations using the Materials Studio CASTEP (CAmbridge Serial Total Energy Package) +version 2021 HF1, ab initio Total Energy Program (first principles methods using CASTEP)1. Prior to the +band structure calculation, we performed the geometry optimization (GO) for the bulk crystal structure +using DFT-D (GGA + dispersion correction) method ˗ Perdew-Bruke-Ernzerhof (PBE) GGA functional2 +along with the dispersion correction (van der Waals correction accounted employing the dispersion +correction for DFT) by Tkatchenko-Scheffler (TS) method3, which was performed using the DFT Semi- +Empirical Dispersion Interaction Correction (DFT-SEDC) module4. We obtained the electron relativistic +correction using the DSPP (DFT-Semicore Pseudopotential)5. During the GO of the bulk structure, +symmetry constrained was imposed considering the International Table #194 (hexagonal, symmetry group +P63/MMC, crystal class 6/m m m) for the bulk MoS2 and WSe2. Following the bulk geometry optimization, +crystal was cleaved parallel to the layer (c* terminated) and then a vacuum slab > 20 Å was added along +the c* to make the 1L TMD structures. Final GO for the atomic arrangement within the 1L and the in-plane +lattice parameters were further optimized constraining the 2D lattice symmetry employing the identical +GGA functional and dispersion correction as above but also including the spin-orbit coupling in the total +energy calculations. In order to include the spin-orbit coupling, norm-conserving potentials in CASTEP +were generated using the kinetic energy optimization scheme developed by Lin et al.6. The spin orbit +coupling was included using the j-dependent pseudopotentials developed for CASTEP based on the work +by ref.7. Following the final step of GO, band structure calculation was performed considering the ultra- +fine k-spacing (k-spacing in single point energy calculation corresponding to 50x50x1 supercell or better +and spectral k-spacing of 0.0005Å-1). +After computation of the electronic band structure in CASTEP, scissors have applied to the band structure +plot to match with the bandgap obtained from the PL spectroscopy measurements. + + +22 + +References: +1. Clark, S. J. et al. First principles methods using CASTEP. Zeitschrift für Kristallographie - +Crystalline Materials 220, 567–570 (2005). +2. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. +Rev. Lett. 77, 3865–3868 (1996). +3. Tkatchenko, A. & Scheffler, M. Accurate Molecular Van Der Waals Interactions from Ground-State +Electron Density and Free-Atom Reference Data. Phys. Rev. Lett. 102, 073005 (2009). +4. McNellis, E. R., Meyer, J. & Reuter, K. Azobenzene at coinage metal surfaces: Role of dispersive +van der Waals interactions. Phys. Rev. B 80, 205414 (2009). +5. Delley, B. Hardness conserving semilocal pseudopotentials. Phys. Rev. B 66, 155125 (2002). +6. Lin, J. S., Qteish, A., Payne, M. C. & Heine, V. Optimized and transferable nonlocal separable ab +initio pseudopotentials. Phys. Rev. B 47, 4174–4180 (1993). +7. Corso, A. D. & Conte, A. M. Spin-orbit coupling with ultrasoft pseudopotentials: Application to Au +and Pt. Phys. Rev. B 71, 115106 (2005). + + + + +23 + + + + +Figure S1: (a) Optical micrograph of the HS. Black line indicates the line region of the AFM image. (b) +AFM height profile of the interlayer hBN shows the thickness of ~9 nm. + + + + + + +Figure S2: (a)-(b) PLE maps of the HS and MoS2 at 200 K, respectively. MoS2 PL emission does not +increase in the HS area. WSe2 emission in the HS data is saturated to visualize the MoS2 emission. Both the +plots have the same intensity range. + + + + +(a) +(b) +Height Profile (nm) +12 +HS +MoS +9 +1LWSe2 +6 +3 +Interlayer +0 +hBN +0 +2 +3 +4 +5 +6 +7 +Distance (um)(a) +(b) +200 K, HS +200 K, MoS2 +40 +40 +Energy (eV) +3.0 +3.0 +30 +30 +2.8 +Intensity (a.u.) +2.8 +Intensity (a.u.) +2.6 +20 +2.6 +20 +Excitation +2.4 +2.4 +10 +10 +2.2 +2.2 +0 +1.6 +1.7 +1.8 +1.9 +2.0 +1.6 +1.7 +1.8 +1.9 +2.0 +Emission Energy (eV) +Emission Energy (eV)24 + + +Figure S3: (a) Top and bottom panel shows PL emission of the MoS2 in the HS area under excitation at +2.83 eV and 2.2 eV, respectively. (b) PL emission profile from the 1L MoS2 area under same excitation +conditions. MoS2 PL emission in the HS area shows similar enhancement factor of ~ 1.6 at both excitation +energies. + + + + + +Figure S4: Calculated spin-resolved momentum-space optical absorption energy landscape of 1L (a) MoS2 +and (b) WSe2 along the Γ-K direction in the Brillouin zone. + + + +(a) +(q) +25 K, HS +25 K, MoS2 +Exc. 2.83 eV +Exc. 2.83 eV +120 +120 +80 +80. +ensity (a.u.) +(a.u.) +40 +40 +ensity +0 +0 +V120 +Inte +120 +80 +80 +40 +40 +0 +0 +1.6 +1.7 +1.8 +1.9 +2.0 +1.6 +1.7 +1.8 +1.9 +2.0 +Emission Energy (eV) +Emission Energy (eV)(a) +(b) +MoS,: CB1-VB2 +WSe2: CB2-VB2 +: MoS2: CB2-VB1 +WSe2: CB1-VB1 +3.6 +3.3 +(eV) +3.2 +3.0 +2.8 +2.7 +2.4 +2.4 +CB +CB +2.1 +2.0 +E +E +1.8 +1.6 +K +K25 + + + + +Figure S5: PL Intensity maps at the resonant WSe2 C excitation (~ 2.56 eV). (a) MoS2 does not show any +PL emission at this excitation energy. Only system noise was detected in this condition. (b) WSe2 PL +emission map does not show any intensity variation in the HS area as compared to the 1L region. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Molas1# 1 Division of Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Institute of Experimental Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' University of Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Pasteura 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 02-093 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Poland 2 International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Ibaraki 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Japan 3 Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Ibaraki 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Japan 4 Center for Functional Nanomaterials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' USA * arka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='karmakar@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='pl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' ⸸ aalmahboo@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' # maciej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='molas@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='pl Keywords: 2D material, MoS2, WSe2, heterostructure, excitons, energy transfer, band-nesting High light absorption (~15%) and strong photoluminescence (PL) emission in monolayer (1L) transition- metal dichalcogenide (TMD) makes it an ideal candidate for optoelectronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Competing interlayer charge (CT) and energy transfer (ET) processes control the photocarrier relaxation pathways in TMD heterostructures (HSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In TMDs, long-distance ET can survive up to several tens of nm, unlike the CT process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our experiment shows that an efficient ET occurs from the 1L WSe2 to 1L MoS2 with ~9 nm interlayer hBN, due to the resonant overlapping of the high-lying excitonic states between the two TMDs, resulting in enhanced HS MoS2 PL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' This type of ET from the lower-to-higher optical bandgap material has never been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' With increasing temperature, the ET process becomes weaker due to the 2 increased electron-phonon scattering, destroying the enhanced MoS2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our work provides a new insight into the long-distance ET process and its effect on the photocarrier relaxation pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Introduction: Group-VI semiconducting transition metal dichalcogenides (TMDs) are formed by stacking of strongly bonded two-dimensional (2D) X-M-X layers (M = transition metals such as Mo, W etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and X = chalcogens such as S, Se, and Te etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' ), which are separated by the weakly bond interlayer van der Waals forces1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The first mechanical exfoliation of the monolayer (1L) molybdenum disulfide (MoS2) film from the bulk crystal in 2010 led us to observe a strong photoluminescence (PL) emission2,3 due to the indirect-to-direct bandgap transition from the bulk-to-1L regime4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Since then, researchers have been exploring the exciting excitonic properties6–11 in these 1L TMD materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Strong light-matter interactions and high light absorption of up to ~15% in the solar spectrum12 helped researchers to realize the future prospects of 1L TMD-based optoelectronic device applications13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Heterostructures (HSs) made by the vertical stacking of different layered materials have shown positive promise for future ultrathin14–16 and flexible17 optoelectronic device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Recent advances in direct and patterned growth of 2D HSs18,19 to obtain a clean large-area interface have also pushed the effort to make commercially available TMD-based device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' However, one of the major challenges in commercializing the promised optoelectronic device applications is the lack of comprehensive understanding in the competing interlayer processes and their role in the photocarrier recombination mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The interlayer charge (CT) and energy transfers (ET) are the two main carrier relaxation pathways in the semiconductor HSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Interlayer CT occurs due to the energy band offset in the HS20 and the 'traditional' ET process happens when nonradiative energy from the excited donor material gets transferred to the acceptor material via dipole-dipole interactions accompanied by a fluorescence emission from the acceptor material21,22." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' ET is observed as a reduction of the donor fluorescence intensity and carrier lifetime followed by an enhancement of the acceptor fluorescence intensity22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The interlayer CT can be stopped by placing a thin layer of dielectric material in between the two semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Britnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='23 showed that only four 3 atomic-layer thick hexagonal boron nitride (hBN) is sufficient as a dielectric medium to block the electron tunneling between the two graphene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Unlike the CT process, in TMD HSs the long-distance interlayer ET process can survive up to several tens of nm24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Thus, developing a comprehensive understanding of the long-distance interlayer ET process is absolute necessity to create practical device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In this work, we study the result of the excitation energy matching with the 1L tungsten diselenide (WSe2) high-lying excitonic levels and its effect on the interlayer ET process to alter the photocarrier relaxation pathways in 1L MoS2 with a ~9 nm thick hBN interlayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Both these TMD materials have overlapping higher energy B and C (MoS2)/D (WSe2) absorption features26,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We show that resonant excitations at the WSe2 B and D absorption regions results in MoS2 PL enhancement in the HS area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We report that this PL enhancement is due to the interlayer ET process from the WSe2-to-MoS2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' This type of long-range ET process from the lower-to-higher optical bandgap material was never observed before, since ET typically happens from the higher-to-lower bandgap materials28–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In this work, we employ multiple optical spectroscopic techniques at cryogenic temperature (8 K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' such as µ-PL, µ-photoluminescence excitation (PLE) and differential reflectance contrast (RC), complemented by the density functional theory (DFT) calculation of spin-resolved band structures to study the ET process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our work reveals a unique interlayer ET process in the TMD HSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' This will significantly contribute to creating a comprehensive understanding of the long-range interlayer ET process and its role to influence the photocarrier radiative recombination processes in these semiconducting HSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Results and discussion: Figure 1a shows the optical micrograph of the fabricated MoS2-hBN-WSe2 HS (see methods for the fabrication details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The inset of Figure 1a shows the schematic illustration of the side view of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We introduce ~9 nm thick interlayer hBN (see Supplementary Figure S1) to eliminate any effect related to the interlayer CT in our system23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The optical absorption of the TMD materials reflects their single-electron energy band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The low temperature RC spectra (see methods for the details) measured at 8 K show strong overlaps between the B peaks of both materials and the WSe2 D peak with the MoS2 C peak (shaded 4 areas in Figure 1b), which agrees well with the previously published reports26,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In the later sections, we discuss how these strong overlaps help us to observe the reported ET from the lower-to-higher bandgap (WSe2-to-MoS2) material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' HS spectrum (Figure 1b) shows similar RC resonance positions as compared to the individual 1Ls, indicating no major strain-induced effect37 in the HS area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" A and B excitonic peaks occur due to the excitonic transitions at the K+/K- valley in the k-space2,3 and higher energy excitonic transitions, such as C and D, are the results of the 'band nesting'38,39 in the Brillouin zone." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" 'Band-nested' regions occur due to the identical dispersion in valence (VB) and conduction band (CB) over a region in the Brillouin zone." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' For 1L MoS2, both the VB maximum and the CB minimum are located at the K+/K- point in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In the case of WSe2, while the VB maximum is located at the K+/K- point, the CB minimum is situated at the Λ point2,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" The 'band-nesting' region happens in between the Γ and Λ point38,39." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Figure 1c shows the DFT calculated electronic band structures (see Supplementary Information for the details) along the Γ-K+ direction in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' For both the band structures, we match the optical bandgaps with the corresponding PL energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' All types of optical transitions are shown with different colors of arrow (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' PLE maps (see methods for the experimental details) taken at 8 K show the emission landscapes of the three individual areas (Figures 1d-1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' After saturating the WSe2 emission in the HS to visualize the MoS2 emission, we observe the significantly enhanced MoS2 PL emission in the HS area as compared to the 1L region (Figures 2a-2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The horizontal cuts at the excitation energies of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV (black dotted lines in Figures 2a-2b) reveal that the MoS2 PL emission in the HS is enhanced by a factor of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 and ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7, respectively as compared to the 1L area (Figures 2c-2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The PL enhancement factor is defined here as the ratio of PL intensity in the HS area to the 1L area under the same excitation (and accumulation) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Similarly, PLE (vertical cut along the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92 eV emission energy in Figures 2a-2b) shows an overall increase of the HS MoS2 PL emission throughout the entire excitation range as compared to the 1L MoS2 region (Figure 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We conclude that the MoS2 PL enhancement in the HS area is a result of an interlayer ET process from the WSe2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' It is important to mention that the total optical absorption in the HS area did not change much as compared to the each 1L areas (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' However, the enhancement in the HS PLE (Figure 2e) suggests that the internal PL 5 efficiency of the HS system was increased due to the ET process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We rule out the possibility of the observed PL enhancement in the MoS2 emission due to the interference of the backscattering light, because the entire measured MoS2 area (including the HS) is placed onto the same hBN thickness (inset of Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 1L WSe2 (thickness <1 nm) in the HS area cannot modulate the interference pattern considering the ~9 nm interlayer and thick substrate hBNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We also rule out the possible contribution of ET from the hBN defect states41 in the HS MoS2 PL enhancement process, as the ET from same hBN thickness cannot result in more HS PL emission as compared to the 1L MoS2 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Strong overlaps between the higher energy absorptions in both the investigated materials (Figure 1b) help us to study the effect of the interlayer ET process under those 'resonant' excitation conditions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' PL intensity map taken at 8 K under the excitation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV (B resonances overlap region) shows an overall enhanced MoS2 emission in the HS area (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Similarly, an excitation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV energy (WSe2 D and MoS2 C peaks overlap region) shows an increased MoS2 PL emission throughout the HS area (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Thus, proving that at both the excitation energies an efficient ET happened from the WSe2-to-MoS2 layer as discussed in the later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The PL intensity maps (Figures 3a-3b) also show that the observed enhancement of the MoS2 PL emission in the HS area is not a localized phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We note that although there is some non-uniformity in the HS PL intensity due to the typical inhomogeneous nature of the exfoliated samples, but the HS PL emission is always higher than the 1L MoS2 area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In order to study the effect of increasing temperature in our experiments, we performed PLE maps at 25 K, 100 K, and 200 K (Figure 4 and Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' At 25 K, MoS2 emissions in the HS area under both the excitation energies at ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='83 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 eV show a similar enhancement factor of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 (Figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' These values are a slight reduction from the 8 K data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The PLE also shows a similar overall enhancement in the MoS2 HS emission at 25 K (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Upon further increasing the temperature at 100 K and 200 K, we observe a complete vanishing of the MoS2 PL enhancement in the HS (Figures 4d-4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A slight quenching of the HS MoS2 PLE at 100 K (Figure 4f) could be due to the traditional type-II HS ET28 from the higher- to-lower bandgap material (MoS2-to-WSe2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 6 For MoS2 and WSe2, the schematics of the A and B transitions based on the VB and CB splitting are shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In these TMD monolayers, VB (VB1 and VB2) and CB (CB1 and CB2) spin splitting occurs due to the spin-orbit coupling and lack of inversion symmetry10,42, allowing possible absorptions based on the optical selection rule43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In these TMDs, PL emission, which comes from the direct radiative recombination at the optical bandgap, strongly depends on the spin-state of CB (CB1 or CB2) electron and VB (VB1 or VB2) hole at the K+/K- point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Based on the allowed electron recombination from the CB1 or CB2 to the hole situated at the top of VB (VB2), the materials are divided into two categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" 'bright' or 'dark'10, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The calculated momentum-space energy landscape for the allowed optical transitions from VB2-to-CB1 and VB1-to-CB2 in the MoS2 layer shows a smaller separation of ~150 meV at the K+/K- point due to the spin splitting (Figures 5b-5c, Figure S4a), which matches well with the previous results45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' WSe2 shows a comparatively larger separation of ~500 meV at the K+/K- point47,48 for the VB2-to-CB2 and VB1-to-CB1 transitions (Figures 5d-5e, Figure S4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Optical excitation at the 'band-nested' region (MoS2 C and WSe2 D peak), excites electrons in the valley in between the Γ-Λ point in MoS2 CB and around the Λ valley in WSe2 CB." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' These excited photocarriers (electron and hole) instantly relax to their immediate band extreme points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Λ valley for electron and Γ hill for hole26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' These carriers then further transfer to the band extrema via the extremely fast (<500 fs) intravalley scattering (kiv)49–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In our HS, to describe the PL intensity map under the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV excitation (Figure 3a), the only possible mechanism is shown as a schematic illustration in Figure 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Upon excitation with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV photons, photoexcited carries are generated at the WSe2 B excitonic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Due to the resonant overlap with the MoS2 B level (Figure 1b), WSe2 B excitonic energy immediately transfer to the MoS2 B and A band, resulting in more carriers in the MoS2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The extra carriers at the MoS2 B level transfer to the subsequent band extremum via intervalley transition (kv), followed by a radiative recombination (kr) process to the ground state (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Thus, we obtain an enhanced MoS2 PL emissions in the HS area with an excitation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' However, at an excitation energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV (MoS2 C and WSe2 D peak overlap region, Figure 1b), two possible ET channels can play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' First, ET 7 from the WSe2 D level can directly generate more carriers at the MoS2 C level due to the resonant overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' These extra carriers radiatively recombine at the band extremum via intravalley transition (kiv), and giving rise to more MoS2 PL emissions in the HS area, as shown in the schematic of Figure 5g (grey colored ET process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Another possibility is that upon excitation with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV photon carriers generated at the WSe2 D level scatter to the WSe2 B level via the intravalley transition (kiv) and then transfer to the MoS2 B and A level via ET process giving rise to the MoS2 PL emission similar as the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV excitation process (black colored ET process in Figure 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Interestingly, an excitation at the WSe2 C absorption peak (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='56 eV) does not result in any MoS2 PL emission (Figure S5), indicating that interlayer coupling between the suitable levels was not possible at this excitation due to the immediate photoexcited carrier transfer to the WSe2 A level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Hence, no enhancement in the MoS2 HS PL emission due to the ET process is also apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our model to describe the enhanced MoS2 PL emission from the HS area also supports the temperature- dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Photocarriers go through a series of phonon scattering before relaxing to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' At low temperature, electron-phonon scattering dominates52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' With the increasing temperature, other types of scattering processes such as anharmonic phonon–phonon scattering and phonon structure scattering53 start to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Thus, with the increasing temperature, the intravalley transition becomes weaker due to the multiple-phonon scattering and eventually a minor fraction of the photocarriers generated at the 'band- nested' region can be transferred to the K+/K- point for radiative recombination." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Furthermore, the thermal activation should make the 'hot' carrier transfer to the band extremum extremely faster (<100 fs)54, preventing the coupling between the materials' corresponding energy levels." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' These eventually result in a complete disappearance of the MoS2 PL enhancement in the HS area at higher temperatures (100 K and 200 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Considering the temperature-dependent data we can conclude that at higher excitation energy (~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV) ET process via WSe2 B to MoS2 B and A level dominates (black colored ET process in Figure 5h) in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Otherwise, with increasing the temperature we should observe an enhanced MoS2 HS PL 8 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' At cryogenic temperature, the fast intravalley scattering (kiv) in TMDs occur at ~100-500 fs timescale49–51,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Whereas, intervalley transitions (kv) occur at a longer timescale of a few ps range55,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our study suggests that the reported ET happened at a faster timescale than the intervalley transition and slower than the intravalley transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Otherwise, the ET from the lower optical bandgap WSe2 cannot excite more carriers in the higher bandgap MoS2, resulting in an enhanced HS MoS2 PL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Finding the 'true' ET timescale in our experiment will require an ultrafast study, which is beyond the scope of this work." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' It is also important to mention that with the increasing temperature the effect of band renormalization in the ET process to alter the radiative recombination pathway of the photocarriers cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A thorough investigation of the band renormalization effect in the ET process is required in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" In conclusion, our study shows that strong light matter interaction in the 1L MoS2 and WSe2 'band-nested' region allows us to observe an unusual ET process from the lower-to-higher bandgap (WSe2-to-MoS2) material." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' All the previous reports28–36 showed that ET always occurs from the higher-to-lower bandgap (all types of) low-dimensional materials (such as quantum dots, nanotubes, TMDs, perovskites, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') irrespective of the type of band alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' This is in a stark contrast to the observed ET process in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The excitation-dependent PL intensity maps prove that the reported HS MoS2 PL enhancement is not a localized phenomenon due to the materials local property change, the entire HS area shows this enhanced PL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Finally, the temperature-dependent study proves that with the increasing temperature due to the growing electron-phonon scattering, the carriers transfer to the band extremum become faster, preventing ET from the WSe2 (smaller gap) to the MoS2 (larger gap) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Our findings provide a unique insight into the interlayer ET process in these layered materials and will help to build a comprehensive understanding about the competing interlayer processes for developing future TMD-based optoelectronic device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Methods: HS fabrication 9 Bottom hBN layer was directly cleaved on the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2-hBN-WSe2 layers were exfoliated onto the Gel-Pak (PDMS) films and were stacked layer-by-layer (in reverse order) onto each other using a home-built semiautomatic transfer stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2, WSe2 and hBN bulk crystals for exfoliation were obtained from the Graphene Supermarket, HQ Graphene and National Institute for Materials Science, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=" Characterization We used Bruker Dimension Icon with NanoScope 6 controller in 'ScanAsyst' (peak force tapping) mode to obtain high resolution AFM image." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The differential RC measurements were performed using a super-continuum light source (without a monochromator) focused by a Nikon L Plan 100x (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7) objective and directed into a spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Sample was loaded in a cryostat and cooled with continuous flow of liquid helium (LHe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The differential reflectance is defined by (Rs-Rsub)/(Rs+Rsub), where Rs is the reflected light intensity from the TMD sample areas and Rsub from the hBN/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We performed the µ-PL/PLE experiments by using a super-continuum light source coupled with a monochromator as an excitation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The incident light was focused using a Mitutoyo M Plan 50x (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='75) objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The excitation power was constant throughout the measurements and the average power on the sample was kept ~50 µW (spot diameter ~1 µm) to avoid any high power induced nonlinear effects from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' For PLE experiment sample was loaded in a LHe cryostat to reach the minimum temperature of ~5 K during the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Data availability: All the data necessary to conclude the results are presented in the manuscript and supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Acknowledgements: 10 The work has been supported by the National Science Centre, Poland (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2017/27/B/ST3/00205 and 2018/31/B/ST3/02111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' acknowledge support from the JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Authors acknowledge the help received from the research staffs at the Center of New Technologies (CeNT) in University of Warsaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Author contributions: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' conceived the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' designed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' did the sample fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' performed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' performed the theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' interpreted the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' provided the bulk hBN for exfoliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' wrote the manuscript with feedback taken from all the coauthors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Competing interests: Authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Mattheiss, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' & Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 108, 196802 (2012).' metadata={'source': 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coherence lifetime in monolayer transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Nature Communications 7, 13279 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' & Dery, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Transport Theory of Monolayer Transition-Metal Dichalcogenides through Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 111, 026601 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Glazov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' B 89, 201302 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 15 Figure 1: Optical characterization of the MoS2-hBN-WSe2 heterostructure (HS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) Optical micrograph of the HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Inset is the schematic illustration of the sample cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (b) Differential reflectance contrast (RC) spectra from the three areas on the sample taken at 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Shaded areas indicate the higher energy excitonic resonances between MoS2 and WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' HS shows the characteristics lower energy absorptions from both the WSe2 (AW) and MoS2 (AM) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (c) Single particle band structure of MoS2 and WSe2 along the Γ- K direction indicating the different optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Optical bandgaps were matched with the PL energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (d)-(f) Photoluminescence excitation (PLE) maps of the three areas taken at 8 K showing the change of emission intensity as a function of excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) 8 K, WSe2 52 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 HS MoS, Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 MoS2 26 wu Interlayer WSe2 hBN hBN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 9 13 hBN SiO2/Si 10 μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 A↓ EmissionEnergy(eV) 0 D B 8 K, MoS2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 WSe2 8 K Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 A 9 0 C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 B 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 3 MoS2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 0 Emission Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 (f) 8 K, HS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 (eV) 120 HS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Excitation Energy Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 10 90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 Energy (eV) (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 60 K K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 30 K C B D B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 K+ K 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 V K『 V K Emission Energy (eV) MoS2 WSe216 Figure 2: MoS2 PLE intensity comparison between the HS and monolayer area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a)-(b) PLE maps of the HS and MoS2 area with the same intensity range taken at 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' WSe2 emission intensity in the HS map is kept saturated to visualize the MoS2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (c)-(d) (MoS2 in) HS and MoS2 PL emission intensities at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV excitation energies, respectively (along the horizontal dotted lines in Figures 2(a)- (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Under both the excited energies, MoS2 emissions in the HS are significantly enhanced as compared to the 1L area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (e) Comparison of HS and MoS2 excitation profile at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92 eV emission energy (along the vertical solid lines in Figures 2(a)-(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Overall MoS2 shows enhanced PLE intensity in the HS area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (q) 8 K, HS 8 K, MoS2 Excitation Energy (eV) 15 Excitation Energy (eV) 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 12 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 9 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 6 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) Emission Energy (eV) (c) (d) (e) Excitation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV Excitation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV Emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92 eV Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') × 102 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') x 102 15 8 15 S HS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='. MoS2 MoS2 6 MoS2 10 10 4 5 5 2 0 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) Emission Energy (eV) Excitation Energy (eV)17 Figure 3: MoS2 PL intensity maps at WSe2 B and D resonant excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a)-(b) MoS2 photoluminescence (PL) intensity maps at 8 K under 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV excitation energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2 emission in the HS area shows an overall increased PL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The scale bars represent 5 µm length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (b) 8 K, MoS2 8 K, MoS2 180 180 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 135 120 90 HS 60 45 5 μm Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='12 eV Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='85 eV 0 018 Figure 4: MoS2 PLE intensity comparison with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a)-(b) HS and MoS2 PLE maps at 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (c) HS and MoS2 PLE comparison along the vertical lines in (a)-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' HS shows a slightly reduced MoS2 PLE enhancement as compared to the 8 K map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (d)-(e) HS and MoS2 PLE maps taken at 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (f) Similar HS and MoS2 PLE comparison at 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2 in the HS area does not show any intensity enhancement at 100 K as compared to the 1L area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In all the HS maps, WSe2 emission intensities are kept saturated to visualize the MoS2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (b) (c) HS Emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='89 eV 25 K, HS 25 K, MoS2 140 (eV) MoS, Emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='89 eV 140 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content="0 (n'e) 120 Energy 25 K Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 105 90 Intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 70 60 Excitation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 30 35 35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) EmissionEnergy (eV) Excitation Energy (eV) (d) (e) (f) HS Emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='88 eV 100 K, MoS2 100 K, HS (eV) 60 60 MoS, Emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='88 ev 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 60 Excitation Energy 100 K Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 45 Intensity 40 30 30 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 15 15 2 2 PL 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) Emission Energy (eV) ExcitationEnergv(eV19 Figure 5: Calculated spin-resolved energy landscape of MoS2 and WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) Schematic illustration of the valence (VB) and conduction band (CB) splitting at the K valley in MoS2 and WSe2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (b)-(c) Calculated MoS2 optical transitions along the K--Γ-K+ direction from VB2 to CB1 and VB1 to CB2 (as shown in (a)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (d)-(e) Similar calculated WSe2 momentum-space energy landscape along the K--Γ-K+ direction from VB2 to CB2 and VB1 to CB1 (as shown in (a)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (f)-(g) Schematic illustration of the photocarrier relaxation pathways from the higher energy levels to the ground state (GS) in MoS2 due to the energy transfer (ET) from WSe2 after resonant excitation at (WSe2) B and D excitonic level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Different types of transition are shown in the MoS2 layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' such as intravalley scattering (kiv), intervalley transition (kv), and radiative recombination (kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) MoS2 WSe2 CB2 CB2 (f) CB1 CB1 DCB BTT A B VB2 A VB2 ET A Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' VB1 VB1 E K K GS GS (b) (c) MoS2: CB1 VB2 MoS2: CB2 VB1 MoS2 WSe2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 (g) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='5 C DCB K K+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 K K+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 ET Vkiv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 B (d) A WSe2: CB2 VB2 WSe2: CB1 VB1 ET E A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 WM (eV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 GS GS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 MoS2 WSe2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='5 K K+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='1 K K+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='720 SUPPORTING INFORMATION Excitation-Dependent High-Lying Excitonic Exchange via Interlayer Energy Transfer from Lower-to-Higher Bandgap 2D Material Arka Karmakar1*, Tomasz Kazimierczuk1, Igor Antoniazzi1, Mateusz Raczyński1, Takashi Taniguchi2, Kenji Watanabe3, Adam Babiński1, Abdullah Al-Mahboob4⸸, Maciej R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Molas1# 1 Division of Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Institute of Experimental Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' University of Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Pasteura 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 02-093 Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Poland 2 International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Ibaraki 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Japan 3 Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Ibaraki 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Japan 4 Center for Functional Nanomaterials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' NY 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' USA * arka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='karmakar@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='pl ⸸ aalmahboo@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='gov # maciej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='molas@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='pl 21 Details of the theoretical calculations: We computed the ground state band structure of 1Ls MoS2 and WSe2 employing the density functional theory (DFT) calculations using the Materials Studio CASTEP (CAmbridge Serial Total Energy Package) version 2021 HF1, ab initio Total Energy Program (first principles methods using CASTEP)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Prior to the band structure calculation, we performed the geometry optimization (GO) for the bulk crystal structure using DFT-D (GGA + dispersion correction) method ˗ Perdew-Bruke-Ernzerhof (PBE) GGA functional2 along with the dispersion correction (van der Waals correction accounted employing the dispersion correction for DFT) by Tkatchenko-Scheffler (TS) method3, which was performed using the DFT Semi- Empirical Dispersion Interaction Correction (DFT-SEDC) module4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' We obtained the electron relativistic correction using the DSPP (DFT-Semicore Pseudopotential)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' During the GO of the bulk structure, symmetry constrained was imposed considering the International Table #194 (hexagonal, symmetry group P63/MMC, crystal class 6/m m m) for the bulk MoS2 and WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Following the bulk geometry optimization, crystal was cleaved parallel to the layer (c* terminated) and then a vacuum slab > 20 Å was added along the c* to make the 1L TMD structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Final GO for the atomic arrangement within the 1L and the in-plane lattice parameters were further optimized constraining the 2D lattice symmetry employing the identical GGA functional and dispersion correction as above but also including the spin-orbit coupling in the total energy calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' In order to include the spin-orbit coupling, norm-conserving potentials in CASTEP were generated using the kinetic energy optimization scheme developed by Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' The spin orbit coupling was included using the j-dependent pseudopotentials developed for CASTEP based on the work by ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Following the final step of GO, band structure calculation was performed considering the ultra- fine k-spacing (k-spacing in single point energy calculation corresponding to 50x50x1 supercell or better and spectral k-spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0005Å-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' After computation of the electronic band structure in CASTEP, scissors have applied to the band structure plot to match with the bandgap obtained from the PL spectroscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 22 References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Clark, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' First principles methods using CASTEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Zeitschrift für Kristallographie - Crystalline Materials 220, 567–570 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' & Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Generalized Gradient Approximation Made Simple.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 102, 073005 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' B 80, 205414 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Delley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Hardness conserving semilocal pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' B 66, 155125 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', Qteish, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=', Payne, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' & Heine, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Optimized and transferable nonlocal separable ab initio pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' B 47, 4174–4180 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Corso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' & Conte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Spin-orbit coupling with ultrasoft pseudopotentials: Application to Au and Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' B 71, 115106 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 23 Figure S1: (a) Optical micrograph of the HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Black line indicates the line region of the AFM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (b) AFM height profile of the interlayer hBN shows the thickness of ~9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Figure S2: (a)-(b) PLE maps of the HS and MoS2 at 200 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2 PL emission does not increase in the HS area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' WSe2 emission in the HS data is saturated to visualize the MoS2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Both the plots have the same intensity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (b) Height Profile (nm) 12 HS MoS 9 1LWSe2 6 3 Interlayer 0 hBN 0 2 3 4 5 6 7 Distance (um)(a) (b) 200 K, HS 200 K, MoS2 40 40 Energy (eV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 30 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 20 Excitation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 10 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) Emission Energy (eV)24 Figure S3: (a) Top and bottom panel shows PL emission of the MoS2 in the HS area under excitation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='83 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (b) PL emission profile from the 1L MoS2 area under same excitation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' MoS2 PL emission in the HS area shows similar enhancement factor of ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 at both excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Figure S4: Calculated spin-resolved momentum-space optical absorption energy landscape of 1L (a) MoS2 and (b) WSe2 along the Γ-K direction in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (q) 25 K, HS 25 K, MoS2 Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='83 eV Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='83 eV 120 120 80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' ensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 40 40 ensity 0 0 V120 Inte 120 80 80 40 40 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 Emission Energy (eV) Emission Energy (eV)(a) (b) MoS,: CB1 VB2 WSe2: CB2 VB2 : MoS2: CB2 VB1 WSe2: CB1 VB1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='3 (eV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='4 CB CB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='0 E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='6 K K25 Figure S5: PL Intensity maps at the resonant WSe2 C excitation (~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='56 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) MoS2 does not show any PL emission at this excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Only system noise was detected in this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (b) WSe2 PL emission map does not show any intensity variation in the HS area as compared to the 1L region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' Scale bars represent 5 µm length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' (a) (b) 36 240 MoS2 WSe2 26 180 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=') 16 120 6 60 8 K 4 8 K Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='56 eV Exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} +page_content='56 eV 14 0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE5T4oBgHgl3EQfhw_s/content/2301.05644v1.pdf'} diff --git a/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/2301.08346v1.pdf.txt b/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/2301.08346v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d32c9ae52725a10b889bdc869d4f015287984c0 --- /dev/null +++ b/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/2301.08346v1.pdf.txt @@ -0,0 +1,1669 @@ +arXiv:2301.08346v1 [math-ph] 19 Jan 2023 +A critical survey of twisted spectral triples +beyond the Standard Model +Manuele Filaci +University of Cracovia, Institute fo Physics, Jagiellonian University +prof. Stanis�lawa �Lojasiewicza 11, 30-348 Krakow, Poland +E-mail: manuele.filaci@uj.edu.pl +Pierre Martinetti +Universit`a di Genova (dpto di matematica) & INFN, +via Dodecaneso, 16146 Genova, Italia +E-mail: martinetti@dima.unige.it +Abstract. +We review the applications of twisted spectral triples to the Standard +Model. The initial motivation was to generate a scalar field, required to stabilise the +electroweak vacuum and fit the Higgs mass, while respecting the first-order condition. +Ultimately, it turns out that the truest interest of the twist lies in a new – and +unexpected – field of 1-forms, which is related to the transition from Euclidean to +Lorentzian signature. +1. Introduction +From the pioneering work of [35] till the full formalism of Connes [16], noncommutative +geometry provides a unified description of the Lagrangian of the Standard Model of +fundamental interactions (electromagnetism, weak and strong interactions) [21][9][8]; +minimally coupled to the Einstein-Hilbert action of General Relativity [18]; including +right handed neutrinos [12]; where the Higgs boson comes out naturally on the same +footing as the other bosons, i.e. as the local expression of a connection 1-form. +The approach works very well on Riemannian manifolds. The generalisation to +pseudo-Riemannian geometry, in particular Lorentzian manifolds, is far from obvious +(there are various attempts in this direction, see for instance [1][2][38][53][3] and reference +within). +In addition, noncommutative geometry offers possibilities to go beyond the +Standard Model, by modifying the rules of the game in various ways: one may enlarge +the space of fermions [51, 52], or get rid of the first-order condition (defined below) +[14, 13], modify the real structure (also defined below) [7, 6], switch to non-associative +geometry [4, 5], use some structure of Clifford bundle in order to modify some of the +mathematical requirements defining a noncommutative geometry [26]. +For a recent +review of “beyond Standard Model” propositions in the framework of noncommutative +geometry, see [15]. +Here we focus on another class of variations around Connes’ initial model, obtained +by twisting the noncommutative geometry by an algebra automorphism [32][34][47]. + +A critical survey of twisted spectral triples +beyond the Standard Model +2 +All the possibilities above but the first are minimal extensions of the Stan- +dard Model, in that they yield an extra scalar field σ – suggested by particle physi- +cists to stabilize the electroweak vacuum – but do not touch the fermionic content. The +novelty of the twist is to generate another additional piece: a new field of 1-forms, which +suprisingly turns out to be related to the transition from Euclidean to Lorentzian sig- +nature [30]. In particular, in the example of electrodynamics, this field identifies with +the (dual) of the 4-momentum vector in Lorentzian signature, even though one started +with a Riemannian manifold [47]. +All this is explained as follows. In the next section we begin by some recalling on +the spectral description of the Standard Model [12]. We stress the process of fluctuation +of the metric, which is the way to generate bosonic fields in noncommutative geometry +by turning the constant parameters of the model into fields. +In section 3 we describe the model of grand algebra developed in [32], which aimed +at generating the extra scalar field σ, while respecting the first-order condition. The idea +is to start with an algebra bigger than the one of the Standard Model, in order to have +more “space” to generate bosonic fields by fluctuations of the metric. This model does +indeed generate the expected field σ, by letting the Majorana mass of the neutrinos +fluctuate. Even though the first-order condition associated with this Majorana mass +is preserved, the problem moves to the free Dirac operator: not only does the latter +break the first-order condition, but its commutator with the algebra is unbounded, in +contradiction with the very definition of spectral triple. This kind of problem is typically +solved by twisting the spectral triple in the sense of Connes and Moscovici [24]. +A +twisting of the grand algebra down to the Standard Model has been worked out in [34], +but we show in §3.3 that this does not define stricto-sensu a twisted spectral triple, for +only the part of the algebra relevant for the extra scalar field has been twisted. +Applying the twist to the whole algebra suggests a general procedure to twist any +graded spectral triple, as recalled in section 4. +It consists in doubling the algebra +one is beginning with, rather than finding it from the reduction of a bigger algebra. +Such a “twisting up” procedure has been studied in a couple of papers [41][42]. There +is some freedom in the construction, especially in the choice of the twisting operator +whose eigenspaces determine the representation of the doubled algebra. By choosing +the grading as the twisting operator, one automatically satisfies the twisted first-order +condition. However, when applied to the spectral triple of the Standard Model, this +twist-by-grading does not generate any extra scalar field. +Some preliminary results, +mentioned in §4.3, indicate that this is a general feature of the twisting-up procedure: +the twisted first-order condition and the extra scalar field are mutually exclusive. Hence +the necessity to adapt to the twisted case the fluctuations without first-order condition +introduced in [14]. This has been done in [49] and is summarised in §4.3. +Section 5 deals with what might be the truest interest of the twist, namely +the unexpected field of 1-forms arising from the twisted fluctuation. In the example +of electrodynamics [47],[54], this field identifies with the dual of the 4-momentum in +Lorentzian signature, even though one started with a Riemannian spectral triple. + +A critical survey of twisted spectral triples +beyond the Standard Model +3 +2. The spectral description of the Standard Model +We begin with the definition of spectral triple, which is the central tool in Connes’ +noncommutative geometry, emphasising how the bosonic fields – including the Higgs +field – are obtained as connection 1-forms, through the process of fluctuation of the +metric. We then summarise the spectral description of the Standard Model. +2.1. Spectral triple +A spectral triple [16] consists of an algebra A acting on a Hilbert space H together with a +selfadjoint operator D with compact resolvent, whose commutator [D, a] is bounded for +any a ∈ A. It is graded if it comes with a selfadjoint operator Γ on H which squares +to the identity operator I, anticommutes with D and commutes with the algebra. A +spectral triple is real [17] if in addition there is an antilinear operator J on H satisfying +J2 = ǫI, +JD = ǫ′DJ, +JΓ = ǫ′′ΓJ +(1) +where ǫ, ǫ′, ǫ′′ = ±1 define the KO-dimension k ∈ [0, 7]. This real structure implements +a map a → a◦ := Ja∗J−1 from A to the opposite algebra A◦. This yields a right action +of A on H, ψa := a◦ψ, which is asked to commute with the left action +[a, Jb∗J−1] = 0 +∀a ∈ A +(order zero condition). +(2) +There is also a first-order condition [18], +[[D, a], Jb∗J−1] = 0 +∀a, b ∈ A +(3) +which is there to guarantee that the operator D be a first-order differential operator. +All these properties are satisfied by the triple +(C∞(M), L2(M, S), /∂) +(4) +where C∞(M) is the (commutative) algebra of smooth functions on a closed Riemannian +manifold M of dimension m, acting by multiplication on the Hilbert space L2(M, S) of +square-integrable spinors on M, and +/∂ = −i +m +� +µ=1 +γµ(∂µ + ωµ), +with +γµγν + γνγµ = 2gµνI +(5) +is the Dirac operator (ωµ is the spin connection, γµ the Dirac matrices and gµν the +Riemannian metric on M, all given in local coordinates). For m even, this spectral +triple has grading the product of the Dirac matrices (in dimension 4, the matrix γ5) and +real structure J the charge conjugation operator. Adding other conditions [20] (which +are satisfied by the triple (4)), one gets Connes’ reconstruction theorem, that extends +Gelfand duality (between compact topological spaces and C∗-commutative algebras) +beyond topology. +Namely, given any real spectral triple (A, H, D) satisfying these +conditions, with A commutative, then there exists a closed Riemannian manifold M +such that A ≃ C∞(M). +A noncommutative geometry is then defined as a spectral triple (A, H, D) where +A is non (necessarily) commutative. This gives access to new geometrical objects, that +can be intended as “spaces” whose algebra of functions A is not commutative. + +A critical survey of twisted spectral triples +beyond the Standard Model +4 +2.2. Connection +Take a gauge theory with gauge group G. From a mathematical point of view, the +fermionic fields form sections of a G-bundle E over the spacetime M, while the bosonic +fields are described as connections on E. +In noncommutative geometry the spacetime M is substituted by a spectral triple +(A, H, D), where A plays the role of “algebra of functions” on the noncommutative +space. +To understand what plays the role of a gauge bundle, recall that the set of +sections of any bundle on a manifold M forms a finite projective C∞(M)-module. +Conversely, by Serre-Swan theorem, any such module actually is the module of sections +of a bundle on M. That is why, in noncommutative geometry, the role of gauge bundles +is played by finite projective A-modules E. +Connections on these modules are, by definition, objects associated with a +derivation. +Recall that a derivation δ on an algebra A is a map from A to some +A-bimodule Ω satisfying the Leibniz rule +δ(ab) = aδ(b) + δ(a)b +∀a, b ∈ A. +(6) +A connection on a (right) A-module E associated with δ is a map from E to E ⊗A Ω +such that the following Leibniz rule holds, +∇(ηa) = ∇(η)a + η ⊗ δ(a) +∀η ∈ E, a ∈ A, +(7) +where +Ω = +�� +i +aiδ(bi), ai, bi ∈ A +� +(8) +is the A-bimodule generated by the derivation δ, while ∇(η)a is a shorthand notation +for η(0)a ⊗ η(1), using Sweedler notations ∇η = η(0) ⊗ η(1) with η(0) ∈ E and η(1) ∈ Ω. +Example: The exterior derivative d is a derivation on the algebra C∞(M). It generates +the C∞(M)-bimodule of section s of the cotangent bundle, +Ω1(M) := +�� +i +fidgi with fi, gi ∈ C∞(M) +� +. +(9) +A connection on the tangent bundle TM associated with d is a map +∇ : Γ∞(TM) → Γ∞(TM) ⊗ Ω1(M), +(10) +∂ν +�→ Γρ +µν∂ρ ⊗ dxµ, +(11) +where Γ∞(TM) denotes the set of smooth sections of TM. One retrieves the usual +notion of connection, as a map from Γ∞(TM) × Γ∞(TM) to Γ∞(TM) as +∇ : (∂η, ∂ν) �→ ∇η∂ν := Γρ +µν∂ρ ⊗C∞(M) ⟨dxµ, ∂η⟩ ≃ ⟨dxµ, ∂η⟩Γρ +µν∂ρ = Γρ +ην∂ρ, +where ⟨. , .⟩ is the C∞(M)-valued dual product between the cotangent and the tangent +bundles and the last equation is the isomorphism between E ⊗C∞(M) C∞(M) and E. + +A critical survey of twisted spectral triples +beyond the Standard Model +5 +2.3. Fluctuation of the metric +To understand when two algebras are “similar”, a relevant notion is Morita equivalence. +This is more flexible than isomorphism of algebras for, roughly speaking, two algebras +A and B are Morita equivalent if they have the same representation theory. Technically, +it means that there exists an Hermitian finite projective A-module E such that B is +isomorphic to the algebra EndA(E) of A-linear, adjointable, endormorphisms of E (for +details see e.g. [50] or [40]). +The idea of fluctuation of the metric comes from the following question: how does +one export a real spectral triple (A, H, D) to a Morita equivalent algebra B ? We recall +the construction of [18], whose details may be found in [23] and even more details in [42]. +Assume E = ER is a right A-module. The algebra B acts on HR := ER ⊗A H as +b(η ⊗ ψ) = bη ⊗ ψ +∀b ∈ B, η ∈ ER, ψ ∈ H. +(12) +However, the “natural” action of D on HR, +DR(η ⊗ ψ) := η ⊗ Dψ, +(13) +is not compatible with the tensor product, for +DR(ηa ⊗ ψ) − DR(η ⊗ aψ) = −η ⊗ [D, a]ψ +(14) +has no reason to vanish. This is cured by equipping ER with a connection ∇ with value +in the A-bimodule of generalised 1-forms +Ω1 +D(A) := +�� +i +ai[D, bi], ai, bi ∈ A +� +(15) +generated by the derivation δ(.) = [D, .]. Indeed, the following operator, +DR(η ⊗ ψ) := η ⊗ Dψ + (∇η)ψ +(16) +is well defined on HR, and selfadjoint as soon as ∇ is an hermitian connection. Moreover +one checks that the commutator [DR, b] is bounded for any b ∈ B, so that (B, HR, DR) +is a spectral triple. +In particular, if one considers self-Morita equivalence, that is +B = ER = A, then the operator (16) with ∇ hermitian reads +DR = D + AR +(17) +with AR = A∗ +R ∈ Ω1 +D(A) a selfadjoint generalised 1-form. +A similar construction holds if one implements Morita equivalence with a left +module EL. Then HL = H ⊗A EL is a Hilbert space and the operator +DL(ψ ⊗ η) := Dψ ⊗ η + (∇◦η)ψ +(18) +with ∇◦ a connection with value in the bimodule +Ω1 +D(A◦) = +�� +i +a◦ +i [D, b◦ +i ], +a◦ +i , b◦ +i ∈ A◦ +� + +A critical survey of twisted spectral triples +beyond the Standard Model +6 +is well defined on HL. For ∇◦ hermitian, one obtains a spectral triple (B, HL, DL). For +self-Morita equivalence, one gets +DL = D + A◦ = D + ǫ′J AL J−1 +(19) +with A◦ ∈ Ω1 +D(A◦) and AL ∈ Ω1 +D(A). +To take into account the real structure, one combines the two constructions, using +an A-bimodule E to implement Morita equivalence. For self-Morita equivalence, one +then obtains the operator D′ = D + AR + ǫ′J ALJ−1 acting on H. +Requiring this +operator to be selfadjoint and J to be a real structure amounts to the existence of a +generalised selfadjoint 1-form A ∈ Ω1 +D(A) such that +D′ = DA := D + A + ǫ′J AJ−1. +(20) +Then (A, H, DA) is a real spectral triple. The operator DA is called a covariant +Dirac operator, and the substitution of D with a DA is a fluctuation of the metric. +The name is motivated by the existing relation between the Dirac operator and the +metric. This relation is obvious on a spin manifold, from the very definition of the Dirac +matrices ( γνγν+γνγµ = 2gµν), and it still makes sense for an arbitrary noncommutative +geometry, via the definition of the spectral distance [22]. On a manifold, this distance +gives back the geodesic distance associated with the Riemannian structure of M, while +on an arbitrary spectral triple it may be seen as a generalisation of the Wasserstein +distance of order 1 in the theory of optimal transport (cf [28, 46] and references therein). +By exporting a noncommutative geometry to a Morita equivalent algebra, one passes +from D to the covariant operator DA and modifies accordingly the spectral distance. +In particular, for the Standard Model, such a fluctuation provides a purely metric +interpretation to the Higgs field (which is one of the components of the generalised +1-form A, see below) [18, 48]. The metric interpretation of the other components of A +has been worked out in [48, 44], in relation with the Carnot-Carath´eodory distance in +sub-Riemannian geometry. +2.4. Gauge transformation +A gauge transformation is a change of connection on the Morita-equivalence bimodule E. +In case of self-Morita equivalence, it is implemented by the conjugate action on H of +the group U(A) of unitaries element of A (i.e. u ∈ A such that u∗u = uu∗ = I): +Ad(u) : ψ → uψu∗ = u(u∗)◦ψ = uJuJ−1ψ +∀ψ ∈ H. +(21) +This action maps the covariant Dirac operator DA to +Ad(u) DA Ad(u)−1 +(22) +and one checks that this operator coincides with the operator DAu, defined as in (20) +with +Au := u[D, u∗] + uAu∗. +(23) +This is the formula of transformation of the gauge potential in noncommutative +geometry, which generalises the usual one of gauge theories. + +A critical survey of twisted spectral triples +beyond the Standard Model +7 +2.5. Standard Model +The spectral triple of the Standard Model [12] is the product +A = C∞(M) ⊗ AF, +H = L2(M, S) ⊗ HF, +D = /∂ ⊗ I96 + γ5 ⊗ DF +(24) +of the spectral triple (4) of a 4-dimensional Riemannian closed spin manifold M with a +finite dimensional spectral triple +AF = C ⊕ H ⊕ M3(C), +HF = C96, +DF = +� +D0 +048 +048 +D† +0 +� +� +�� +� +DY ++ +� +048 +DR +D† +R +048 +� +� +�� +� +DM +(25) +where H is the algebra of quaternions and M3(C) the algebra of complex 3×3 matrices. +The dimension of HF is the number of fermions in the Standard Model (including +right-handed neutrinos). Its entries are labelled by a multi-index C I α n where +• C = 0, 1 labels particles (C = 0) or anti-particles (C = 1); +• I = 0, i with i = 1, 2, 3 is the lepto-colour index: it takes value I = 0 for a lepton +and I = 1, 2, 3 for a quark with its three possible colours; +• α = ˙1, ˙2, 1, 2 is the flavour index (with dot indicating the chirality): +˙1 = νR, ˙2 = eR, 1 = νL, 2 = eL for leptons (I = 0), +(26) +˙1 = uR, ˙2 = dR, 1 = qL, 2 = dL for quarks (I = i); +(27) +• n = 1, 2, 3 is the generation index. +The details of the representation of AF is in [12]. The important point for our +matter is that the quaternions act only on the particle subspace of HF (C = 0), trivially +on the lepto-colour index I, and through their fundamental representation on the last +two flavour indices α; whereas M3(C) acts only on antiparticle subspace of HF (C = 1), +trivially on the flavour index α and through their fundamental representation on the +lepto-colour index i. The algebra C acts both on particles together with the quaternions +(but on the first two flavour indices), and on antiparticles together with M3(C) (on +I = 0). +The grading of the finite dimensional spectral triple is the 96 × 96 matrix ΓF with +entries +1 on left particles/right antiparticles, −1 on right particles/left antiparticles. +The real structure is the matrix JF that exchanges particles with antiparticles. The +spectral triple (24) is real, with grading Γ = γ5 ⊗ ΓF and real structure J = J ⊗ JF. +In the particles/antiparticles indices, +the Dirac operator DF +of the finite +dimensional spectral triple is the sum of a block diagonal matrix DY which contains +the Yukawa couplings of the fermions, the Cabibbo-Kobayashi-Maskawa mixing matrix +for the quarks and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix for the left- +handed neutrinos, and a block off-diagonal matrix DM which contains the Majorana +masses kn +R, n = 1, 2, 3 of the right-handed neutrinos and the corresponding mixing +matrix (notations are those of [36], they differ from the ones of [32] and [34]). + +A critical survey of twisted spectral triples +beyond the Standard Model +8 +The generalised 1-forms (15) for a product of spectral triples (24) decompose as [25] +A = γ5 ⊗ H − i +� +µ +γµ ⊗ Aµ +(28) +where H is a scalar field on M with values in AF, while Aµ is a 1-form field on M with +values in the Lie algebra of the group U(AF) of unitary elements of AF (differently said: +a connection 1-form on a U(AF)-bundle on TM). In particular, for the spectral triple +of the Standard Model, one has +U(AF) = U(C ⊕ H ⊕ M3(C)) ≃ U(1) × SU(2) × U(3), +(29) +which is reduced to the gauge group U(1) × SU(2) × SU(3) of the Standard Model by +imposing a unimodularity condition (which also guarantees that the model is anomaly +free, see e.g [12, §2.5]). +The action of this group on H is a matrix whose components are the hypercharges +of the fermions of the Standard Model [12, Prop. 2.16]. This allows to identify the basis +elements of HF with the 96 fermions of the Standard Model, and the corresponding +elements in H with the fermionic fields. Moreover, the action of A+JAJ−1 corresponds +to the bosonic hypercharges, and allows to identify the components of Aµ with the +bosonic fields of the Standard Model [12, Prop. 3.9]. One also checks that (23) yields +the expected gauge transformation. +The interpetation of the scalar field H follows from the computation of the spectral +action [8, 9], namely the asymptotic expansion Λ → ∞ of Tr f( D2 +A +Λ2 ) where f is +a smooth approximation of the characteristic function of the interval [0, 1]. +One +obtains the bosonic Lagrangian of the Standard Model coupled with Einstein-Hilbert +action in Euclidean signature, where H is the Higgs field. The coupling constants of +the electroweak and strong interactions satisfy the relation expected in grand unified +theories, and are related to the value at 0 of the function f. +The spectral action provides some relations between the parameters of the Standard +Model at a putative unification scale. The physical predictions are obtained by running +down the parameters of the theory under the renormalisation group equation, taking +these relations as initial conditions. +Assuming there is no new physics between the +unification scale and the electroweak scale, one finds a value for the Higgs mass around +170 GeV, in disagrement with the measured value 125, 1 GeV. +However, for a Higgs boson with mass mH ≤ 130 Gev, the quartic coupling λ of +the Higgs field becomes negative at high energy, meaning the electroweak vacuum is +meta-stable rather than stable [29]. This instability can be cured by a new scalar field +σ which couples to the Higgs field. In the spectral description of the Standard Model, +such a field is obtained by turning into a field the neutrino Majorana mass kR which +appears in the off-diagonal part DR of the finite dimensional Dirac operator DF: +kR → kRσ, +Furthermore, by altering the running of the parameters under the equations of the +group of renormalization, this extra scalar field makes the computation of the mass of +the Higgs boson compatible with its experimental value [11]. + +A critical survey of twisted spectral triples +beyond the Standard Model +9 +3. Grand algebra beyond the Standard Model +The point in the above is to justify the turning of the constant kR into a field kRσ. This +cannot be obtained by fluctuation of the metric, for one checks that +[γ5 ⊗ DM, a] = 0 +∀a, b ∈ A = C∞(M) ⊗ AF. +(30) +In other terms, the constant kR is transparent under fluctuation. The solution proposed +in [14] is to remove the first-order condition. This gives more flexibility, and permits +to obtain the extra scalar field as a fluctuation without the first-order condition. The +latter is retrieved dynamically, by minimising the spectral action [13]. In this way the +field σ is the “Higgs” boson associated with the breaking of the first-order condition. +3.1. Grand algebra +At the same time, an alternative process was described in [32] where one mixes the +internal degrees of freedom per generation of the finite dimensional Hilbert space HF, +that is HF ≃ C32, with the 4 spinorial degrees of freedom of L2(M, S). This provides +exactly the 4 × 32 = 128 degrees of freedom required to represent the “second next +algebra” in the classification of finite dimensional spectral triples made in [19, 10]. +In this classification, the smallest algebra – H⊕M2(C) – is too small to accomodate +the Standard Model; the second smallest one – ASM = M2(H)⊕M4(C) – reduces to the +left-right algebra ALR = HL ⊕ HR ⊕ M4(C) by imposing the grading condition, which +breaks to the algebra AF of the Standard Model by the first-order condition. The next +one is M3(H)⊕M6(C) and has not found any physical interpretation so far. Then comes +the grand algebra [32] +AG = M4(H) ⊕ M8(C). +(31) +It is too big to be represented on the Hilbert space HF in a way compatible with the +axioms of noncommutative geometry: the latter require a space of dimension d = 2(2a)2, +where a is the dimension of the quaternionic matrix algebra. For ASM one has a = 2, +which corresponds to d = 2(2 · 2)2 = 32, that is the dimension of HF. For the grand +algebra AG, a = 4 and one needs a space four times bigger. +This bigger space is obtained by allowing C∞(M) to act independently on the left +and right components of spinors. This permits to represent on L2(M, S) ⊗ HF the +algebra C∞(M) ⊗ AG – viewed as functions on M with value in AG – in such a way +that for any a ∈ C∞(M) ⊗ AG and x ∈ M, then a(x) ∈ AG acts on HF in agreement +with the classification of [10]. +Within the tensorial notation of §2.5, the components M4(H) and M8(C) of the +grand algebra are 2 × 2 matrices Q, M with entries in M2(H) and M4(C) that act on +HF as ASM. The difference with the spectral triple of the Standard Model is that, once +tensorised by C∞(M), the 2×2 matrices Q, M have a non-trivial action on the spinorial +degrees of freedom of L2(M, S). We denote the latter by two indices: s = l, r for the +left/right components of spinors; ˙s = ˙0, ˙1 for the particle/antiparticle subspaces. + +A critical survey of twisted spectral triples +beyond the Standard Model +10 +In [32] one makes C∞(M) ⊗ M4(H) ∋ Q, resp. C∞(M) ⊗ M8(C) ∋ M, act non +trivially on the ˙s, resp s, index. Omitting all the indices on which the action is trivial, +Q = +� +Q +˙0β +˙0α +Q +˙1β +˙0α +Q +˙0β +˙1α +Q +˙1β +˙1α +� +˙s˙t +, +M = +� +MrJ +rI +MlJ +rI +MrJ +lI +MlJ +lI +� +st +, +(32) +where β, J, t and ˙t are summation indices within the same range as α, I, s, t (the +indices after the closing parenthesis are those labelling the matrix entries). +Since γ5 acts non trivially on the spinorial chiral index, the grading condition +forces M to be diagonal in the st indices: MlJ +rI = MlJ +lr = 0. Since ΓF is non trivial +only in the flavour index α, in which the remaining entries MlJ +lI , MrJ +rI ∈ M4(C) act +trivially, the grading does not induce any further breaking in the complex sector. On +the contrary, since γ5 is trivial in the ˙s index but quaternions act non trivially on the +α index, the grading forces Q to be diagonal in the flavour index, with components +QL +˙t +˙s, QR +˙t +˙s ∈ C∞(M) ⊗ M2(H) acting on the left/right subspaces of the internal Hilbert +space HF. In other terms, the grading condition breaks the grand algebra in +A′ +G = (M2(H)L ⊕ M2(H)R) ⊕ (M4(C)l ⊕ M4(C)r) . +(33) +To guarantee the first-order condition for the Majorana component γ5⊗DR of the Dirac +operator, a solution is to further break A′ +G to +A′′ +G = (HL ⊕ H′ +L ⊕ CR ⊕ C′ +R) ⊕ (Cl ⊕ M3(C)l ⊕ Cr ⊕ M3(C)r) +(34) +with CR = Cr = Cl. In the first term, the unprimed algebras act on the particle subspace +˙s = ˙0, while the primed ones act on the antiparticle subspace ˙s = ˙1. A fluctuation of +the metric of γ5 ⊗ DR then yields an extra scalar field σ, which lives in the difference +between CR and C′ +R, and fixes the Higgs mass as expected [33]. In this grand algebra +model, the fermionic content is not altered, since the total Hilbert space H is untouched. +One also checks the order zero condition. +The first-order condition for the free part /∂ ⊗ I of the Dirac operator forces the +components acting on the chiral spinorial index to be equal, as well as those acting on the +particle/antiparticle index, meaning H′ +L = HL, C′ +R = CR and M3(C)l = M3(C)r. Thus +A′′ +G reduces to HL⊕CR⊕M3(C), namely the algebra of the Standard Model. The field σ +thus appears as the Higgs field related to the breaking of the first-order condition by /∂⊗I, +whereas in [14] it is related with the first-order condition for γ5 ⊗ DR. By enlarging the +algebra, one has moved the symmetry breaking from the internal space to the manifold. +However, the price to pay for a non trivial action on spinors is the unboundedness of +the commutator of /∂ ⊗I with the grand algebra: for a = f ⊗m ∈ C∞(M)⊗AG one has +[/∂ ⊗ I, a] = [/∂, f] ⊗ m = −i[γµ∂µ, f] ⊗ m − i[γµωµ, f] ⊗ m. +(35) +The second term is always bounded. By the Leibniz rule, the first one is +−i[γµ, f]∂µ − iγµ(∂µf), +(36) +which is bounded iff the component ∂µ vanishes. Since the only matrix that commutes +with all the Dirac matrices is the identity matrix, the commutator (35) is bounded if +and only if f acts on L2(M, S) as a multiple of the identity matrix, that is on the same +way on the left and right components of spinors. + +A critical survey of twisted spectral triples +beyond the Standard Model +11 +3.2. Twisted spectral triples +Mixing the spinorial and internal degrees of freedom of the Hilbert space H - in order +to represent an algebra bigger than the one of the Standard Model - turns out to be +incompatible with the very definition of spectral triple. As explained at the end of the +preceding section, this does not depend on the details of the representation: as soon as +the grand algebra acts non trivially on spinors, then the commutator with the free part +of the Dirac operator is unbounded [45], no matter if the representation is (32) or not. +The unboundedness of the commutator is the kind of problems adressed by Connes +and Moscovici when they define twisted spectral triples in [24]. Their motivation had +little to do with physics, but were purely mathematical (building spectral triples with +type III algebras). Given a triple (A, H, D), instead of asking the commutators [D, a] +to be bounded, one asks the boundedness of the twisted commutators +[D, a]ρ := Da − ρ(a)D +(37) +for some fixed automorphism ρ ∈ Aut(A). +The whole process of fluctuation of the metric has been adapted to the twisted case +in [41, 42]. One obtains the covariant Dirac operator +DAρ := D + Aρ + J Aρ J−1 +(38) +where Aρ is an element of the set of twisted 1-forms +Ω1 +D(A, ρ) := +�� +i +ai[D, Jb∗ +i J−1]ρ◦, ai, bi ∈ A +� +(39) +with ρ◦ := ρ(a∗)◦ is the automorphism of the opposite algebra A◦ induced by ρ. There +is also twisted version of the first-order condition [34, 41] +[[D, a]ρ, Jb∗J−1]ρ◦ = 0 +∀a, b ∈ A. +(40) +A gauge transformation is implemented by the twisted action of the operator Adu (22), +ρ(Adu) DAρ Adu−1, +(41) +with ρ(Adu) := ρ(u)Jρ(u)J−1 . Such a transformation maps DAρ to DAuρ where +Au +ρ = ρ(u)[D, u∗]ρ + ρ(u)Aρu∗. +(42) +This is the twisted version of the gauge transformation (23). +3.3. Twisting the grand algebra +To resolve the unboundedness of the commutator arising in the grand algebra model, +the idea is to find an automorphism of C∞(M) ⊗ AG such that the twisted commutator +(37) of any element (Q, M) ∈ C∞(M) ⊗ AG with /∂ ⊗ I be bounded. This must be +true in particular for (Q, 0) and (0, M). +Repeating the computation (35) (36), and +taking into account only the spinorial indices s, ˙s (since /∂ ⊗ I acts as the identity on all + +A critical survey of twisted spectral triples +beyond the Standard Model +12 +the other indices, the corresponding sector of the algebra must be invariant under the +automorphism, for Ia − ρ(a)I = 0 iff a = ρ(a)), one finds that ρ should be such that +γµQ − ρ(Q)γµ = 0 and γµM − ρ(M)γµ = 0 +∀µ = 1, ..., dim M +(43) +for any Q ∈ M4(H) ⊗ C∞(M) and M ∈ M8(C) ⊗ C∞(M). By easy computation, using +the explicit form of the γ matrices in the chiral basis, +γµ = +� +02 +σµ +¯σµ +02 +� +st +σµ = +� +I, σi� +, ¯σµ = +� +I, iσi� +, +(44) +where σi are the Pauli matrices, one checks that any two 4 × 4 complex matrices A, B +such that Aγµ = γµB for any µ are necessarily of the form +A = +� +λI2 +02 +02 +λ′I2 +� +B = +� +λ′I2 +02 +02 +λI2 +� +for some λ, λ′ ∈ C. +(45) +Thus (43) implies that both M and Q must be trivial in the ˙s index, diagonal in the +chiral index s, with ρ the autormorphism that exchanges the left and right components. +Therefore the representation (32) of the grand algebra is not suitable to build a twisted +spectral triple. +In order to find a good representation, remember that the field σ has its origin +in the two copies CR, C′ +R of C in A′′ +G (34), which come from the non-trivial action of +C∞(M) ⊗ M4(H) on the ˙s index. Since the latter is no longer allowed, it seems natural +to make C∞(M) ⊗ M4(H) act non trivially on the chiral index s. On the contrary, +the complex sector plays no obvious role in the generation of the field σ, so one lets +C∞(M) ⊗ M8(C) act trivially on both the s, ˙s indices. On the other indices, the action +of M4(H), M8(C) is as in the Standard Model. The grading condition now breaks M4(H) +to Hl +L ⊕ Hr +L ⊕ Hl +R ⊕ Hr +R but leaves M8(C) untouched. Reducing the latter “by hand” to +M4(C), one gets the algebra [34] +B′ = Hl +L ⊕ Hr +L ⊕ Hl +R ⊕ Hr +R ⊕ M4(C). +(46) +Let ρ be the automorphism of C∞(M) ⊗ B′ that flips the chiral spinorial degrees of +freedom, +ρ(ql +L, qr +L, ql +R, qr +R, m) := (qr +L, ql +L, qr +R, ql +R, m) +(47) +where each of the q is a quaternionic function with value in its respective copy of H and +m ∈ C∞(M) ⊗ M4(C). Then +(C∞(M) ⊗ B′, L2(M, S) ⊗ C32, /∂ ⊗ I) +(48) +is a twisted spectral triple which satisfies the first-order condition [34, Prop. 3.4]. +Regarding the Majorana Dirac operator, let us consider the subalgebra of B′ +˜B = Hl +L ⊕ Hr +L ⊕ Cl +R ⊕ Cr +R ⊕ (C ⊕ M3(C)). +(49) +Given two of its elements (ql +L, qr +L, cl +R, cr +R, c, m), (rl +L, rr +L, dl +R, dr +R, d, n) with c, d, cl +R, cr +R, dl +R, dr +R +complex functions, ql +L, qr +L, rl +L, rr +L quaternionic functions and m, n functions with values +in M3(C), denoting π′ the representation of B′ in the spectral triple (48), one finds that +[γ5 ⊗ DR, π′(ql +L, qr +L, cl +R, cr +R, c, m)]ρ, π′(rl +L, rr +L, dl +R, dr +R, d, n)]ρ +(50) +vanishes as soon as c = cl +R and d = dl +R (or c = cr +R and d = dr +R). + +A critical survey of twisted spectral triples +beyond the Standard Model +13 +In [34], this was improperly interpreted as a breaking of B′ to +B = Hl +L ⊕ Hr +L ⊕ Cl +R ⊕ Cr +R ⊕ M3(C). +(51) +acting as ˜B with C = Cl +R, namely the representation π of B is +π(ql +L, qr +L, cl +R, cr +R, m) := π′(ql +L, qr +L, cl +R, cr +R, cl +R, m). +(52) +But ρ exchanges the left/right components in the quaternionic sector only, so that +π′(ρ(ql +L, qr +L, cl +R, cr +R, cl +R, m)) = π′(qr +L, ql +L, cr +R, cl +R, cl +R, m) +(53) +is not the representation (52) of any element in C∞(M) ⊗ B (the latter requires the +identification of the first and third complex functions, whereas in (53) the second and +third are identified), unless cr +R = cl +R. This means that the breaking from B′ to B is not +compatible with the twist unless C = Cl +R identifies with Cr +R. In that case, B′ actually +breaks to Hl +L ⊕ Hr +L ⊕ C ⊕ M3(C). This algebra contains only one copy of C and so does +not generate the field σ by twisted fluctuation of γ5 ⊗ DR. +In other terms, the model developed in [34] does not allow to generate the extra +scalar field while preserving the first-order condition (even in a twisted form), as opposed +to what was claimed. The error is due to not noticing that the reduction from ˜B to +B, imposed by the twisted first-order condition of the Majorana Dirac operator, is not +invariant under the twist. So it does not make sense to try to build a spectral triple +with C∞(M) ⊗ B. +Nevertheless all the expressions computed in [34] of the form +Tπ′(a) − π′(ρ(a))T +(54) +for T = /∂ ⊗ I or γ5 ⊗ DR are algebraically correct. The point is that they are twisted +commutators (37) for a in C∞(M) ⊗ ˜B, but not for a in C∞(M) ⊗ B. Indeed, although +(53) does define a representation of C∞(M) ⊗ B, +ˆπ(ql +L, qr +L, cl +R, cr +R, m) := π′(qr +L, ql +L, cr +R, cl +R, cl +R, m), +(55) +there is no automorphism η of C∞(M) ⊗ B such that ˆπ would equal π ◦ η. What the +results of [34] show is that starting with the twisted spectral triple +(C∞(M) ⊗ ˜B, L2(M, S) ⊗ HF, /∂ ⊗ I + γ5 ⊗ DF), +(56) +whose Majorana part violates the twisted first-order condition, then a twisted fluctuation +of the Dirac operator by the subalgebra C∞(M) ⊗ B yields the field σ. Minimising the +spectral action (suitably generalised to the twisted case) breaks the algebra to the one +of the Standard Model, which satisfies the first-order condition. +As noticed at the end of [41], an alternative way to interprete (54) for a in +C∞(M) ⊗ B is to view it as a twisted commutator for the represented algebra. Namely +defining the inner automorphism αU(B) := UBU∗ of B(H) ⊃ B that exchanges the +l, r components in the particle sector C = 0 of HF (it is implemented by the unitary +U = γ0 ⊗ P + I ⊗ (I − P) with P the projection on the particle subspace of HF), then +(54) reads as +Tπ(a) − αU(π(a))T +for +a ∈ C∞(M) ⊗ B. +(57) +It is not yet clear whether this observation is of interest. + +A critical survey of twisted spectral triples +beyond the Standard Model +14 +3.4. Twisting down +In the light of the preceding section, the conclusion of [34] should be corrected: twisted +spectral triples do resolve the unboundedness of the commutator arising in the grand +algebra model, but the extra scalar field breaks the first-order condition, even in its +twisted form. The latter is retrieved dynamically by minimising the spectral action. +Therefore, twisting the grand algebra down to the Standard Model produces results +similar to the ones of [14]. This raises questions on the interest of the twist. As explained +in section 5, there is an added value in twists, even if not the one expected! But before +coming to that, let us try to generalize the twisting of the grand algebra to arbitrary +spectral triples. +4. Minimal twist +4.1. Twisting up +The algebra B is not invariant under the twisting automorphism ρ because the grand +algebra has been only partially twisted: only the quaternionic sector acts non-trivially +on the chiral index s. If one also makes the complex sector non trivial on the chiral +index, then the grading condition breaks the grand algebra to +� +Hl +L ⊕ Hr +L ⊕ Hl +R ⊕ Hr +R +� +⊕ +� +Ml +4(C) ⊕ Mr +4(C) +� +, +(58) +which is invariant under ρ. This is twice the left-right algebra ALR of §3.1, which is +broken to the algebra ASM of the Standard Model by the first-order condition of γ5⊗DF. +This suggests another approach to twisting the Standard Model while preserving +the first-order condition. Rather than twisting down a bigger algebra to ASM, one may +double ASM to +ASM ⊗ C2 ≃ ASM ⊕ ASM, +(59) +then make each copy of ASM act independently on the left/right components of spinors, +and finally twist the commutator to avoid unboundedness problems. +This is a “twisting up” procedure, in which the idea is to use the flexibility +introduced by twisted spectral triples to enlarge the algebra – hopefully preserving the +grading and the first-order conditions – rather than using these conditions to constrain a +bigger algebra. The rule of the game is to leave the Hilbert space and the Dirac operator +untouched, in order not to alter the fermionic content of the model. As a side remark, +there exist some models in noncommutative geometry that introduce new fermions, as +mentioned in the introduction, but since there is no phenomenological indications of +new fermions so far, we limit ourselves to models that preserve the fermionic sector. +Given a spectral triple (A, H, D), the idea is thus to build a twisted spectral triple +(A′, H, D), ρ with the same Hilbert space and Dirac operator, in such a way that the +initial triple is retrieved as a “non-twisted” limit of the twisted one. This led in [41] to +define the minimal twist of a spectral triple (A, H, D) by a unital algebra B as a twisted + +A critical survey of twisted spectral triples +beyond the Standard Model +15 +spectral triple (A ⊗ B, H, D), ρ such that the representation of A ⊗ IB coincides with +the initial representation of A. +One may think of other ways to implement the idea of “non-twisted limit”, for +instance by simply asking that A′ contains A as a subalgebra invariant under the twist. +More elaborate procedure for untwisting a twisted spectral triple have been proposed, +for instance in [39, 7]. +An advantage of minimal twists is to allow to play with the Standard Model, +remaining close to it. +For almost commutative geometries – i.e. +the product of a +manifold by a finite dimensional spectral triple as in (24) – then the only possible +minimal twist by a finite dimensional algebra is with B = Cl ⊗ C2, with ρ the flip +automorphism of C2 and l ∈ N a measure of the non irreducibility of the representation +of AF on HF [41, Prop. 4.4]. +4.2. Twist by grading +The twisting up procedure is easily applicable to any graded spectral triple (A, H, D). +Indeed, by definition, the grading Γ commutes with the representation of A, so the +latter actually is the direct sum of two independent – commuting – representations of +A on the eigenspaces H+, H− of Γ, +π+(a) = 1 +2 (I + Γ) a, +π−(a) = 1 +2 (I − Γ) a. +(60) +In other words, decomposing H as the sum of the two eigenspaces of Γ, the representation +of A is block diagonal. Thus there is enough space on H to represent A ⊗ C2 as +π((a, a′)) = π+(a) + π−(a′) +∀(a, a′) ∈ A ⊗ C2. +(61) +Let +ρ((a, a′)) = (a′, a) +∀(a, a′) ∈ A ⊗ C2 +(62) +denote the flip automorphism. Then the triple +(A ⊗ C2, H, D), ρ +(63) +with representation (61) is a graded twisted spectral triple [41, Prop. 3.8]. In addition, +if the initial triple is real with real structure J, then the latter is also a real structure +for the twisted spectral triple (61). In particular the twisted first-order condition is +automatically satisfied. +This twist by grading procedure associates a twisted partner to any graded spectral +triple, preserving a first-order condition. This seems the ideal way to twist the Standard +Model. Unfortunately, this does not generate the extra scalar field. Indeed, one has that +ΓF anticommutes independently with DY and DM (see e.g. [32, §4.1] for the computation +in tensorial notations) so in particular γ5 ⊗ DM anticommutes with Γ = γ5 ⊗ ΓF. This +means that +(γ5 ⊗ DM)π+(a) = π−(a)(γ5 ⊗ DM) + 1 +2(I − Γ)[γ5 ⊗ DM, a], +(64) +(γ5 ⊗ DM)π−(a) = π+(a)(γ5 ⊗ DM) + 1 +2(I + Γ)[γ5 ⊗ DM, a]. +(65) + +A critical survey of twisted spectral triples +beyond the Standard Model +16 +So +[γ5 ⊗ DM, π((a, a′))]ρ = (γ5 ⊗ DM)(π+(a) + π−(a′)) − (π+(a′) + π−(a))(γ5 ⊗ DM), += [γ5 ⊗ DM, a] + [γ5 ⊗ DM, a′]. +(66) +The right hand side is zero since γ5 ⊗ DM commutes with the representation of A. +Therefore γ5 ⊗ DM twist-commutes with the representation of A ⊗ C2. Hence the twist +by grading does not modify the situation: γ5 ⊗ DM is transparent under under twisted +fluctuations, just like it was under usual fluctuations. +4.3. Twisted fluctuation without the first-order condition +The twist by grading is not the only possibility for twisting up the Standard Model. As +explained in [41, below Prop.3.8], in order to minimally twist a spectral triple (A, H, D) +by C2, one may repeat the construction of the precedent section using, instead of the +grading Γ, any operator ˜Γ that +• squares to I and commutes with A: this condition is sufficient to guarantee that +π+, π− in (60) are two representations of A commuting with each other, and it +becomes necessary as soon as A is unital; +• is selfadjoint: this is to guarantee that π+ and π− are involutive representations; +• has both eigenvalues +1, −1 of non-zero multiplicity, so that neither π+ nor π− is +zero. +But there is no need for ˜Γ to anticommute with the Dirac operator. This means that ˜Γ +is not necessarily a grading for the spectral triple. +A classification of all such twisting operators ˜Γ for almost commutative geometries +is on its way [37]. The anticommutation with the Dirac operator seems to be required +to have the twisted first-order condition. This would imply that the extra scalar field +and the twisted first-order condition be mutually exclusive. +Therefore it becomes relevant to extend to the twisted case the results of [14] +regarding inner fluctuations without the first-order condition. This has been done in +[49], where it was shown that the removal of the twisted first-order condition yields a +second order term in the twisted fluctuation (38), which is a straightforward adaptation +of the term worked out in the non-twisted case. +Following this path, a minimal twist of the Standard Model has been worked out +in great details in [36], that does not preserve the twisted first-order condition and +generates the extra scalar field. The gauge part of this model is similar to the Standard +Model’s, and the Higgs sector is made of two Higgs doublets which are expected to +combine in a single doublet in the action. +There is the extra scalar field with two +components σl, σr acting independently on the chiral components of spinors, and finally, +there is also an unexpected new field of 1-forms Xµ, whose interpretation is discussed +in the next section. + +A critical survey of twisted spectral triples +beyond the Standard Model +17 +5. Twist and change of signature +At this point of our journey through twisted spectral triples, one seems to be back to +the starting point: twisted spectral triples solve the unboundeness of the commutator of +the grand algebra with /∂ ⊗ I, but they do not permit to generate the extra scalar field, +unless one violates the twisted first-order condition. What is then their added value? +The interest of the twist is not so much in the generation of the extra scalar field +than in the new field of 1-form Xµ mentioned above. This field was already observed in +[34], and its appearance actually does not depend on the details of the model [45]: it +is intrinsic to minimal twists of almost commutative geometries. Even in the simplest +case of a minimally twisted four dimensional manifold (without any product by a finite +dimensional structure), a twisted fluctuation of the Dirac operator /∂ yields a field of +1-forms, in contrast with the non twisted case where /∂ does not fluctuate. +The physical sense of this fluctuation remained obscure, until it was confronted with +an observation made in [30]: a twist induces on the Hilbert space a new inner product +with Lorentzian signature. Furthermore, this product permits to define a twisted version +of the fermionic action. In some example detailed below, in this action formula the field +Xµ identifies with the (dual of) the 4-momentum in Lorentzian signature [47]. +5.1. Twisted inner product +A gauge transformation (22), DA → Ad(u) DA Ad(u)−1, preserves the selfadjointness +of the covariant Dirac operator DA, for Ad(u)−1 = Ju∗J−1u∗ = Ad(u)∗. A twisted +gauge transformation (41) +DAρ → ρ(Ad(u)) DAρ Ad(u)−1 +(67) +does not. Is there some selfadjointness which is preserved by (67)? +There is a natural inner product associated with a twisted spectral triple, as soon +as the twisting autormorphism ρ extends to an inner automorphism of B(H): +ρ(O) = ROR† +∀O ∈ B(H) +(68) +for some unitary operator R on H. Namely, the ρ-inner product [30] +⟨Ψ, Φ⟩ρ := ⟨Ψ, RΦ⟩. +(69) +Since ⟨Ψ, OΦ⟩ρ = ⟨ρ(O)†Ψ, Φ⟩ρ, the adjoint of O with respect to this new product is +O+ := ρ(O)†. +(70) +If the unitary R commutes or anticommutes with the real structure, then ρ(Ad(u)) +as defined before (42) coincides with RAd(u)R∗ (making the notation ρ(Ad(u)) +unambiguous). In addition, +� +Ad(u)−1�+ = +� +RJu∗J−1u∗R∗�† = RuJuJ−1R∗ = ρ(Ad(u)). +(71) +Therefore a twisted gauge transformation (67) preserves the selfadjointness with respect +to the ρ-inner product. + +A critical survey of twisted spectral triples +beyond the Standard Model +18 +Example: The minimal twist of a Riemannian spin manifold M of even dimension +2m is +A = C∞(M) ⊗ C2, +H = L2(M, S), +D = /∂; +ρ +(72) +with twisting automorphism the flip ρ(f, g) = (g, f) for f, g in C∞(M). +The +representation is +π(f, g) = +� +f I2m−1 +0 +0 +gI2m−1 +� +∀(f, g) ∈ A. +(73) +The flip ρ extends to the inner automorphism of B(H) that exchanges the element on +the diagonal and on the off-diagonal, implemented for instance by R = γ0 the first Dirac +matrix. Then the ρ-product (69) +⟨Ψ, Φ⟩ρ = +� +M +Ψ†γ0Φ d4x +(74) +coincides pointwise with the Krein product for the space of spinors on a Lorentzian +manifold (only pointwise, for the manifold on which one integrates is still Riemannian). +This example points towards a link between twists and a kind of transition from +Euclidean to Lorentzian signatures: by fluctuating a twisted Riemannian manifold, one +ends up preserving a Lorentzian product! However, the twist is not an implementation +of Wick rotation in noncommutative geometry (for this, see [27]): a twisted fluctuation +(67) does not turn the operator DAρ, selfadjoint for the initial (Euclidean) inner product, +into an operator DAuρ selfadjoint for the Lorentzian product.‡ A better understanding of +the link between twist and Lorentzian signature follows from the study of the fermionic +action. +5.2. Fermionic action +Given a real spectral triple (A, H, D), the fermionic action for the covariant operator +DA is [12] +Sf(DA) = ADA(˜ξ, ˜ξ) +(75) +with ˜ξ the Grassman variables associated to ξ ∈ H+ = {ξ ∈ H, Γξ = ξ} and +ADA(ξ, ξ′) = ⟨Jξ, DAξ′⟩ +(76) +the antisymmetric bilinear form defined by DA and the real structure J. The latter +is needed to make the form antisymmetric (hence applicable on Grassman variables). +One restricts to the eigenspace H+ of the grading because of the fermion doubling [43]. +This also makes sense physically, for H+ is the subspace of H generated by the elements +ψ ⊗ Ψ with a well defined chirality (that is ψ ∈ L2(M, S) and Ψ ∈ HF are eigenvectors +of γ5, ΓF with the same eigenvalue). +‡ If one were starting with an operator selfadjoint for the twisted product, much in the vein of [53], +then this selfadjointness would be preserved by twisted fluctuation. + +A critical survey of twisted spectral triples +beyond the Standard Model +19 +For a twisted spectral triple (A, H, D), ρ as in §5.1, the fermionic action is [30] +Sf(DAρ) = TDAρ(˜ξ, ˜ξ) +(77) +for ξ ∈ Hr := {ξ ∈ H, Rξ = ξ}, ˜. the Grassmann variables and +TDAρ(ξ, ξ′) := ⟨Jξ, RDAρξ′⟩. +One inserts the matrix R in the bilinear form in order to make the action (77) invariant +under a twisted gauge transformation (41) (the same is true in case there is no first- +order condition [49]). +The restriction to Hr guarantees that the bilinear form be +antisymmetric. +5.3. Twisted fluctuation as Lorentzian 4-momentum +We begin with the minimal twist (72) of a 4-dimensional manifold. The +1 eigenspace +of R = γ0 is spanned by Dirac spinors of the form ξ = +� +ζ +ζ +� +with ζ a Weyl spinor. A +selfadjoint twisted fluctuation (38) sends /∂ to the covariant operator +/∂Aρ = /∂ − i Xµγµ, +(78) +parametrised by the 1-form field +Xµ = fµγ5 +with +fµ ∈ C∞(M, R). +(79) +The twisted fermionic action is [47, Prop. 3.5] +Sf(/∂Aρ) = 2 +� +M +dµ ¯˜ζ +†σ2 (if0 − +3 +� +j=1 +σj∂j) ˜ζ. +(80) +The integrand reminds of the Weyl Lagrangian – which lives in Lorentzian signature +iψ† +l ˜σµ +M ∂µψl +where +˜σµ +M := {I2, −σj} , +(81) +except that the ∂0 derivative is missing. It can be restored assuming that ζ is a plane +wave function of energy f0 (in unit ℏ = 1) with spatial part ζ(x), that is +ζ(x0, x) = eif0x0ζ(x). +(82) +Then the integrand reads (modulo an irrelevant factor 2) as ¯˜ζ +† +σ2 ˜σµ +M∂µ ˜ζ. However, this +cannot be identified with the Weyl Lagrangian (81) because of the extra σ2 matrix which +forbids the simultaneous identification of ˜ζ with ψl and ¯˜ζ +† +σ2 with iψ† +l . In other terms, +there are not enough degrees of freedom to identify the fermionic action of a twisted +manifold with the Weyl Lagrangian. +This can be cured by doubling the manifold. Namely one considers the product +(C∞(M) ⊗ C2, L2(M, S) ⊗ C2, /∂ ⊗ I2). +(83) +of M by a finite dimensional spectral triple (C2, C2, 0). Its minimal twist is +A = +� +C∞(M) ⊗ C2� +⊗ C2, +H = L2(M, S) ⊗ C2, +D = /∂ ⊗ I2 +(84) + +A critical survey of twisted spectral triples +beyond the Standard Model +20 +with representation +π(a, a′) = + + + + + +fI2 +0 +0 +0 +0 +f ′I2 +0 +0 +0 +0 +g′I2 +0 +0 +0 +0 +gI2 + + + + + +a = (f, g), a′ = (f ′, g′) ∈ A +(85) +and twist ρ(a, a′) = (a′, a). The latter is implemented by the unitary R = γ0 ⊗I2, whose ++1 eigenspace Hr is now spanned by {ξ ⊗ e, φ ⊗ ¯e} where {e, ¯e} is a basis of C2 and +ξ = +� +ζ +ζ +� +, +φ = +� +ϕ +ϕ +� +(86) +are Dirac spinors with ζ and ϕ Weyl spinors. A selfadjoint twisted fluctuation of D, +DAρ = D − iXµγµ ⊗ I2 + gµγµ ⊗ ΓF +(87) +with ΓF the grading of the finite dimensional spectral triple [47, Prop. +4.3], is +parametrised by the same field Xµ as before and a second 1-form field +gµI4 +with +gµ ∈ C∞(M). +(88) +For a vanishing gµ, the fermionic action is the integral of [47, Prop. 4.4] +L := 4¯˜ϕ†σ2 +� +if0 − �3 +j=1 σj∂j +� +˜ζ. +(89) +One retrieves the Weyl Lagrangian (81) by identifying the physical Weyl spinors as +ψl := ˜ζ and ψ† +l := −i¯˜ϕ†σ2, then assuming ψl be of the form (82), that is ∂0ψl = if0ψl. +Thus the fermionic action for a twisted doubled Riemannian manifold describes a plane +wave solution of Weyl equation, in Lorentzian signature, whose 0th component of the +4-momentum is p0 = −f0. The result also holds for the right-handed Weyl equation +(see [47, Prop. 4.5]). +A similar analysis holds for the spectral triple of electrodynamics proposed in [54]. +Its minimal twist is +AED = +� +C∞(M) ⊗ C2� +⊗ C2, H = L2(M, S) ⊗ C4, +D = /∂ ⊗ I4 + γ5 ⊗ DF +where the internal Dirac operator and the representation are +DF = + + + + + +0 +d +0 +0 +¯d +0 +0 +0 +0 +0 +0 +¯d +0 +0 +d +0 + + + + + , π(a, a′) = + + + + + + + + + + + + + + +fI2 +0 +0 +0 +0 +0 +0 +0 +0 +f ′I2 +0 +0 +0 +0 +0 +0 +0 +0 +f ′I2 +0 +0 +0 +0 +0 +0 +0 +0 +fI2 +0 +0 +0 +0 +0 +0 +0 +0 +g′I2 +0 +0 +0 +0 +0 +0 +0 +0 +gI2 +0 +0 +0 +0 +0 +0 +0 +0 +gI2 +0 +0 +0 +0 +0 +0 +0 +0 +g′I2 + + + + + + + + + + + + + + + +A critical survey of twisted spectral triples +beyond the Standard Model +21 +with d ∈ C, a = (f, g), a′ = (f ′, g′) in C∞(M) ⊗ C2. The twist ρ(a, a′) = (a′, a) extends +to an inner automorphism of B(H) generated by the unitary γ0 ⊗ I4. Its +1-eigenspace +is generated by +ξ1 ⊗ el, +ξ2 ⊗ er, +φ1 ⊗ el, +φ2 ⊗ er, +(90) +where ξk, φk (k = 1, 2) are Dirac spinors of the form (86) while {el, er, el, er} is an +orthonormal basis of C4. +A selfadjoint twisted fluctuation of D is parametrized by the same two 1-form fields +as before [47, Prop. 5.3] +DAρ = D − iXµγµ ⊗ I′ + gµγµ ⊗ I′′ +(91) +where I′ = diag(1, −1, 1, −1), I′′ = diag(1, 1, −1, −1) (the part γ5 ⊗ DF is transparent +under twisted fluctuation: there is no Higgs field in classical electrodynamics!). Under +a gauge transformation (41), one has that fµ is invariant while gµ trasforms as the U(1) +gauge potential of electrodynamics. +The spectral action is the integral of [47, Prop. 5.12] +Lf +ρ = ¯˜ϕ† +1σ2 +� +if0 − +� +j +σjDj +� +˜ζ1−¯˜ϕ† +2σ2 +� +if0 + +� +j +σjDj +� +˜ζ2+ +� +¯d¯˜ϕ† +1σ2¯ζ2 + d¯˜ϕ† +2σ2¯ζ1 +� +(92) +where Dµ = ∂µ − igµ is the covariant derivative associated to the electromagnetic 4- +potential. Defining the physical spinors as +ψ = +� +ψl +ψr +� +:= +� ˜ζ1 +˜ζ2 +� +, +ψ† = +� +ψ† +l , ψ† +r +� +:= +� +−i¯˜ϕ† +1σ2, i¯˜ϕ† +2σ2 +� +(93) +then assuming that ∂0ψ = if0ψ and imposing d = −im with m > 0 to be purely +imaginary, the Lagrangian (92) reads +LM = iψ† +l +� +D0 − +� +j +σjDj +� +ψl+iψ† +r +� +D0 + +� +j +σjDj +� +ψr−m +� +ψ† +l ψr + ψ† +rψl +� +.(94) +This is the Dirac Lagrangian in Minkowski spacetime, for a mass m, in the temporal +gauge (that is D0 = ∂0). Hence the fermionic action for the minimal twist of the spectral +triple of electrodynamics describes a plane wave solution of the Dirac equation in Lorentz +signature, with 0th component of the 4-momentum p0 = −f0. +By implementing the action of boosts on L2(M, S) ⊗C2, one is able to identify the +other components of the fluctuation fµ with the other components of the 4-momentum. +However this is obtained at the level of the equation of motion, not for the Lagrangian +density (see [47, §6.1]). +6. Conclusion and outlook +The idea of using twisted spectral triples in high-energy physics was born with the hope +of generating the extra scalar field needed to stabilise the electroweak vacuum (and to fit +the Higgs mass), respecting the axioms of noncommutative geometry. More specifically + +A critical survey of twisted spectral triples +beyond the Standard Model +22 +it was thought that the first-order condition could be twisted, rather than abandoned. +We have shown in this note that this is not possible. This moves the interest of the +twist towards what seemed at first sight a side effect, namely the non-zero twisted +fluctuation of the free Dirac operator /∂. It yields a new field of 1-forms, whose physical +meaning becomes clear by computing the fermionic action. For the minimal twist of a +doubled manifold, and the minimal twist of the spectral triple of electrodynamics, this +fields identifies with (the dual of) the 4-momentum in Lorentzian signature. Preliminary +computations indicate that a similar result also holds for the twist of the Standar Model +presented in [36]. +It remains to be understood why one ends up in the temporal gauge. +And +more importantly, does the identification between twisted fluctuation of /∂ and the 4- +momentum still makes sense for the bosonic part of the action, given by the spectral +action? Not to mention that the definition of the latter in a twisted context has not +been estabilised yet [31]. +Acknowledgments +The first author is supported by the POLONEZ BIS program cofunded by a Marie Curie +action. This work is part of the second author’s activity in the mathematical physics +group of INDAM. +Bibliography +[1] J. W. Barrett. A Lorentzian version of the non-commutative geometry of the standard model of +particle physics. J. Math. Phys., 48:012303, 2007. +[2] F. Besnard and C. Brouder. Noncommutative geometry, the Lorentzian Standard Model and its +B-L extension. Phys. Rev. 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Geom., 7:433–456, 2013. + diff --git a/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/load_file.txt b/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..959c0babaa630e7cde333efa660879687229c646 --- /dev/null +++ b/hNE_T4oBgHgl3EQf3Rzj/content/tmp_files/load_file.txt @@ -0,0 +1,895 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf,len=894 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='08346v1 [math-ph] 19 Jan 2023 A critical survey of twisted spectral triples beyond the Standard Model Manuele Filaci University of Cracovia, Institute fo Physics, Jagiellonian University prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Stanis�lawa �Lojasiewicza 11, 30-348 Krakow, Poland E-mail: manuele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='filaci@uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='pl Pierre Martinetti Universit`a di Genova (dpto di matematica) & INFN, via Dodecaneso, 16146 Genova, Italia E-mail: martinetti@dima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='it Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We review the applications of twisted spectral triples to the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The initial motivation was to generate a scalar field, required to stabilise the electroweak vacuum and fit the Higgs mass, while respecting the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Ultimately, it turns out that the truest interest of the twist lies in a new – and unexpected – field of 1-forms, which is related to the transition from Euclidean to Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Introduction From the pioneering work of [35] till the full formalism of Connes [16], noncommutative geometry provides a unified description of the Lagrangian of the Standard Model of fundamental interactions (electromagnetism, weak and strong interactions) [21][9][8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' minimally coupled to the Einstein-Hilbert action of General Relativity [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' including right handed neutrinos [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' where the Higgs boson comes out naturally on the same footing as the other bosons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' as the local expression of a connection 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The approach works very well on Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The generalisation to pseudo-Riemannian geometry, in particular Lorentzian manifolds, is far from obvious (there are various attempts in this direction, see for instance [1][2][38][53][3] and reference within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In addition, noncommutative geometry offers possibilities to go beyond the Standard Model, by modifying the rules of the game in various ways: one may enlarge the space of fermions [51, 52], or get rid of the first-order condition (defined below) [14, 13], modify the real structure (also defined below) [7, 6], switch to non-associative geometry [4, 5], use some structure of Clifford bundle in order to modify some of the mathematical requirements defining a noncommutative geometry [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For a recent review of “beyond Standard Model” propositions in the framework of noncommutative geometry, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Here we focus on another class of variations around Connes’ initial model, obtained by twisting the noncommutative geometry by an algebra automorphism [32][34][47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 2 All the possibilities above but the first are minimal extensions of the Stan- dard Model, in that they yield an extra scalar field σ – suggested by particle physi- cists to stabilize the electroweak vacuum – but do not touch the fermionic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The novelty of the twist is to generate another additional piece: a new field of 1-forms, which suprisingly turns out to be related to the transition from Euclidean to Lorentzian sig- nature [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In particular, in the example of electrodynamics, this field identifies with the (dual) of the 4-momentum vector in Lorentzian signature, even though one started with a Riemannian manifold [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' All this is explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In the next section we begin by some recalling on the spectral description of the Standard Model [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We stress the process of fluctuation of the metric, which is the way to generate bosonic fields in noncommutative geometry by turning the constant parameters of the model into fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In section 3 we describe the model of grand algebra developed in [32], which aimed at generating the extra scalar field σ, while respecting the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The idea is to start with an algebra bigger than the one of the Standard Model, in order to have more “space” to generate bosonic fields by fluctuations of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This model does indeed generate the expected field σ, by letting the Majorana mass of the neutrinos fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Even though the first-order condition associated with this Majorana mass is preserved, the problem moves to the free Dirac operator: not only does the latter break the first-order condition, but its commutator with the algebra is unbounded, in contradiction with the very definition of spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This kind of problem is typically solved by twisting the spectral triple in the sense of Connes and Moscovici [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A twisting of the grand algebra down to the Standard Model has been worked out in [34], but we show in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3 that this does not define stricto-sensu a twisted spectral triple, for only the part of the algebra relevant for the extra scalar field has been twisted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Applying the twist to the whole algebra suggests a general procedure to twist any graded spectral triple, as recalled in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It consists in doubling the algebra one is beginning with, rather than finding it from the reduction of a bigger algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Such a “twisting up” procedure has been studied in a couple of papers [41][42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' There is some freedom in the construction, especially in the choice of the twisting operator whose eigenspaces determine the representation of the doubled algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By choosing the grading as the twisting operator, one automatically satisfies the twisted first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However, when applied to the spectral triple of the Standard Model, this twist-by-grading does not generate any extra scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Some preliminary results, mentioned in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3, indicate that this is a general feature of the twisting-up procedure: the twisted first-order condition and the extra scalar field are mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Hence the necessity to adapt to the twisted case the fluctuations without first-order condition introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This has been done in [49] and is summarised in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Section 5 deals with what might be the truest interest of the twist, namely the unexpected field of 1-forms arising from the twisted fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In the example of electrodynamics [47],[54], this field identifies with the dual of the 4-momentum in Lorentzian signature, even though one started with a Riemannian spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The spectral description of the Standard Model We begin with the definition of spectral triple, which is the central tool in Connes’ noncommutative geometry, emphasising how the bosonic fields – including the Higgs field – are obtained as connection 1-forms, through the process of fluctuation of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We then summarise the spectral description of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Spectral triple A spectral triple [16] consists of an algebra A acting on a Hilbert space H together with a selfadjoint operator D with compact resolvent, whose commutator [D, a] is bounded for any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It is graded if it comes with a selfadjoint operator Γ on H which squares to the identity operator I, anticommutes with D and commutes with the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A spectral triple is real [17] if in addition there is an antilinear operator J on H satisfying J2 = ǫI, JD = ǫ′DJ, JΓ = ǫ′′ΓJ (1) where ǫ, ǫ′, ǫ′′ = ±1 define the KO-dimension k ∈ [0, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This real structure implements a map a → a◦ := Ja∗J−1 from A to the opposite algebra A◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This yields a right action of A on H, ψa := a◦ψ, which is asked to commute with the left action [a, Jb∗J−1] = 0 ∀a ∈ A (order zero condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (2) There is also a first-order condition [18], [[D, a], Jb∗J−1] = 0 ∀a, b ∈ A (3) which is there to guarantee that the operator D be a first-order differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' All these properties are satisfied by the triple (C∞(M), L2(M, S), /∂) (4) where C∞(M) is the (commutative) algebra of smooth functions on a closed Riemannian manifold M of dimension m, acting by multiplication on the Hilbert space L2(M, S) of square-integrable spinors on M, and /∂ = −i m � µ=1 γµ(∂µ + ωµ), with γµγν + γνγµ = 2gµνI (5) is the Dirac operator (ωµ is the spin connection, γµ the Dirac matrices and gµν the Riemannian metric on M, all given in local coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For m even, this spectral triple has grading the product of the Dirac matrices (in dimension 4, the matrix γ5) and real structure J the charge conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Adding other conditions [20] (which are satisfied by the triple (4)), one gets Connes’ reconstruction theorem, that extends Gelfand duality (between compact topological spaces and C∗-commutative algebras) beyond topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Namely, given any real spectral triple (A, H, D) satisfying these conditions, with A commutative, then there exists a closed Riemannian manifold M such that A ≃ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A noncommutative geometry is then defined as a spectral triple (A, H, D) where A is non (necessarily) commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This gives access to new geometrical objects, that can be intended as “spaces” whose algebra of functions A is not commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Connection Take a gauge theory with gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' From a mathematical point of view, the fermionic fields form sections of a G-bundle E over the spacetime M, while the bosonic fields are described as connections on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In noncommutative geometry the spacetime M is substituted by a spectral triple (A, H, D), where A plays the role of “algebra of functions” on the noncommutative space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' To understand what plays the role of a gauge bundle, recall that the set of sections of any bundle on a manifold M forms a finite projective C∞(M)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Conversely, by Serre-Swan theorem, any such module actually is the module of sections of a bundle on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' That is why, in noncommutative geometry, the role of gauge bundles is played by finite projective A-modules E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Connections on these modules are, by definition, objects associated with a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Recall that a derivation δ on an algebra A is a map from A to some A-bimodule Ω satisfying the Leibniz rule δ(ab) = aδ(b) + δ(a)b ∀a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (6) A connection on a (right) A-module E associated with δ is a map from E to E ⊗A Ω such that the following Leibniz rule holds, ∇(ηa) = ∇(η)a + η ⊗ δ(a) ∀η ∈ E, a ∈ A, (7) where Ω = �� i aiδ(bi), ai, bi ∈ A � (8) is the A-bimodule generated by the derivation δ, while ∇(η)a is a shorthand notation for η(0)a ⊗ η(1), using Sweedler notations ∇η = η(0) ⊗ η(1) with η(0) ∈ E and η(1) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Example: The exterior derivative d is a derivation on the algebra C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It generates the C∞(M)-bimodule of section s of the cotangent bundle, Ω1(M) := �� i fidgi with fi, gi ∈ C∞(M) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (9) A connection on the tangent bundle TM associated with d is a map ∇ : Γ∞(TM) → Γ∞(TM) ⊗ Ω1(M), (10) ∂ν �→ Γρ µν∂ρ ⊗ dxµ, (11) where Γ∞(TM) denotes the set of smooth sections of TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One retrieves the usual notion of connection, as a map from Γ∞(TM) × Γ∞(TM) to Γ∞(TM) as ∇ : (∂η, ∂ν) �→ ∇η∂ν := Γρ µν∂ρ ⊗C∞(M) ⟨dxµ, ∂η⟩ ≃ ⟨dxµ, ∂η⟩Γρ µν∂ρ = Γρ ην∂ρ, where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='⟩ is the C∞(M)-valued dual product between the cotangent and the tangent bundles and the last equation is the isomorphism between E ⊗C∞(M) C∞(M) and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Fluctuation of the metric To understand when two algebras are “similar”, a relevant notion is Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This is more flexible than isomorphism of algebras for, roughly speaking, two algebras A and B are Morita equivalent if they have the same representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Technically, it means that there exists an Hermitian finite projective A-module E such that B is isomorphic to the algebra EndA(E) of A-linear, adjointable, endormorphisms of E (for details see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' [50] or [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The idea of fluctuation of the metric comes from the following question: how does one export a real spectral triple (A, H, D) to a Morita equivalent algebra B ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We recall the construction of [18], whose details may be found in [23] and even more details in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Assume E = ER is a right A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The algebra B acts on HR := ER ⊗A H as b(η ⊗ ψ) = bη ⊗ ψ ∀b ∈ B, η ∈ ER, ψ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (12) However, the “natural” action of D on HR, DR(η ⊗ ψ) := η ⊗ Dψ, (13) is not compatible with the tensor product, for DR(ηa ⊗ ψ) − DR(η ⊗ aψ) = −η ⊗ [D, a]ψ (14) has no reason to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This is cured by equipping ER with a connection ∇ with value in the A-bimodule of generalised 1-forms Ω1 D(A) := �� i ai[D, bi], ai, bi ∈ A � (15) generated by the derivation δ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=') = [D, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Indeed, the following operator, DR(η ⊗ ψ) := η ⊗ Dψ + (∇η)ψ (16) is well defined on HR, and selfadjoint as soon as ∇ is an hermitian connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Moreover one checks that the commutator [DR, b] is bounded for any b ∈ B, so that (B, HR, DR) is a spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In particular, if one considers self-Morita equivalence, that is B = ER = A, then the operator (16) with ∇ hermitian reads DR = D + AR (17) with AR = A∗ R ∈ Ω1 D(A) a selfadjoint generalised 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A similar construction holds if one implements Morita equivalence with a left module EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Then HL = H ⊗A EL is a Hilbert space and the operator DL(ψ ⊗ η) := Dψ ⊗ η + (∇◦η)ψ (18) with ∇◦ a connection with value in the bimodule Ω1 D(A◦) = �� i a◦ i [D, b◦ i ], a◦ i , b◦ i ∈ A◦ � A critical survey of twisted spectral triples beyond the Standard Model 6 is well defined on HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For ∇◦ hermitian, one obtains a spectral triple (B, HL, DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For self-Morita equivalence, one gets DL = D + A◦ = D + ǫ′J AL J−1 (19) with A◦ ∈ Ω1 D(A◦) and AL ∈ Ω1 D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' To take into account the real structure, one combines the two constructions, using an A-bimodule E to implement Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For self-Morita equivalence, one then obtains the operator D′ = D + AR + ǫ′J ALJ−1 acting on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Requiring this operator to be selfadjoint and J to be a real structure amounts to the existence of a generalised selfadjoint 1-form A ∈ Ω1 D(A) such that D′ = DA := D + A + ǫ′J AJ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (20) Then (A, H, DA) is a real spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The operator DA is called a covariant Dirac operator, and the substitution of D with a DA is a fluctuation of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The name is motivated by the existing relation between the Dirac operator and the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This relation is obvious on a spin manifold, from the very definition of the Dirac matrices ( γνγν+γνγµ = 2gµν), and it still makes sense for an arbitrary noncommutative geometry, via the definition of the spectral distance [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' On a manifold, this distance gives back the geodesic distance associated with the Riemannian structure of M, while on an arbitrary spectral triple it may be seen as a generalisation of the Wasserstein distance of order 1 in the theory of optimal transport (cf [28, 46] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By exporting a noncommutative geometry to a Morita equivalent algebra, one passes from D to the covariant operator DA and modifies accordingly the spectral distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In particular, for the Standard Model, such a fluctuation provides a purely metric interpretation to the Higgs field (which is one of the components of the generalised 1-form A, see below) [18, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The metric interpretation of the other components of A has been worked out in [48, 44], in relation with the Carnot-Carath´eodory distance in sub-Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Gauge transformation A gauge transformation is a change of connection on the Morita-equivalence bimodule E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In case of self-Morita equivalence, it is implemented by the conjugate action on H of the group U(A) of unitaries element of A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' u ∈ A such that u∗u = uu∗ = I): Ad(u) : ψ → uψu∗ = u(u∗)◦ψ = uJuJ−1ψ ∀ψ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (21) This action maps the covariant Dirac operator DA to Ad(u) DA Ad(u)−1 (22) and one checks that this operator coincides with the operator DAu, defined as in (20) with Au := u[D, u∗] + uAu∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (23) This is the formula of transformation of the gauge potential in noncommutative geometry, which generalises the usual one of gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Standard Model The spectral triple of the Standard Model [12] is the product A = C∞(M) ⊗ AF, H = L2(M, S) ⊗ HF, D = /∂ ⊗ I96 + γ5 ⊗ DF (24) of the spectral triple (4) of a 4-dimensional Riemannian closed spin manifold M with a finite dimensional spectral triple AF = C ⊕ H ⊕ M3(C), HF = C96, DF = � D0 048 048 D† 0 � � �� � DY + � 048 DR D† R 048 � � �� � DM (25) where H is the algebra of quaternions and M3(C) the algebra of complex 3×3 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The dimension of HF is the number of fermions in the Standard Model (including right-handed neutrinos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Its entries are labelled by a multi-index C I α n where C = 0, 1 labels particles (C = 0) or anti-particles (C = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' I = 0, i with i = 1, 2, 3 is the lepto-colour index: it takes value I = 0 for a lepton and I = 1, 2, 3 for a quark with its three possible colours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' α = ˙1, ˙2, 1, 2 is the flavour index (with dot indicating the chirality): ˙1 = νR, ˙2 = eR, 1 = νL, 2 = eL for leptons (I = 0), (26) ˙1 = uR, ˙2 = dR, 1 = qL, 2 = dL for quarks (I = i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (27) n = 1, 2, 3 is the generation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The details of the representation of AF is in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The important point for our matter is that the quaternions act only on the particle subspace of HF (C = 0), trivially on the lepto-colour index I, and through their fundamental representation on the last two flavour indices α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' whereas M3(C) acts only on antiparticle subspace of HF (C = 1), trivially on the flavour index α and through their fundamental representation on the lepto-colour index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The algebra C acts both on particles together with the quaternions (but on the first two flavour indices), and on antiparticles together with M3(C) (on I = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The grading of the finite dimensional spectral triple is the 96 × 96 matrix ΓF with entries +1 on left particles/right antiparticles, −1 on right particles/left antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The real structure is the matrix JF that exchanges particles with antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The spectral triple (24) is real, with grading Γ = γ5 ⊗ ΓF and real structure J = J ⊗ JF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In the particles/antiparticles indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' the Dirac operator DF of the finite dimensional spectral triple is the sum of a block diagonal matrix DY which contains the Yukawa couplings of the fermions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' the Cabibbo-Kobayashi-Maskawa mixing matrix for the quarks and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix for the left- handed neutrinos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' and a block off-diagonal matrix DM which contains the Majorana masses kn R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3 of the right-handed neutrinos and the corresponding mixing matrix (notations are those of [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' they differ from the ones of [32] and [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 8 The generalised 1-forms (15) for a product of spectral triples (24) decompose as [25] A = γ5 ⊗ H − i � µ γµ ⊗ Aµ (28) where H is a scalar field on M with values in AF, while Aµ is a 1-form field on M with values in the Lie algebra of the group U(AF) of unitary elements of AF (differently said: a connection 1-form on a U(AF)-bundle on TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In particular, for the spectral triple of the Standard Model, one has U(AF) = U(C ⊕ H ⊕ M3(C)) ≃ U(1) × SU(2) × U(3), (29) which is reduced to the gauge group U(1) × SU(2) × SU(3) of the Standard Model by imposing a unimodularity condition (which also guarantees that the model is anomaly free, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='g [12, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The action of this group on H is a matrix whose components are the hypercharges of the fermions of the Standard Model [12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This allows to identify the basis elements of HF with the 96 fermions of the Standard Model, and the corresponding elements in H with the fermionic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Moreover, the action of A+JAJ−1 corresponds to the bosonic hypercharges, and allows to identify the components of Aµ with the bosonic fields of the Standard Model [12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One also checks that (23) yields the expected gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The interpetation of the scalar field H follows from the computation of the spectral action [8, 9], namely the asymptotic expansion Λ → ∞ of Tr f( D2 A Λ2 ) where f is a smooth approximation of the characteristic function of the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One obtains the bosonic Lagrangian of the Standard Model coupled with Einstein-Hilbert action in Euclidean signature, where H is the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The coupling constants of the electroweak and strong interactions satisfy the relation expected in grand unified theories, and are related to the value at 0 of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The spectral action provides some relations between the parameters of the Standard Model at a putative unification scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The physical predictions are obtained by running down the parameters of the theory under the renormalisation group equation, taking these relations as initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Assuming there is no new physics between the unification scale and the electroweak scale, one finds a value for the Higgs mass around 170 GeV, in disagrement with the measured value 125, 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However, for a Higgs boson with mass mH ≤ 130 Gev, the quartic coupling λ of the Higgs field becomes negative at high energy, meaning the electroweak vacuum is meta-stable rather than stable [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This instability can be cured by a new scalar field σ which couples to the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In the spectral description of the Standard Model, such a field is obtained by turning into a field the neutrino Majorana mass kR which appears in the off-diagonal part DR of the finite dimensional Dirac operator DF: kR → kRσ, Furthermore, by altering the running of the parameters under the equations of the group of renormalization, this extra scalar field makes the computation of the mass of the Higgs boson compatible with its experimental value [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Grand algebra beyond the Standard Model The point in the above is to justify the turning of the constant kR into a field kRσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This cannot be obtained by fluctuation of the metric, for one checks that [γ5 ⊗ DM, a] = 0 ∀a, b ∈ A = C∞(M) ⊗ AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (30) In other terms, the constant kR is transparent under fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The solution proposed in [14] is to remove the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This gives more flexibility, and permits to obtain the extra scalar field as a fluctuation without the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The latter is retrieved dynamically, by minimising the spectral action [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In this way the field σ is the “Higgs” boson associated with the breaking of the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Grand algebra At the same time, an alternative process was described in [32] where one mixes the internal degrees of freedom per generation of the finite dimensional Hilbert space HF, that is HF ≃ C32, with the 4 spinorial degrees of freedom of L2(M, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This provides exactly the 4 × 32 = 128 degrees of freedom required to represent the “second next algebra” in the classification of finite dimensional spectral triples made in [19, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In this classification, the smallest algebra – H⊕M2(C) – is too small to accomodate the Standard Model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' the second smallest one – ASM = M2(H)⊕M4(C) – reduces to the left-right algebra ALR = HL ⊕ HR ⊕ M4(C) by imposing the grading condition, which breaks to the algebra AF of the Standard Model by the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The next one is M3(H)⊕M6(C) and has not found any physical interpretation so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Then comes the grand algebra [32] AG = M4(H) ⊕ M8(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (31) It is too big to be represented on the Hilbert space HF in a way compatible with the axioms of noncommutative geometry: the latter require a space of dimension d = 2(2a)2, where a is the dimension of the quaternionic matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For ASM one has a = 2, which corresponds to d = 2(2 · 2)2 = 32, that is the dimension of HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For the grand algebra AG, a = 4 and one needs a space four times bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This bigger space is obtained by allowing C∞(M) to act independently on the left and right components of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This permits to represent on L2(M, S) ⊗ HF the algebra C∞(M) ⊗ AG – viewed as functions on M with value in AG – in such a way that for any a ∈ C∞(M) ⊗ AG and x ∈ M, then a(x) ∈ AG acts on HF in agreement with the classification of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Within the tensorial notation of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='5, the components M4(H) and M8(C) of the grand algebra are 2 × 2 matrices Q, M with entries in M2(H) and M4(C) that act on HF as ASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The difference with the spectral triple of the Standard Model is that, once tensorised by C∞(M), the 2×2 matrices Q, M have a non-trivial action on the spinorial degrees of freedom of L2(M, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We denote the latter by two indices: s = l, r for the left/right components of spinors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' ˙s = ˙0, ˙1 for the particle/antiparticle subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 10 In [32] one makes C∞(M) ⊗ M4(H) ∋ Q, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' C∞(M) ⊗ M8(C) ∋ M, act non trivially on the ˙s, resp s, index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Omitting all the indices on which the action is trivial, Q = � Q ˙0β ˙0α Q ˙1β ˙0α Q ˙0β ˙1α Q ˙1β ˙1α � ˙s˙t , M = � MrJ rI MlJ rI MrJ lI MlJ lI � st , (32) where β, J, t and ˙t are summation indices within the same range as α, I, s, t (the indices after the closing parenthesis are those labelling the matrix entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Since γ5 acts non trivially on the spinorial chiral index, the grading condition forces M to be diagonal in the st indices: MlJ rI = MlJ lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Since ΓF is non trivial only in the flavour index α, in which the remaining entries MlJ lI , MrJ rI ∈ M4(C) act trivially, the grading does not induce any further breaking in the complex sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' On the contrary, since γ5 is trivial in the ˙s index but quaternions act non trivially on the α index, the grading forces Q to be diagonal in the flavour index, with components QL ˙t ˙s, QR ˙t ˙s ∈ C∞(M) ⊗ M2(H) acting on the left/right subspaces of the internal Hilbert space HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In other terms, the grading condition breaks the grand algebra in A′ G = (M2(H)L ⊕ M2(H)R) ⊕ (M4(C)l ⊕ M4(C)r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (33) To guarantee the first-order condition for the Majorana component γ5⊗DR of the Dirac operator, a solution is to further break A′ G to A′′ G = (HL ⊕ H′ L ⊕ CR ⊕ C′ R) ⊕ (Cl ⊕ M3(C)l ⊕ Cr ⊕ M3(C)r) (34) with CR = Cr = Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In the first term, the unprimed algebras act on the particle subspace ˙s = ˙0, while the primed ones act on the antiparticle subspace ˙s = ˙1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A fluctuation of the metric of γ5 ⊗ DR then yields an extra scalar field σ, which lives in the difference between CR and C′ R, and fixes the Higgs mass as expected [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In this grand algebra model, the fermionic content is not altered, since the total Hilbert space H is untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One also checks the order zero condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The first-order condition for the free part /∂ ⊗ I of the Dirac operator forces the components acting on the chiral spinorial index to be equal, as well as those acting on the particle/antiparticle index, meaning H′ L = HL, C′ R = CR and M3(C)l = M3(C)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Thus A′′ G reduces to HL⊕CR⊕M3(C), namely the algebra of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The field σ thus appears as the Higgs field related to the breaking of the first-order condition by /∂⊗I, whereas in [14] it is related with the first-order condition for γ5 ⊗ DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By enlarging the algebra, one has moved the symmetry breaking from the internal space to the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However, the price to pay for a non trivial action on spinors is the unboundedness of the commutator of /∂ ⊗I with the grand algebra: for a = f ⊗m ∈ C∞(M)⊗AG one has [/∂ ⊗ I, a] = [/∂, f] ⊗ m = −i[γµ∂µ, f] ⊗ m − i[γµωµ, f] ⊗ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (35) The second term is always bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By the Leibniz rule, the first one is −i[γµ, f]∂µ − iγµ(∂µf), (36) which is bounded iff the component ∂µ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Since the only matrix that commutes with all the Dirac matrices is the identity matrix, the commutator (35) is bounded if and only if f acts on L2(M, S) as a multiple of the identity matrix, that is on the same way on the left and right components of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisted spectral triples Mixing the spinorial and internal degrees of freedom of the Hilbert space H - in order to represent an algebra bigger than the one of the Standard Model - turns out to be incompatible with the very definition of spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' As explained at the end of the preceding section, this does not depend on the details of the representation: as soon as the grand algebra acts non trivially on spinors, then the commutator with the free part of the Dirac operator is unbounded [45], no matter if the representation is (32) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The unboundedness of the commutator is the kind of problems adressed by Connes and Moscovici when they define twisted spectral triples in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Their motivation had little to do with physics, but were purely mathematical (building spectral triples with type III algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Given a triple (A, H, D), instead of asking the commutators [D, a] to be bounded, one asks the boundedness of the twisted commutators [D, a]ρ := Da − ρ(a)D (37) for some fixed automorphism ρ ∈ Aut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The whole process of fluctuation of the metric has been adapted to the twisted case in [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One obtains the covariant Dirac operator DAρ := D + Aρ + J Aρ J−1 (38) where Aρ is an element of the set of twisted 1-forms Ω1 D(A, ρ) := �� i ai[D, Jb∗ i J−1]ρ◦, ai, bi ∈ A � (39) with ρ◦ := ρ(a∗)◦ is the automorphism of the opposite algebra A◦ induced by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' There is also twisted version of the first-order condition [34, 41] [[D, a]ρ, Jb∗J−1]ρ◦ = 0 ∀a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (40) A gauge transformation is implemented by the twisted action of the operator Adu (22), ρ(Adu) DAρ Adu−1, (41) with ρ(Adu) := ρ(u)Jρ(u)J−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Such a transformation maps DAρ to DAuρ where Au ρ = ρ(u)[D, u∗]ρ + ρ(u)Aρu∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (42) This is the twisted version of the gauge transformation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisting the grand algebra To resolve the unboundedness of the commutator arising in the grand algebra model, the idea is to find an automorphism of C∞(M) ⊗ AG such that the twisted commutator (37) of any element (Q, M) ∈ C∞(M) ⊗ AG with /∂ ⊗ I be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This must be true in particular for (Q, 0) and (0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Repeating the computation (35) (36), and taking into account only the spinorial indices s, ˙s (since /∂ ⊗ I acts as the identity on all A critical survey of twisted spectral triples beyond the Standard Model 12 the other indices, the corresponding sector of the algebra must be invariant under the automorphism, for Ia − ρ(a)I = 0 iff a = ρ(a)), one finds that ρ should be such that γµQ − ρ(Q)γµ = 0 and γµM − ρ(M)γµ = 0 ∀µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=', dim M (43) for any Q ∈ M4(H) ⊗ C∞(M) and M ∈ M8(C) ⊗ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By easy computation, using the explicit form of the γ matrices in the chiral basis, γµ = � 02 σµ ¯σµ 02 � st σµ = � I, σi� , ¯σµ = � I, iσi� , (44) where σi are the Pauli matrices, one checks that any two 4 × 4 complex matrices A, B such that Aγµ = γµB for any µ are necessarily of the form A = � λI2 02 02 λ′I2 � B = � λ′I2 02 02 λI2 � for some λ, λ′ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (45) Thus (43) implies that both M and Q must be trivial in the ˙s index, diagonal in the chiral index s, with ρ the autormorphism that exchanges the left and right components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Therefore the representation (32) of the grand algebra is not suitable to build a twisted spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In order to find a good representation, remember that the field σ has its origin in the two copies CR, C′ R of C in A′′ G (34), which come from the non-trivial action of C∞(M) ⊗ M4(H) on the ˙s index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Since the latter is no longer allowed, it seems natural to make C∞(M) ⊗ M4(H) act non trivially on the chiral index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' On the contrary, the complex sector plays no obvious role in the generation of the field σ, so one lets C∞(M) ⊗ M8(C) act trivially on both the s, ˙s indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' On the other indices, the action of M4(H), M8(C) is as in the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The grading condition now breaks M4(H) to Hl L ⊕ Hr L ⊕ Hl R ⊕ Hr R but leaves M8(C) untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Reducing the latter “by hand” to M4(C), one gets the algebra [34] B′ = Hl L ⊕ Hr L ⊕ Hl R ⊕ Hr R ⊕ M4(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (46) Let ρ be the automorphism of C∞(M) ⊗ B′ that flips the chiral spinorial degrees of freedom, ρ(ql L, qr L, ql R, qr R, m) := (qr L, ql L, qr R, ql R, m) (47) where each of the q is a quaternionic function with value in its respective copy of H and m ∈ C∞(M) ⊗ M4(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Then (C∞(M) ⊗ B′, L2(M, S) ⊗ C32, /∂ ⊗ I) (48) is a twisted spectral triple which satisfies the first-order condition [34, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Regarding the Majorana Dirac operator, let us consider the subalgebra of B′ ˜B = Hl L ⊕ Hr L ⊕ Cl R ⊕ Cr R ⊕ (C ⊕ M3(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (49) Given two of its elements (ql L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' qr L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cr R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (rl L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' rr L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dr R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' n) with c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cr R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dr R complex functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' ql L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' qr L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' rl L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' rr L quaternionic functions and m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' n functions with values in M3(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' denoting π′ the representation of B′ in the spectral triple (48),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' one finds that [γ5 ⊗ DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' π′(ql L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' qr L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' cr R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' m)]ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' π′(rl L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' rr L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dl R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' dr R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' n)]ρ (50) vanishes as soon as c = cl R and d = dl R (or c = cr R and d = dr R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 13 In [34], this was improperly interpreted as a breaking of B′ to B = Hl L ⊕ Hr L ⊕ Cl R ⊕ Cr R ⊕ M3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (51) acting as ˜B with C = Cl R, namely the representation π of B is π(ql L, qr L, cl R, cr R, m) := π′(ql L, qr L, cl R, cr R, cl R, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (52) But ρ exchanges the left/right components in the quaternionic sector only, so that π′(ρ(ql L, qr L, cl R, cr R, cl R, m)) = π′(qr L, ql L, cr R, cl R, cl R, m) (53) is not the representation (52) of any element in C∞(M) ⊗ B (the latter requires the identification of the first and third complex functions, whereas in (53) the second and third are identified), unless cr R = cl R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This means that the breaking from B′ to B is not compatible with the twist unless C = Cl R identifies with Cr R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In that case, B′ actually breaks to Hl L ⊕ Hr L ⊕ C ⊕ M3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This algebra contains only one copy of C and so does not generate the field σ by twisted fluctuation of γ5 ⊗ DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In other terms, the model developed in [34] does not allow to generate the extra scalar field while preserving the first-order condition (even in a twisted form), as opposed to what was claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The error is due to not noticing that the reduction from ˜B to B, imposed by the twisted first-order condition of the Majorana Dirac operator, is not invariant under the twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' So it does not make sense to try to build a spectral triple with C∞(M) ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Nevertheless all the expressions computed in [34] of the form Tπ′(a) − π′(ρ(a))T (54) for T = /∂ ⊗ I or γ5 ⊗ DR are algebraically correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The point is that they are twisted commutators (37) for a in C∞(M) ⊗ ˜B, but not for a in C∞(M) ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Indeed, although (53) does define a representation of C∞(M) ⊗ B, ˆπ(ql L, qr L, cl R, cr R, m) := π′(qr L, ql L, cr R, cl R, cl R, m), (55) there is no automorphism η of C∞(M) ⊗ B such that ˆπ would equal π ◦ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' What the results of [34] show is that starting with the twisted spectral triple (C∞(M) ⊗ ˜B, L2(M, S) ⊗ HF, /∂ ⊗ I + γ5 ⊗ DF), (56) whose Majorana part violates the twisted first-order condition, then a twisted fluctuation of the Dirac operator by the subalgebra C∞(M) ⊗ B yields the field σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Minimising the spectral action (suitably generalised to the twisted case) breaks the algebra to the one of the Standard Model, which satisfies the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' As noticed at the end of [41], an alternative way to interprete (54) for a in C∞(M) ⊗ B is to view it as a twisted commutator for the represented algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Namely defining the inner automorphism αU(B) := UBU∗ of B(H) ⊃ B that exchanges the l, r components in the particle sector C = 0 of HF (it is implemented by the unitary U = γ0 ⊗ P + I ⊗ (I − P) with P the projection on the particle subspace of HF), then (54) reads as Tπ(a) − αU(π(a))T for a ∈ C∞(M) ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (57) It is not yet clear whether this observation is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisting down In the light of the preceding section, the conclusion of [34] should be corrected: twisted spectral triples do resolve the unboundedness of the commutator arising in the grand algebra model, but the extra scalar field breaks the first-order condition, even in its twisted form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The latter is retrieved dynamically by minimising the spectral action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Therefore, twisting the grand algebra down to the Standard Model produces results similar to the ones of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This raises questions on the interest of the twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' As explained in section 5, there is an added value in twists, even if not the one expected!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' But before coming to that, let us try to generalize the twisting of the grand algebra to arbitrary spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Minimal twist 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisting up The algebra B is not invariant under the twisting automorphism ρ because the grand algebra has been only partially twisted: only the quaternionic sector acts non-trivially on the chiral index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' If one also makes the complex sector non trivial on the chiral index, then the grading condition breaks the grand algebra to � Hl L ⊕ Hr L ⊕ Hl R ⊕ Hr R � ⊕ � Ml 4(C) ⊕ Mr 4(C) � , (58) which is invariant under ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This is twice the left-right algebra ALR of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1, which is broken to the algebra ASM of the Standard Model by the first-order condition of γ5⊗DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This suggests another approach to twisting the Standard Model while preserving the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Rather than twisting down a bigger algebra to ASM, one may double ASM to ASM ⊗ C2 ≃ ASM ⊕ ASM, (59) then make each copy of ASM act independently on the left/right components of spinors, and finally twist the commutator to avoid unboundedness problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This is a “twisting up” procedure, in which the idea is to use the flexibility introduced by twisted spectral triples to enlarge the algebra – hopefully preserving the grading and the first-order conditions – rather than using these conditions to constrain a bigger algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The rule of the game is to leave the Hilbert space and the Dirac operator untouched, in order not to alter the fermionic content of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' As a side remark, there exist some models in noncommutative geometry that introduce new fermions, as mentioned in the introduction, but since there is no phenomenological indications of new fermions so far, we limit ourselves to models that preserve the fermionic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Given a spectral triple (A, H, D), the idea is thus to build a twisted spectral triple (A′, H, D), ρ with the same Hilbert space and Dirac operator, in such a way that the initial triple is retrieved as a “non-twisted” limit of the twisted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This led in [41] to define the minimal twist of a spectral triple (A, H, D) by a unital algebra B as a twisted A critical survey of twisted spectral triples beyond the Standard Model 15 spectral triple (A ⊗ B, H, D), ρ such that the representation of A ⊗ IB coincides with the initial representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One may think of other ways to implement the idea of “non-twisted limit”, for instance by simply asking that A′ contains A as a subalgebra invariant under the twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' More elaborate procedure for untwisting a twisted spectral triple have been proposed, for instance in [39, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' An advantage of minimal twists is to allow to play with the Standard Model, remaining close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For almost commutative geometries – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' the product of a manifold by a finite dimensional spectral triple as in (24) – then the only possible minimal twist by a finite dimensional algebra is with B = Cl ⊗ C2, with ρ the flip automorphism of C2 and l ∈ N a measure of the non irreducibility of the representation of AF on HF [41, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twist by grading The twisting up procedure is easily applicable to any graded spectral triple (A, H, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Indeed, by definition, the grading Γ commutes with the representation of A, so the latter actually is the direct sum of two independent – commuting – representations of A on the eigenspaces H+, H− of Γ, π+(a) = 1 2 (I + Γ) a, π−(a) = 1 2 (I − Γ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (60) In other words, decomposing H as the sum of the two eigenspaces of Γ, the representation of A is block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Thus there is enough space on H to represent A ⊗ C2 as π((a, a′)) = π+(a) + π−(a′) ∀(a, a′) ∈ A ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (61) Let ρ((a, a′)) = (a′, a) ∀(a, a′) ∈ A ⊗ C2 (62) denote the flip automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Then the triple (A ⊗ C2, H, D), ρ (63) with representation (61) is a graded twisted spectral triple [41, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In addition, if the initial triple is real with real structure J, then the latter is also a real structure for the twisted spectral triple (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In particular the twisted first-order condition is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This twist by grading procedure associates a twisted partner to any graded spectral triple, preserving a first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This seems the ideal way to twist the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Unfortunately, this does not generate the extra scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Indeed, one has that ΓF anticommutes independently with DY and DM (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' [32, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1] for the computation in tensorial notations) so in particular γ5 ⊗ DM anticommutes with Γ = γ5 ⊗ ΓF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This means that (γ5 ⊗ DM)π+(a) = π−(a)(γ5 ⊗ DM) + 1 2(I − Γ)[γ5 ⊗ DM, a], (64) (γ5 ⊗ DM)π−(a) = π+(a)(γ5 ⊗ DM) + 1 2(I + Γ)[γ5 ⊗ DM, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (65) A critical survey of twisted spectral triples beyond the Standard Model 16 So [γ5 ⊗ DM, π((a, a′))]ρ = (γ5 ⊗ DM)(π+(a) + π−(a′)) − (π+(a′) + π−(a))(γ5 ⊗ DM), = [γ5 ⊗ DM, a] + [γ5 ⊗ DM, a′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (66) The right hand side is zero since γ5 ⊗ DM commutes with the representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Therefore γ5 ⊗ DM twist-commutes with the representation of A ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Hence the twist by grading does not modify the situation: γ5 ⊗ DM is transparent under under twisted fluctuations, just like it was under usual fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisted fluctuation without the first-order condition The twist by grading is not the only possibility for twisting up the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' As explained in [41, below Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='8], in order to minimally twist a spectral triple (A, H, D) by C2, one may repeat the construction of the precedent section using, instead of the grading Γ, any operator ˜Γ that squares to I and commutes with A: this condition is sufficient to guarantee that π+, π− in (60) are two representations of A commuting with each other, and it becomes necessary as soon as A is unital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' is selfadjoint: this is to guarantee that π+ and π− are involutive representations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' has both eigenvalues +1, −1 of non-zero multiplicity, so that neither π+ nor π− is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' But there is no need for ˜Γ to anticommute with the Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This means that ˜Γ is not necessarily a grading for the spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A classification of all such twisting operators ˜Γ for almost commutative geometries is on its way [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The anticommutation with the Dirac operator seems to be required to have the twisted first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This would imply that the extra scalar field and the twisted first-order condition be mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Therefore it becomes relevant to extend to the twisted case the results of [14] regarding inner fluctuations without the first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This has been done in [49], where it was shown that the removal of the twisted first-order condition yields a second order term in the twisted fluctuation (38), which is a straightforward adaptation of the term worked out in the non-twisted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Following this path, a minimal twist of the Standard Model has been worked out in great details in [36], that does not preserve the twisted first-order condition and generates the extra scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The gauge part of this model is similar to the Standard Model’s, and the Higgs sector is made of two Higgs doublets which are expected to combine in a single doublet in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' There is the extra scalar field with two components σl, σr acting independently on the chiral components of spinors, and finally, there is also an unexpected new field of 1-forms Xµ, whose interpretation is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twist and change of signature At this point of our journey through twisted spectral triples, one seems to be back to the starting point: twisted spectral triples solve the unboundeness of the commutator of the grand algebra with /∂ ⊗ I, but they do not permit to generate the extra scalar field, unless one violates the twisted first-order condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' What is then their added value?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The interest of the twist is not so much in the generation of the extra scalar field than in the new field of 1-form Xµ mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This field was already observed in [34], and its appearance actually does not depend on the details of the model [45]: it is intrinsic to minimal twists of almost commutative geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Even in the simplest case of a minimally twisted four dimensional manifold (without any product by a finite dimensional structure), a twisted fluctuation of the Dirac operator /∂ yields a field of 1-forms, in contrast with the non twisted case where /∂ does not fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The physical sense of this fluctuation remained obscure, until it was confronted with an observation made in [30]: a twist induces on the Hilbert space a new inner product with Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Furthermore, this product permits to define a twisted version of the fermionic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In some example detailed below, in this action formula the field Xµ identifies with the (dual of) the 4-momentum in Lorentzian signature [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisted inner product A gauge transformation (22), DA → Ad(u) DA Ad(u)−1, preserves the selfadjointness of the covariant Dirac operator DA, for Ad(u)−1 = Ju∗J−1u∗ = Ad(u)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A twisted gauge transformation (41) DAρ → ρ(Ad(u)) DAρ Ad(u)−1 (67) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Is there some selfadjointness which is preserved by (67)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' There is a natural inner product associated with a twisted spectral triple, as soon as the twisting autormorphism ρ extends to an inner automorphism of B(H): ρ(O) = ROR† ∀O ∈ B(H) (68) for some unitary operator R on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Namely, the ρ-inner product [30] ⟨Ψ, Φ⟩ρ := ⟨Ψ, RΦ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (69) Since ⟨Ψ, OΦ⟩ρ = ⟨ρ(O)†Ψ, Φ⟩ρ, the adjoint of O with respect to this new product is O+ := ρ(O)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (70) If the unitary R commutes or anticommutes with the real structure, then ρ(Ad(u)) as defined before (42) coincides with RAd(u)R∗ (making the notation ρ(Ad(u)) unambiguous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In addition, � Ad(u)−1�+ = � RJu∗J−1u∗R∗�† = RuJuJ−1R∗ = ρ(Ad(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (71) Therefore a twisted gauge transformation (67) preserves the selfadjointness with respect to the ρ-inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 18 Example: The minimal twist of a Riemannian spin manifold M of even dimension 2m is A = C∞(M) ⊗ C2, H = L2(M, S), D = /∂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' ρ (72) with twisting automorphism the flip ρ(f, g) = (g, f) for f, g in C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The representation is π(f, g) = � f I2m−1 0 0 gI2m−1 � ∀(f, g) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (73) The flip ρ extends to the inner automorphism of B(H) that exchanges the element on the diagonal and on the off-diagonal, implemented for instance by R = γ0 the first Dirac matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Then the ρ-product (69) ⟨Ψ, Φ⟩ρ = � M Ψ†γ0Φ d4x (74) coincides pointwise with the Krein product for the space of spinors on a Lorentzian manifold (only pointwise, for the manifold on which one integrates is still Riemannian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This example points towards a link between twists and a kind of transition from Euclidean to Lorentzian signatures: by fluctuating a twisted Riemannian manifold, one ends up preserving a Lorentzian product!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However, the twist is not an implementation of Wick rotation in noncommutative geometry (for this, see [27]): a twisted fluctuation (67) does not turn the operator DAρ, selfadjoint for the initial (Euclidean) inner product, into an operator DAuρ selfadjoint for the Lorentzian product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='‡ A better understanding of the link between twist and Lorentzian signature follows from the study of the fermionic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Fermionic action Given a real spectral triple (A, H, D), the fermionic action for the covariant operator DA is [12] Sf(DA) = ADA(˜ξ, ˜ξ) (75) with ˜ξ the Grassman variables associated to ξ ∈ H+ = {ξ ∈ H, Γξ = ξ} and ADA(ξ, ξ′) = ⟨Jξ, DAξ′⟩ (76) the antisymmetric bilinear form defined by DA and the real structure J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The latter is needed to make the form antisymmetric (hence applicable on Grassman variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One restricts to the eigenspace H+ of the grading because of the fermion doubling [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This also makes sense physically, for H+ is the subspace of H generated by the elements ψ ⊗ Ψ with a well defined chirality (that is ψ ∈ L2(M, S) and Ψ ∈ HF are eigenvectors of γ5, ΓF with the same eigenvalue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' ‡ If one were starting with an operator selfadjoint for the twisted product, much in the vein of [53], then this selfadjointness would be preserved by twisted fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A critical survey of twisted spectral triples beyond the Standard Model 19 For a twisted spectral triple (A, H, D), ρ as in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1, the fermionic action is [30] Sf(DAρ) = TDAρ(˜ξ, ˜ξ) (77) for ξ ∈ Hr := {ξ ∈ H, Rξ = ξ}, ˜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' the Grassmann variables and TDAρ(ξ, ξ′) := ⟨Jξ, RDAρξ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' One inserts the matrix R in the bilinear form in order to make the action (77) invariant under a twisted gauge transformation (41) (the same is true in case there is no first- order condition [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The restriction to Hr guarantees that the bilinear form be antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Twisted fluctuation as Lorentzian 4-momentum We begin with the minimal twist (72) of a 4-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The +1 eigenspace of R = γ0 is spanned by Dirac spinors of the form ξ = � ζ ζ � with ζ a Weyl spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A selfadjoint twisted fluctuation (38) sends /∂ to the covariant operator /∂Aρ = /∂ − i Xµγµ, (78) parametrised by the 1-form field Xµ = fµγ5 with fµ ∈ C∞(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (79) The twisted fermionic action is [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='5] Sf(/∂Aρ) = 2 � M dµ ¯˜ζ †σ2 (if0 − 3 � j=1 σj∂j) ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (80) The integrand reminds of the Weyl Lagrangian – which lives in Lorentzian signature iψ† l ˜σµ M ∂µψl where ˜σµ M := {I2, −σj} , (81) except that the ∂0 derivative is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It can be restored assuming that ζ is a plane wave function of energy f0 (in unit ℏ = 1) with spatial part ζ(x), that is ζ(x0, x) = eif0x0ζ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (82) Then the integrand reads (modulo an irrelevant factor 2) as ¯˜ζ † σ2 ˜σµ M∂µ ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However, this cannot be identified with the Weyl Lagrangian (81) because of the extra σ2 matrix which forbids the simultaneous identification of ˜ζ with ψl and ¯˜ζ † σ2 with iψ† l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' In other terms, there are not enough degrees of freedom to identify the fermionic action of a twisted manifold with the Weyl Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This can be cured by doubling the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Namely one considers the product (C∞(M) ⊗ C2, L2(M, S) ⊗ C2, /∂ ⊗ I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (83) of M by a finite dimensional spectral triple (C2, C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Its minimal twist is A = � C∞(M) ⊗ C2� ⊗ C2, H = L2(M, S) ⊗ C2, D = /∂ ⊗ I2 (84) A critical survey of twisted spectral triples beyond the Standard Model 20 with representation π(a, a′) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed fI2 0 0 0 0 f ′I2 0 0 0 0 g′I2 0 0 0 0 gI2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 a = (f, g), a′ = (f ′, g′) ∈ A (85) and twist ρ(a, a′) = (a′, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The latter is implemented by the unitary R = γ0 ⊗I2, whose +1 eigenspace Hr is now spanned by {ξ ⊗ e, φ ⊗ ¯e} where {e, ¯e} is a basis of C2 and ξ = � ζ ζ � , φ = � ϕ ϕ � (86) are Dirac spinors with ζ and ϕ Weyl spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A selfadjoint twisted fluctuation of D, DAρ = D − iXµγµ ⊗ I2 + gµγµ ⊗ ΓF (87) with ΓF the grading of the finite dimensional spectral triple [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3], is parametrised by the same field Xµ as before and a second 1-form field gµI4 with gµ ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (88) For a vanishing gµ, the fermionic action is the integral of [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='4] L := 4¯˜ϕ†σ2 � if0 − �3 j=1 σj∂j � ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (89) One retrieves the Weyl Lagrangian (81) by identifying the physical Weyl spinors as ψl := ˜ζ and ψ† l := −i¯˜ϕ†σ2, then assuming ψl be of the form (82), that is ∂0ψl = if0ψl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Thus the fermionic action for a twisted doubled Riemannian manifold describes a plane wave solution of Weyl equation, in Lorentzian signature, whose 0th component of the 4-momentum is p0 = −f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The result also holds for the right-handed Weyl equation (see [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A similar analysis holds for the spectral triple of electrodynamics proposed in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Its minimal twist is AED = � C∞(M) ⊗ C2� ⊗ C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' H = L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' S) ⊗ C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' D = /∂ ⊗ I4 + γ5 ⊗ DF where the internal Dirac operator and the representation are DF = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 d 0 0 ¯d 0 0 0 0 0 0 ¯d 0 0 d 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' π(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' a′) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed fI2 0 0 0 0 0 0 0 0 f ′I2 0 0 0 0 0 0 0 0 f ′I2 0 0 0 0 0 0 0 0 fI2 0 0 0 0 0 0 0 0 g′I2 0 0 0 0 0 0 0 0 gI2 0 0 0 0 0 0 0 0 gI2 0 0 0 0 0 0 0 0 g′I2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 A critical survey of twisted spectral triples beyond the Standard Model 21 with d ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' a = (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' a′ = (f ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' g′) in C∞(M) ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The twist ρ(a, a′) = (a′, a) extends to an inner automorphism of B(H) generated by the unitary γ0 ⊗ I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Its +1-eigenspace is generated by ξ1 ⊗ el, ξ2 ⊗ er, φ1 ⊗ el, φ2 ⊗ er, (90) where ξk, φk (k = 1, 2) are Dirac spinors of the form (86) while {el, er, el, er} is an orthonormal basis of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A selfadjoint twisted fluctuation of D is parametrized by the same two 1-form fields as before [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='3] DAρ = D − iXµγµ ⊗ I′ + gµγµ ⊗ I′′ (91) where I′ = diag(1, −1, 1, −1), I′′ = diag(1, 1, −1, −1) (the part γ5 ⊗ DF is transparent under twisted fluctuation: there is no Higgs field in classical electrodynamics!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Under a gauge transformation (41), one has that fµ is invariant while gµ trasforms as the U(1) gauge potential of electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' The spectral action is the integral of [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='12] Lf ρ = ¯˜ϕ† 1σ2 � if0 − � j σjDj � ˜ζ1−¯˜ϕ† 2σ2 � if0 + � j σjDj � ˜ζ2+ � ¯d¯˜ϕ† 1σ2¯ζ2 + d¯˜ϕ† 2σ2¯ζ1 � (92) where Dµ = ∂µ − igµ is the covariant derivative associated to the electromagnetic 4- potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Defining the physical spinors as ψ = � ψl ψr � := � ˜ζ1 ˜ζ2 � , ψ† = � ψ† l , ψ† r � := � −i¯˜ϕ† 1σ2, i¯˜ϕ† 2σ2 � (93) then assuming that ∂0ψ = if0ψ and imposing d = −im with m > 0 to be purely imaginary, the Lagrangian (92) reads LM = iψ† l � D0 − � j σjDj � ψl+iψ† r � D0 + � j σjDj � ψr−m � ψ† l ψr + ψ† rψl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' (94) This is the Dirac Lagrangian in Minkowski spacetime, for a mass m, in the temporal gauge (that is D0 = ∂0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Hence the fermionic action for the minimal twist of the spectral triple of electrodynamics describes a plane wave solution of the Dirac equation in Lorentz signature, with 0th component of the 4-momentum p0 = −f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' By implementing the action of boosts on L2(M, S) ⊗C2, one is able to identify the other components of the fluctuation fµ with the other components of the 4-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' However this is obtained at the level of the equation of motion, not for the Lagrangian density (see [47, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Conclusion and outlook The idea of using twisted spectral triples in high-energy physics was born with the hope of generating the extra scalar field needed to stabilise the electroweak vacuum (and to fit the Higgs mass), respecting the axioms of noncommutative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' More specifically A critical survey of twisted spectral triples beyond the Standard Model 22 it was thought that the first-order condition could be twisted, rather than abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' We have shown in this note that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This moves the interest of the twist towards what seemed at first sight a side effect, namely the non-zero twisted fluctuation of the free Dirac operator /∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It yields a new field of 1-forms, whose physical meaning becomes clear by computing the fermionic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' For the minimal twist of a doubled manifold, and the minimal twist of the spectral triple of electrodynamics, this fields identifies with (the dual of) the 4-momentum in Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Preliminary computations indicate that a similar result also holds for the twist of the Standar Model presented in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' It remains to be understood why one ends up in the temporal gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' And more importantly, does the identification between twisted fluctuation of /∂ and the 4- momentum still makes sense for the bosonic part of the action, given by the spectral action?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Not to mention that the definition of the latter in a twisted context has not been estabilised yet [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Acknowledgments The first author is supported by the POLONEZ BIS program cofunded by a Marie Curie action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' This work is part of the second author’s activity in the mathematical physics group of INDAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Bibliography [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' Barrett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE_T4oBgHgl3EQf3Rzj/content/2301.08346v1.pdf'} +page_content=' A Lorentzian version of the non-commutative geometry of the standard model of particle physics.' 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-0,0 +1,1881 @@ +Minimal Left-Right Symmetric Model with A4 modular symmetry +Ankita Kakoti,1, ∗ Bichitra Bijay Boruah,1, † and Mrinal Kumar Das1, ‡ +1Department of Physics, Tezpur University, Tezpur 784028, India +Abstract +In this paper, we have realized the left-right symmetric model with modular symmetry. We have used +Γ(3) modular group which is isomorphic to non-abelian discrete symmetry group A4. The advantage of +using modular symmetry is the non-requirement for the use of extra particles called ’flavons’. In this +model, the Yukawa couplings are expressed in terms of modular forms (Y1, Y2, Y3). +In this work, we +have studied minimal Left-Right Symmetric Model for both type-I and type-II dominances. Here, we +have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino +masses. The results obtained are well within the experimental limits for the desired values of sum of +neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on +neutrinoless double beta decay with the new physics contributions within Left-Right Symmetric Model. +∗ ankitak@tezu.ernet.in +† bijay@tezu.ernet.in +‡ mkdas@tezu.ernet.in +1 +arXiv:2301.13552v1 [hep-ph] 31 Jan 2023 + +I. +INTRODUCTION +Despite the huge and continued success of the Standard Model (SM) of particle physics, it leaves +some of the puzzles unanswered like the existence of neutrino masses, baryon asymmetry of the +universe, existence of dark matter etc. The discovery of neutrino oscillation by Sudbury neutrino +observatory and Super-Kamiokande experiments was a milestone discovery in the area of neutrino +physics. The experiments like MINOS [1], T2K [2], Daya-Bay [3], Double-Chooz [4], RENO [5] +etc. provided evidence on the neutrinos being massive which is one of the most compelling rev- +elation that we need to go beyond Standard Model. However inspite of the huge achievements +in determining the neutrino oscillation parameters in solar, atmospheric , reactor and accelerator +neutrino experiments, many questions related to neutrino still remain unsolved. Among these lies +the question regarding the absolute mass scale of neutrinos, exact nature of the particle (Dirac +or Majorana), hierarchical pattern of the mass spectrum (Normal or Inverted) and leptonic CP +violation. The absolute mass scale of the neutrinos is not yet known. However experiments like +Planck has given an upper bound on the sum of the light neutrino masses to be Σ|mνi| < 0.23eV in +2012 [6] and recently the bound has been constarined to Σ|mνi| < 0.11eV [7]. The most successful +data pertaining to neutrino oscillation parameters is found in the 3σ global fit data [8] as shown +in table (1) . +Parameters +Normal +Ordering +Inverted +Ordering +∆ m2 +21 +(10−5eV 2) +6.82 → 8.04 +6.82 → 8.04 +∆ m2 +3l +(10−5eV 2) +2.435 → 2.598 +−2.581 → +−2.414 +sin2 θ12 +0.264 → 0.343 +0.269 → 0.343 +sin2 θ23 +0.415 → 0.616 +0.419 → 0.617 +sin2 θ13 +0.02032 → +0.02410 +0.02052 → +0.02428 +TABLE I: Global fit 3σ values for neutrino oscillation parameters. +2 + +We have used the definition, +∆m2 +3l = ∆m2 +31; ∆m2 +31 > 0; NO +(1.1) +∆m2 +3l = ∆m2 +32; ∆m2 +32 < 0; IO +(1.2) +The simplest way to look for neutrino masses is by the seesaw mechanism. The mechanism may +be of type I [9], [10],type II [11], [12],type III [13] and Inverse Seesaw [14]. These are extensions +of the SM where we incorporate extra particles like right-handed fermions,scalar fermion triplets, +gauge singlet neutral fermions etc. The BSM physics also sheds light upon the phenomena like +baryon asymmetry of the universe (BAU) [15], Lepton Number Violation (LNV) [16], Lepton +Flavor violation (LFV) [17], existence of dark matter [18], [19] etc. A BSM framework which has +been successful in explaining the first three of the phenomenologies is the Left- Right Symmetric +Model (LRSM) [20–24], an extension of the SM corresponding to the addition of SU(2)R group +into the theory. The gauge group of LRSM is SU(3)C ⊗ SU(2)R ⊗ SU(2)L ⊗ U(1)B−L. The type +I and type II seesaw masses appear naturally in the model. The right-handed neutrinos are an +essential part of the model, which acquires Majorana mass when SU(2)R symmetry is broken. +LRSM provides a natural framework to understand the spontaneous breaking of parity and origin +of small neutrino masses by seesaw mechanism [25]. +Another concerning aspect is the ambiguity regarding nature of neutrinos which has not been yet +predicted by the SM of particle physics, that whether neutrinos are Dirac or Majorana fermions. +This problem is directly connected to the issue of lepton number conservation. One of the process +of fundamental importance which arises in almost any extension of the SM is Neutrinoless Double +Beta Decay(NDBD) [26], [27] which when verified can assure that neutrinos are Majorana fermions. +NDBD is a slow, radiative process that transforms a nuclide of atomic number Z into its isobar +with atomic number Z+2 [28], +N(A, Z) → N(A, Z + 2) + e− + e− +(1.3) +The main aim in the search of NDBD (0νββ) is the measurement of effective Majorana neutrino +mass, which is a combination of the neutrino mass eigenstates and neutrino mixing matrix terms +[28]. However, no experimental evidence regarding the decay has been in picture till date. In +3 + +addition to the determination of the effective masses, the half-life of the decay [29] combined with +sufficient knowledge of the nuclear matrix elements (NME), we can set a constraint on the neutrino +masses. The experiments like KamLAND-Zen [30] and GERDA [31] which uses Xenon-136 and +Germanium-76 respectively have improved the lower bound on the half-life of the decay process. +However, KamLAND-Zen imposes the best lower limit on the half life as T 0ν +1/2 > 1.07 × 1026 yr at +90 % CL and the corresponding upper limit of the effective Majorana mass in the range (0.061- +0.165)eV. There are several contributions in LRSM that appear due to additional RH current +interactions, giving rise to sizeable LFV rates for TeV scale RH neutrino that occur at rates +accessible in current experiments. It has been found that the most significant constraints has been +provided by the decays, µ → 3e and µ → γe. In the Standard Model, these LFV decays are +suppressed by the tiny neutrino masses. No experiment has so far observed any flavor violating +processes including charged leptons. However, many experiments are currently going on to set +strong limits on the most relevant LFV observables that will constrain the parameter space of +many new models. The best bounds on the branching ratio for LFV decays of the form µ → γe +comes from MEG experiment and it is set at BR(µ → γe) < 4.2 × 10−13. In case of the decay +µ → 3e, the bound is set by the SINDRUM experiment at BR(µ → 3e) < 1.0 × 10−12. +As mentioned LRSM is an important theory that incorporates the above mentioned phe- +nomenologies, i.e., the phenomenologies related to neutrinos. There are many works where the +authors make use of discrete symmetry groups like A4 [32],S4 [33],Z2 etc. [34] to analyze the prob- +lem of flavor structure of fermions and to study various related phenomenologies. In our work, +we have used A4 modular symmetry to study neutrino masses and mixings and hence study Neu- +trinoless Double Beta Decay within the model. The advantage of using modular symmetry over +discrete flavor symmetries is that the study of the model using symmetries can be done without +the introduction of extra particles called ’flavons’. Hence the model is minimal. +However, in this work we have not done a very detailed analysis of the above mentioned phe- +nomenologies, but only realized the left-right symmetric model with the help of A4 modular sym- +metry and studied the variations of new physics contributions of neutrinoless double beta decay +within LRSM with the range of values for Yukawa couplings, which in our model is expressed as +modular forms. In section (II), we have given a detailed explanation of the left-right symmetric +model, the associated Lagrangian and the mass terms. We begin section (III) by introducing +4 + +modular symmetry and then in section (IV), we incorporate modular symmetry into LRSM +and determine the associated mass matrices. In section (V), we present a very brief discussion +of neutrinoless double beta decay and its associated contributions and their relations with the +modular forms. In section (VI), the numerical analysis and results of this work has been discussed +and the last section reads the conclusion for the present work. +II. +MINIMAL LEFT-RIGHT SYMMETRIC MODEL +The Left-Right Symmetric Model (LRSM) was first introduced around 1974 by Pati and Salam. +Rabindra N. Mohapatra and Goran Senjanovic were also some pioneers of this very elegant theory. +LRSM is an extension of the Standard Model of particle physics, the gauge group being SU(3)C ⊗ +SU(2)R ⊗ SU(2)L ⊗ U(1)B−L, which has been studied by several groups since 1970’s [25], [21–24]. +The usual type-I and type-II seesaw neutrino masses arises naturally in the model. The seesaw +scale is identified by the breaking of SU(2)R. Some other problems are also addressed in LRSM +like parity, CP violation of weak interaction, massive neutrinos, hierarchy problems, etc. LRSM +removes the disparity between the left and right-handed fields by considering the RH fields to be +doublet under the additional SU(2)R keeping the right sector couplings same as the left-one by left- +right symmetry. In this model, the electric charge is given by Q = T3L +T3R + B−L +2 , where T3L and +T3R are the generators of SU(2)L and SU(2)R respectively. B − L refers to baryon number minus +lepton number. The particle content of the model along with their respective charge assignments +are given in table(III). The matrix representation for the scalar sector is given by, +φ = +� +�φ0 +1 φ+ +1 +φ− +2 +φ0 +2 +� +� +(2.1) +∆L,R = +� +� +δ+ +L,R +√ +2 +δ++ +L,R +δ0 +L,R − +δ+ +L,R +√ +2 +� +� +(2.2) +In order for the fermions to attain mass, a Yukawa Lagrangian is necessary which couples to the +bidoublet φ. The Yukawa Lagrangian incorporating the bidoublet is given by, +LD = liL(Y l +ijφ + � +Y l +ij �φ)ljR + QiL(Y q +ijφ + � +Y q +ij �φ)QjR + h.c +(2.3) +5 + +where, lL and lR are the left-handed and right-handed lepton fields, QL and QR are the left- +handed and right-handed quark fields. Y l being the Yukawa coupling corresponding to leptons and +Y q being the Yukawa coupling for the quarks. The Yukawa Lagrangian incorporating the scalar +triplets which play a role in providing Majorana mass to the neutrinos is given by, +LM = fL,ijΨL,i +TCiσ2∆LΨL,j + fR,ijΨR,i +TCiσ2∆RΨR,j + h.c +(2.4) +fL and fR are the Majorana Yukawa couplings and are equal subjected to discrete left-right sym- +metry. The scalar potential in LRSM is a combination of interaction terms consisting the potential +and after spontaneous symmetry breaking the scalars attain VEVs given by, +< ∆L,R >= 1 +√ +2 +� +� 0 +0 +vL,R 0 +� +� +(2.5) +< φ >= +� +�k +0 +0 eiθk′ +� +� +(2.6) +The magnitudes of the VEVs follows the relation, |vL|2 < |k2 + k′2| < |vR|2. The breaking pattern +of the LRSM gauge group takes place in two steps. The LRSM gauge group is first broken down +to the Standard Model gauge group by the vev of the scalar triplet ∆R, and then the Standard +Model gauge group is broken down to the electromagnetic gauge group i.e., U(1)em by the vev of +the bidoublet and a tiny vev of the scalar triplet ∆L. +The Dirac mass terms for the leptons come from the Yukawa Lagrangian, which for the charged +leptons and the neutrinos are given by, +Ml = 1 +√ +2(k′Yl + k ˜Yl) +(2.7) +MD = 1 +√ +2(kYl + k′ ˜Yl) +(2.8) +The light neutrino mass after spontaneous symmetry breaking (SSB), generated within a type +(I+II) seesaw can be written as, +Mν = Mν +I + Mν +II, +(2.9) +Mν = MDMRR +−1MD +T + MLL +(2.10) +6 + +where, +MLL = +√ +2vLfL +(2.11) +and, +MRR = +√ +2vRfR +(2.12) +The first and second terms in corresponds to type-I seesaw and type-II seesaw masses respectively. +It is an interesting fact that in the context of LRSM both type-I and type-II terms can be equally +dominant or either of the two terms can be dominant, but under certain conditions [35, 36]. It +has been demonstrated in the Appendix A. In the context of LRSM however, both the type-I and +type-II mass terms can be expressed in terms of the heavy right-handed Majorana mass matrix, +so equation (2.10) will follow, +Mν = MDM −1 +RRM T +D + γ +� +MW +vR +�2 +MRR +(2.13) +where, γ is a dimensionless parameter which is a function of various couplings, appearing in the +VEV of the triplet Higgs ∆L, i.e., vL = γ( v2 +vR) and here, v = +√ +k2 + k′2, and +γ = β1kk′ + β2k2 + β3k′2 +(2ρ1 − ρ3)(k2 + k′2) +(2.14) +In our model, the dimensionless parameter γ has been fine tuned to γ ≈ 10−6 and vR is of the +order of TeV . +III. +MODULAR SYMMETRY +Modular symmetry has gained much importance in aspects of model building [37], [38]. This +is because it can minimize the extra particle called ’flavons’ while analyzing a model with respect +to a particular symmetry group. An element q of the modular group acts on a complex variable τ +which belongs to the upper-half of the complex plane given as [38] [39] +qτ = aτ + b +cτ + d +(3.1) +where a, b, c, d are integers and ad − bc = 1, Imτ>0. +7 + +The modular group is isomorphic to the projective special linear group PSL(2,Z) = SL(2,Z)/Z2 +where, SL(2,Z) is the special linear group of integer 2 × 2 matrices having determinant unity and +Z2 = (I, −I) is the centre, I being the identity element. The modular group can be represented in +terms of two generators S and T which satisfies S2 = (ST)3 = I. S and T satisfies the following +matrix representations: +S = +� +� 0 +1 +−1 0 +� +� +(3.2) +T = +� +�1 1 +0 1 +� +� +(3.3) +corresponding to the transformations, +S : τ → −1 +τ ; T : τ → τ + 1 +(3.4) +Finite modular groups (N ≤ 5) are isomorphic to non-abelian discrete groups, for example, +Γ(3) ≈ A4, Γ(2) ≈ S3, Γ(4) ≈ S4. While using modular symmetry, the Yukawa couplings can +be expressed in terms of modular forms, and the number of modular forms present depends upon +the level and weight of the modular form. For a modular form of level N and weight 2k, the +table below shows the number of modular forms associated within and the non-abelian discrete +symmetry group to which it is isomorphic [39]. +N No. of modular forms Γ(N) +2 +k + 1 +S3 +3 +2k + 1 +A4 +4 +4k + 1 +S4 +5 +10k + 1 +A5 +6 +12k +7 +28k - 2 +TABLE II: No. of modular forms corresponding to modular weight 2k. +8 + +In our work, we will be using modular form of level 3, that is, Γ(3) which is isomorphic to A4 +discrete symmetry group. The weight of the modular form is taken to be 2, and hence it will have +three modular forms (Y1, Y2, Y3) which can be expressed as expansions of q given by, +Y1 = 1 + 12q + 36q2 + 12q3 + 84q4 + 72q5 + 36q6 + 96q7 + 180q8 + 12q9 + 216q10 +(3.5) +Y2 = −6q1/3(1 + 7q + 8q2 + 18q3 + 14q4 + 31q5 + 20q6 + 36q7 + 31q8 + 56q9) +(3.6) +Y3 = −18q2/3(1 + 2q + 5q2 + 4q3 + 8q4 + 6q5 + 14q6 + 8q7 + 14q8 + 10q9) +(3.7) +where, q = exp(2πiτ). +IV. +MINIMAL LRSM WITH A4 MODULAR SYMMETRY +In particle physics, symmetries have always played a very crucial role. The realization of LRSM +with the help of discrete flavor symmetries have been done in earlier works [40], [41]. +In our +work we have incorporated A4 modular symmetry into LRSM. The advantage of using modular +symmetry rather than flavor symmetry is the minimal use of extra particles (flavons) and hence the +model is minimal. The model contains usual particle content of LRSM [42]. The lepton doublets +transform as triplets under A4 and the bidoublet and scalar triplets transform as 1 under A4 [43]. +As we have considered modular symmetry, we assign modular weights to the particles, keeping +in mind that matter multiplets corresponding to the model can have negative modular weights, +but the modular forms cannot be assigned negative weights. The assignment of these weights are +done in such a way that in the Lagrangian the sum of the modular weights in each term is zero. +Modular weights corresponding to each particle is shown in table (III). The Yukawa Lagrangian +for the leptonic and quark sector in LRSM is given by equation (2.3),(2.4) and with a reference to +that we can write the Yukawa Lagrangian of our A4 modular symmetric LRSM, for the fermionic +sector, by introducing Yukawa coupling in the form of modular forms Y is given as, +LY = lLφlRY + lL ˜φlRY + QLφQRY + QL ˜φQRY + lR +TCiτ2∆RlRY + lL +TCiτ2∆LlLY +(4.1) +9 + +Gauge group QL QR lL lR φ ∆L ∆R +SU(3)C +3 +3 +1 +1 1 +1 +1 +SU(2)L +2 +1 +2 +1 2 +3 +1 +SU(2)R +1 +2 +1 +2 2 +1 +3 +U(1)B−L +1/3 1/3 -1 -1 0 +2 +2 +A4 +3 +3 +3 +3 1 +1 +1 +kI +0 +-2 +0 -2 0 -2 +2 +TABLE III: Charge assignments for the particle content of the model. +The Yukawa couplings Y = (Y1, Y2, Y3) are expressed as modular forms of level 3. +Y (modular forms) +A4 +3 +kI +2 +TABLE IV: Charge assignment and modular weight for the corresponding modular Yukawa form +for the model. +In our work, we are concerned with the mass of the neutrinos and as such, using A4 modular +symmetry and using the multiplication rules for A4 group, we construct the Dirac and Majorana +mass matrices as given below. The Dirac mass matrix is given by, +MD = v +� +� +� +� +2Y1 −Y3 −Y2 +−Y2 −Y1 2Y3 +−Y3 2Y2 −Y1 +� +� +� +� +(4.2) +where, v is considered to be the VEV for the Higgs bidoublet. +The right-handed Majorana mass matrix is given by, +MR = vR +� +� +� +� +2Y1 −Y3 −Y2 +−Y3 2Y2 −Y1 +−Y2 −Y1 2Y3 +� +� +� +� +(4.3) +10 + +where, vR is the VEV for the scalar triplet ∆R. As it is seen that the Majorana mass matrix for our +model is found to be symmetric in nature as it should be. Under these assumptions for modular +symmetric LRSM and the basis that we have considered, our charged lepton mass matrix is also +found to be diagonal. +The type-I seesaw mass is then given by, +MνI = MD.MR +−1.MD +T +(4.4) +and, the type-II seesaw mass is given by, +MνII = MLL +(4.5) +As mentioned above, in LRSM type-II seesaw mass can also be expressed in terms of the right- +handed mass MR as, +MνII = γ +� +MW +vR +�2 +MR +(4.6) +A. +Type-I dominanace +In LRSM, the type-I seesaw mass dominates when the vev of the left-handed triplet is taken to +be negligibly small and hence the type-II term is absent. In such a case the lightest neutrino mass +can be given in terms of the type-I seesaw mass term given by, +Mν = MDMR +−1MD +T +(4.7) +and the heavy right-handed Majorana mass term can be given as, +MR = fRvR +(4.8) +where, fR is the right-handed Majorana Yukawa coupling. +In the approximation that k′ << k, and if we consider that our Yukawa coupling Y l corresponding +to the neutrino masses is yD and the coupling � +Y l for the charged fermion masses is denoted by yL, +so considering yDk >> yLk′ we can write the type-I mass term as [44], +Mν = k2 +vR +yDf −1 +R yT +D +(4.9) +11 + +If we consider that UR is a unitary matrix that diagonalizes MR, so since the VEV vR is a constant +the same matrix will also diagonalize the coupling matrix fR. Taking fR = fL = f, so +f = URf diaU T +R +(4.10) +If we take inverse on both sides and taking into account the property of a unitary matrix (U −1 +R = +U T +R), we get, +f −1 = U T +R(f dia)−1UR +(4.11) +Therefore, we get +Mν = k2 +vR +yDU T +R(f dia)−1URyT +D +(4.12) +Multiplying both sides of the equation with U T +R from the right and with UR from left, we finally +arrive at the following equation, +URMνU T +R = (Mν)dia +(4.13) +where we have used URyDU T +R = yD. So, the unitary matrix diagonalizing the matrix MR also +diagonalizes the light neutrino mass matrix. +So in this case it can be determined that if mi +denotes the light neutrino mass and Mi denotes the heavy neutrino mass, then they are related as +mi ∝ 1 +Mi +(4.14) +For our model, the Yukawa couplings are modular forms expressed as expansions of q, and the +mass matrices are expressed in terms of the modular forms (Y1, Y2, Y3). So, the light neutrino mass +matrix, Mν for type-I dominance is given by the equation (4.7). As already stated in equations +(4.2) and (4.3), the Dirac and Majorana mass matrices are determined by the application of +multiplication rules for the A4 group. So, for type-I dominance, our light neutrino mass matrix +will be given by, +Mν = v2 +vR +� +� +� +� +2Y1 −Y2 −Y3 +−Y2 2Y3 −Y1 +−Y3 −Y1 2Y2 +� +� +� +� +(4.15) +As mentioned previously, the value for vR is of the order of TeV and that for v is in GeV . We +have computed the values of the sum of the neutrino masses for type-I dominance and checked the +correctness of our model by plotting it against the Yukawa couplings and the result was found to +match the experimental bounds. +12 + +FIG. 1: Variation of |Y1| with sum of neutrino masses. +FIG. 2: Variation of |Y2| with sum of neutrino masses. +FIG. 3: Variation of |Y3| with sum of neutrino masses. +13 + +.NH +Excluded KamLAND-Zen region +0.100 +0.050 F +(Aa) +0.010F +Em +0.005E +0.001E +Allowed 3oregion +5.10- +1.×10-9 +5.×10-1.×10-s +5.×101.×10 +5.x10 +[Y1]IH +Exciuded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed 3o region +10-5 +2.×10-7 +4. ×10-7 +6.× 10-7 +8.×10-7 +[Y1]NH +Excluded KamLAND-Z +001°0 +egio +0.050 F +(19) +0.010F +Zm, +0.005 +0.001 +Allowed 3gregion +5.10- +5.x10-9 1.× 10-s +5.x10s1.x10 +5.x10-7 1. ×10-6 +[Y2]IH +Exciuded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed 3oregion +10-5 +5.×10-91.×108 +5.×1081.×10-7 +5.×10-71.×10-6 +[Y2]NH +0.100 +Excluded KamLAND +0.050 +(19) +0.010F +Em. +0.005 +0.001E +Allowed3gregion +5.10- +5.x10-1.x10 +5.×101.×10 +5.×10-71.×10-6 +[Y3]IH +Exciuded KamLAND-Zen region +0.100 +0.010 +Aa) +0.001 +10-4 +Allowed Bo region +10-5 +1.×10-8 +5.×10-81.×10-7 +5.×1071.×10-6 +[Y3]B. +Type-II dominance +Type-II seesaw mass in LRSM dominates when the Dirac term connecting the right-handed and +left-handed parts is negligible as compared to that of the type-II term [44]. In that case, our light +neutrino mass mν will given by the type-II seesaw mass term, i.e., +MνL = fLvL +(4.16) +And the heavy mass matrix is given by, +MR = fRvR +(4.17) +Again if we consider that UL and UR diagonalizes MνL and MR respectively, so for the reason +mentioned above the same matrices will also diagonalize fL and fR respectively and since in our +model, fL = fR, so we can consider UL = UR. In such a case, we arrive at an important result that +mi ∝ Mi +(4.18) +Now using modular symmetry the light neutrino mass matrix for type-II dominance in our model +is given by, +mν = vL +� +� +� +� +2Y1 −Y3 −Y2 +−Y3 2Y2 −Y1 +−Y2 −Y1 2Y3 +� +� +� +� +(4.19) +where, vL is the vev for left-handed scalar triplet. The value of vL is taken to be of the order of +eV . The sum of the neutrino masses is computed for type-II dominance and plotting of the sum +is done with the Yukawa couplings which are found to be as shown under, +FIG. 4: Variation of |Y1| with sum of neutrino masses. +14 + +HN +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed 3o region +10-5 +5.x10-911.x10-8 +5.x10-81.×10-7 +5.×10-7 +(Y1]IH +Exciuded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed3o region +10-5 +2.×10-7 +4.×10-7 +6. × 10-7 +8.×10-7 +[YI]FIG. 5: Variation of |Y2| with sum of neutrino masses. +FIG. 6: Variation of |Y3| with sum of neutrino masses. +V. +NEUTRINOLESS DOUBLE BETA DECAY (0νββ) IN MINIMAL LRSM +Neutrinoless double beta decay is a lepton number violating process, which if proven to exist will +directly imply the Majorana nature of neutrinos. +N(A, Z) → N(A, Z + 2) + e− + e− +(5.1) +Many groups have however already done a lot of work on NDBD in the model , [21],[28, 45–50]. In +LRSM [51], there are several contributions to NDBD in addition to the standard contribution via +light Majorana neutrino exchange owing to the presence of several heavy additional scalar,vector +15 + +.IH +Exciuded KamLAND-Zen region +0.100 +1010:0 +0.001 +10-4 +Allowed 3o region +10-5 +5.×10-91.×108 +5.×10-81.×107 +5. × 10-7 1. × 10-6 +[Y2]NH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed3oregion +10-5 +5.×10-91.×10-8 +5.×10-81.×10-7 +5.×10-71.×10-6 +[Y3].IH +Exciuded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed 3o region +10-5 +5.×10-91.×10-8 +5.×10-81.×10-7 +5.×10-71.×10-6 +[Y3]NH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed3gregion +10-5 +5.x10-81.x10-7 +5.x10-7 +1.x10-6 +[Y2]and fermionic fields [52–55]. Various contributions to NDBD transition rate in LRSM are discussed +as follows : +• Standard Model contribution to NDBD where the intermediate particles are the WL bosons +and light neutrinos, the process in which the amplitude depends upon the leptonic mixing +matrix elements and light neutrino masses. +• Heavy right-handed neutrino contribution in which the mediator particles are the WL bosons +and the amplitude depends upon the mixing between light and heavy neutrinos as well as +the mass of the heavy neutrino. +• Light neutrino contribution to NDBD where the intermediate particles are WR bosons and +the amplitude depends upon the mixing between light and heavy neutrinos as well as mass +of the right-handed gauge boson WR. +• Heavy right-handed neutrino contribution where the mediator particles are the WR bosons. +The amplitude of this process is dependent on the elements of the right handed leptonic +mixing matrix and mass of the right-handed gauge boson, WR as well as the mass of the +heavy right handed Majorana neutrino. +• Light neutrino contribution from the Feynman diagram mediated by both WL and WR, +and the amplitude of the process depends upon the mixing between light and heavy neutri- +nos,leptonic mixing matrix elements, light neutrino masses and the mass of the gauge bosons, +WL and WR. +• Heavy neutrino contribution from the Feynman diagram mediated by both WL and WR, +and the amplitude of the process depends upon the right handed leptonic mixing matrix +elements, mixing between the light and heavy neutrinos, also the mass of the gauge bosons, +WL and WR and the mass of the heavy right handed neutrino. +• Scalar triplet contribution (∆L) in which the mediator particles are WL bosons, and the +amplitude for the process depends upon the masses of the WL bosons, left-handed triplet +Higgs, as well as their coupling to leptons. +16 + +• Right-handed scalar triplet contribution (∆R) contribution to NDBD in which the mediator +particles are WR bosons, and the amplitude for the process depends upon the masses of the +WR bosons, right-handed triplet Higgs, ∆R as well as their coupling to leptons. +In our work, where we have incorporated A4 modular symmetry to LRSM and in our present +work we have considered three of the above mentioned contributions, one from the standard light +neutrino contribution and the other two new physics contribution mediated by W − +R and ∆R re- +spectively. For simple approximations, an assumption has been made in the mass scales of heavy +particles, where, +MR ≈ MWR ≈ M∆L++ ≈ M∆R++ ≈ TeV +. Under these assumptions, the amplitude for the light-heavy mixing contribution which is pro- +portional to mD2 +MR remains very small, since mν ≈ mD2 +MR ≈ (0.01 − 0.1)eV, mD ≈ (105 − 106)eV which +implies mD +MR ≈ (10−7 − 10−6)eV . Thus in our model, we ignore the contributions involving the light +and heavy neutrino mixings. +When NDBD is done in the framework of LRSM, the standard light neutrino contribution is +given by, +meff +v += U 2 +Limi +(5.2) +where, ULi are the elements of the first row of the neutrino mixing matrix UPMNS, in which the +elements are dependent on known mixing angles θ13 , θ12 and the Majorana phases κ and η. The +UPMNS matrix is given by, +UPMNS = +� +� +� +� +c12c13 +s12c13 +s13e−iδ +−c23s12 − s23s13c12eiδ −c23c12 − s23s12s13eiδ s23c13 +s23s12 − c23s13c12eiδ +−s23c12 − c23s13s12eiδ +c23c13 +� +� +� +� P +(5.3) +where, P = diag(1, eiκ, eiη). So the effective mass can be parametrized in terms of the elements of +the diagonalizing matrix and the eigenvalues as, +meff +v += m1c2 +12c2 +13 + m2s2 +12c2 +13e2iκ + m3s2 +13e2iη. +(5.4) +VI. +NUMERICAL ANALYSIS AND RESULTS +In our present work, we have modified left-right symmetric model by incorporating A4 modu- +17 + +lar symmetry for both type-I and type-II dominances. As we are using modular symmetry, the +Yukawa couplings are expressed as expansions of q as shown in equations (3.5),(3.6) and (3.7). In +our model, the value of q is found to be of the order of 10−1. The aboslute value of the modulus +should however be greater than 1. +τ = Re(τ) + Im(τ) +(6.1) +Re(τ) +Im(τ) +|τ| +Range [0.715,0.789] [0.8,0.9] [1.073,1.197] +TABLE V: Range of values corresponding to real and imaginary parts of the modulus τ. +Yukawa couplings Normal Hierarchy Inverted hierarchy +Y1(min) +1.29155 × 10−9 +1.32276 × 10−7 +Y1(max) +8.22986 × 10−7 +9.21382 × 10−7 +Y2(min) +3.02229 × 10−9 +2.42826 × 10−9 +Y2(max) +1.32006 × 10−6 +1.45952 × 10−6 +Y3(min) +3.39287 × 10−9 +3.759 × 10−9 +Y3(max) +1.3165 × 10−6 +1.50082 × 10−6 +TABLE VI: Range of Yukawa couplings (Y1, Y2, Y3) for both normal and inverted hierarchy. +From table (V), it is seen that the absolute value of the modulus is greater than unity, which +is the expected result. The range of the Yukawa couplings for our model is shown in the table +above. It is seen from the table that the minimum value of the Yukawa coupling Y1 is in the scale +of 10−10 for normal hierarchy while for inverted hierarchy it is in the scale of 10−7. However, for +the maximum of Y1 both the orderings are in the same scale. For Y2 and Y3, the minimum and +maximum values for both normal and inverted hierarchies are within the same scale. We have +plotted the effective masses against the Yukawa couplings (Y1, Y2, Y3) and it was found to be well +within the bound set by experiments. +As shown both for type-I and type-II dominances, we have plotted the absolute values of the +18 + +Yukawa couplings against the sum of the neutrino masses. The range of the values for sum of +neutrino masses for both the cases are given as under, +� mν +Normal Hierarchy Inverted hierarchy +Type − I(min) +0.000980556 +0.000437758 +Type − I(max) +0.177296 +0.186377 +Type − II(min) +0.000219304 +0.000035 +Type − II(max) +0.0200981 +0.0203081 +TABLE VII: Range of values for sum of neutrino masses for type-I and type-II dominances for +both normal and inverted hierarchy. +A. +Standard Light Neutrino Contribution to 0νββ +As mentioned above, in the standard light neutrino contribution to 0νββ, the intermediate +particles are the WL bosons and light neutrino. The effective mass for the contribution is given +by equation (5.2). Simplifying for the respective elements of ULi and mi, the value of the effective +mass is obtained in terms of the modular forms (Y1, Y2, Y3) as, +meff +ν += meff +1 ++ meff +2 ++ meff +3 +(6.2) +where, +meff +1 += ν(Y2 − Y3)2(Y1 + Y2 + Y3) +νR(Y1 − Y3)2 +meff +2 += ν2(Y1 − Y2)2(Y1 + Y2 + Y3 − +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2νR(Y1 − Y3)2 +meff +3 += ν2(Y1 + Y2 + Y3 − +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2νR +for type-I dominance, and the plots are shown as, +19 + +FIG. 7: Variation of |Y1| with effective neutrino mass for standard light neutrino contribution. +FIG. 8: Variation of |Y2| with effective neutrino mass for standard light neutrino contribution. +FIG. 9: Variation of |Y3| with effective neutrino mass for standard light neutrino contribution. +20 + +1 +.NH +Excluded KamLAND-Zen region +0.100 +0.010 +(Aa) +0.001 +10-4 +Allowed 3o region +10-5 +5.×10-9 +1.×10-8 +5.×108 +1.x10~ +5.x10- +[Y1].H +Excluded KamLAND-Zen region +0.100 +(Aa) +meff +0T0:0 +0.001 +Allowed 3o region +107 +2. ×10-7 +4. × 10-7 +6. ×10-7 +8.×10-7 +[Y1]1 +.NH +Excluded KamLAND-Zen region +0.100 +0.010 +(eV) +0.001 +10-4 +Allowed 3o region +10-5 +5.×1091.×108 +5.×10-81.×10-7 +5.×10-7 +1.×10-6 +[Y2].IH +Excluded KamLAND-Zen region +0.100 +(Aa) +meff +0.010 +0.001 +Allowed 3o region +107 +5.×101.×108 +5.×10-81.×10-7 +5.×10-71.×10-6 +[Y2].NH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +10-4 +Allowed3oregion +10-5 +5.×10-91.×10-8 +5.×10-1.×10 +5.x10-71.x10-6 +[Y3].IH +Excluded KamLAND-Zen region +0.100 +(Aa) +0.010 +0.0015 +Allowed 3o region +10-4 +5.×1091.×10-8 +5.×10-81.×10-7 +5.×10-71.×10-6 +/Y3For type-II dominance, we have +meff +1 += −νL(−Y2 + Y3)(Y1 + Y2 + Y3) +Y1 − Y2 +meff +2 += −νL(Y1 − Y3)(Y1 + Y2 + Y3 − +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2(Y1 − Y2) +meff +3 += νL(Y1 − Y3)(Y1 + Y2 + Y3 + +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2 +FIG. 10: Variation of |Y1| with effective neutrino mass for standard light neutrino contribution. +FIG. 11: Variation of |Y2| with effective neutrino mass for standard light neutrino contribution. +21 + +.NH +Excluded KamLAND-Zen region +0.100 +0.010 +(eV) +0.001 +10-4 +Allowed3gregion +10-5 +5.×10-91.×10-8 +5.×10-1.×10- +5.×10-71.×10-6 +[Y1].IH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001E +Allowed3oregidh +5.0x10~7 +1.0-10-5 +1.5,10-5 +2.0x10-5 +2.5x10-5 +3.010-5 +3.5x10-5 +YINH +Excluded KamLAND-Zen region +0.100 +0.010 +(Aa) +Meff +0.001 +10-4 +Allowed 3g region +10-5 +5.x1091.×108 +5.x101.x107 +5.× 10-71. x10-6 +5.×10-6 +(Y2).IH +Excluded KamLAND-Zen region +0.100 +(A) +O1O1O +0.001 +Allowed 3o region +10 +5.×10-8 +1.×10-7 +5.×10-7 +1.×10-6 +5.×10-6 +[Y2]FIG. 12: Variation of |Y3| with effective neutrino mass for standard light neutrino contribution. +B. +Heavy Right-Handed Neutrino contribution to 0νββ +In our work, we have considered contributions of heavy right-handed neutrino and scalar Higgs +triplet to NDBD. The effective mass for heavy right-handed neutrino is given by, +meff +R += p2 +� +M 4 +WL +M 4 +WR +�� +U ∗2 +Rei +Mi +� +(6.3) +where, p2 is the typical momentum exchange of the process. As it is known that TeV scale LRSM +plays a very important role in the process of neutrinoless double beta decay (0νββ), we have +considered the values as MWR = 10TeV , MWL = 80GeV , M∆R ≈ 3TeV and after calculation, +the value for heavy right-handed neutrino is found to be in the scale of TeV . The allowed value +of p is in the range (100 − 200)MeV and so we consider, p ≈ 180MeV . Thus, we get, +p2 +� +M 4 +WL +M 4 +WR +� += 1010eV 2 +(6.4) +where, URei refers to the first row elements of the diagonalizing matrix of the heavy Majorana mass +matrix and Mi are its eigenvalues. The effective mass corresponding to the heavy right-handed +neutrino can be expressed in terms of the modular forms as, +mR +eff = 1010(mR1 +eff + mR2 +eff + mR3 +eff) +(6.5) +where, +mR1 +eff = +2 +νR(Y1 + Y2 + Y3 + +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +22 + +1 +NH +Excluded KamLAND-Zen region +0.100 +0.010 +(eV) +0.001 +10-4 +Allowed 3oregion +10-5 +1.x10-8 +5.×10-81.x10 +5.×1071.×10-6 +5.×10-6 +[Y3].IH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +Allowed 3oregion +5.x10~9 +1. ×10-S +5.×10-8 +1.×10~7 +5.x10-7 +1.x106 +5.×10~6 +(Y3)mR2 +eff = +2(Y ∗ +1 − Y ∗ +3 )2 +νR(Y1 + Y2 + Y3 − +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 )(Y ∗ +1 − Y ∗ +2 )2 +mR3 +eff = +(−Y ∗ +2 + Y ∗ +3 )2 +νR(Y1 + Y2 + Y3)(Y ∗ +1 − Y ∗ +2 )2 +The total effective mass is also calculated for the standard light and right-handed heavy neutrino +contribution, given by, +|mefftotal +ν +| = |meff +ν ++ mR +eff| +(6.6) +which can be obtained in terms of the modular forms as a summation of the above mentioned +terms. +FIG. 13: Variation of |Y1| with total effective neutrino mass. +FIG. 14: Variation of |Y2| with total effective neutrino mass. +23 + +NH +Excluded KamLAND-Zenregion +0.100 +0.010 +0.001 +10-4 +Ailowed3oregion +10-5 +[Y1]IH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +Allowed3gregion +5.0-10-7 +1.010-5 +1.5,10-6 +2.0x10-5 +2.5x10-5 +3.0±105 +3.5x10-5 +YI.NH +Excluded KamLAND-Zenregion +0.100 +0.010 +(10) +total +0.001 +10-4 +Allowed 3o region +105 +5.×10-91.×10-8 +5.×10-$1.×10-7 +5.×10-71.×10-6 +5.×10-6 +[Y2]H +Exciuded KamLAND-Zen region +0.100 +010'0 +100'0 +Ailowed3oregion +10-4 +1.×10-8 +5.×1081.×107 +5.x10-7 +1.x10-6 +5.×10-6 +/Y2]FIG. 15: Variation of |Y3| with total effective neutrino mass. +The plots above are for type-I dominance. +FIG. 16: Variation of |Y1| with total effective neutrino mass. +FIG. 17: Variation of |Y2| with total effective neutrino mass. +24 + +.NH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +10~4 +Allowed 3g region +10~5 +1.×108 +5.×108 1.×10- +5.×10-7 +1.×10-6 +5.×10-6 +[Y3]H +Excluded KamLAND-Zen region +0.100 +0.010 +0.001 +Ailowed3o region +10-4 +1.× 10-8 +5.x10-8 +1.×10-7 +5.×10-7 +1.x10-6 +5.×106 +[Y3]NH +Excluded KamLAND-Zen.region +0.100 +O1O'0 +0.001 +10-4 +Ailowed3o region +10-5 +5.x10-81.x10-7 +5.×10-71.x10-6 +[Y1]IH +Excluded KamLAND-Zen region +0.100 +0.010 +0.001E +Allowed 3g region +10-4 +5.0x10~7 +1.0-10-5 +1.510-5 +2.0x10-5 +2.5x10-5 +3.010-5 +3.5x10-5 +YI.NH +Excluded KamLAND-Zen.region +0.100 +0.010 +(10) +total +0.001 +10-4 +Allowed 3o region +10~5 +5.×10-91.×10-8 +5.×10-81.×10~ +5.×10-71.×10-6 +5.×10-6 +[Y2]H +Exciuded KamLAND-Zen region +0.100 +nltot +0.010 +100'0 +Ailowed3oregion +10-4 +5.×10-8 +1. × 10-7 +5.×10-7 +1.x10-6 +5.×10-6 +/Y2]FIG. 18: Variation of |Y3| with total effective neutrino mass. +The figures above are the plots for type-II dominance. +C. +Scalar Triplet contribution to 0νββ +The magnitude of ∆R contribution is controlled by the factor +Mi +M∆R [44]. However, scalar triplet +contribution is not included in the total contribution under the assumption +Mi +M∆R < 0.1. But, +some the mixing parameters in the large part of the parameter space may result in a higher +Mi +M∆R +ratio and in such cases we will have to include it in the total contribution. The impact of this +contribution here is studied in the limit, M∆R ≊ Mheaviest. +The effective mass for scalar triplet contribution is given as, +|meff +∆ | = |p2 M 4 +WL +M 4 +WR +U 2 +ReiMi +M 2 +∆R +| +(6.7) +The value of the mass for the right-handed scalar triplet is taken as, M∆R = 3TeV . So, the value +of the coefficient results as, +p2 M 4 +WL +M 4 +WR +1 +M 2 +∆R += +1010 +9 × 1024 +(6.8) +In terms of modular forms, the effective scalar mass can be expressed as, +m∆R +eff = m∆R +eff1 + m∆R +eff2 + m∆R +eff3 +(6.9) +where, +m∆R +eff1 = νR(Y2 + Y3)2(Y1 + Y2Y3) +(Y1 − Y2)2 +25 + +.NH +Excluded KamLAND-Zen region +0.100 +0.010 +ltotal( +0.001 +10-4 +Allowed 3g region +1.×10-8 +5.×10-8 1. ×10 +5.×10-7 +1.×10-6 +5.×10-6 +[Y3]Exciuded KamLAND-Zen region +0.100 +0.010 +Hltot +0.001 +Allowed 3o region +5.×1091.×108 +5.x1081.x10-7 +5.×10-6 +IY3Im∆R +eff2 = νR(Y1 − Y3)2(Y1 + Y2 + Y3 − +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2(Y1 − Y2)2 +m∆R +eff3 = νR(Y1 + Y2 + Y3 + +√ +3 +� +3Y 2 +1 − 2Y1Y2 + 3Y 2 +2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 +3 ) +2 +The plots are shown as under. +FIG. 19: Variation of |Y1| with effective neutrino mass for scalar triplet contribution. +FIG. 20: Variation of |Y2| with effective neutrino mass for scalar triplet contribution. +26 + +. NH +Excluded KamLAND-Zen region +0.100 +0.010 +(Aa) +0.001 +10-4 +Allowed 3o region +10-5 +5.x1091.x108 +5.x10-81.x10-7 +5.×1071.x10-6 +(Y1]·IH +Excluded KamLAND-Zen region +0.100 +0.010 +meffxh +0.01 +104F +Allowed3o region +10-5 +5.010-7 +1.0-10-6 +13:10-6 +2.0-10-5 +25±10-5 +3.0.10-5 +3.5x10-5 +YINH +Excluded KamLAND-Zen region +0.100 +0.0105 +(Aa) +0.001 +10-4 +Allowed 3o region +10-5 +5.×101.×108 +5.×10-1.×10 +5.×10-71.×10-6 +5.×10-6 +[Y2].IH +Exciuded KamLAND-Zen region +0.100 +0.010 +effsclr +0.001 +10-4 +Allowed 3oregion +10-5 +5.×10-81.×10-7 +5.×10-71.× 10-6 +5. × 10-6 +[Y2]FIG. 21: Variation of |Y3| with effective neutrino mass for scalar triplet contribution. +VII. +CONCLUSION +The discovery of neutrino oscillations paved the gateway for physics beyond the Standard Model. +In our paper, we have realized LRSM with the help of modular A4 symmetry for both type-I and +type-II dominance. Using modular symmetry provides the advantage of using no extra particles +called ’flavons’. The Yukawa couplings are represented as modular forms expressed as expansions +of q. The values of the Yukawa couplings (Y1, Y2, Y3) are calculated using ’Mathematica’. The mass +matrices are then determined using the multiplication rules for A4 group stated in the Appendix. +The Majorana mass matrix is found to be symmetric and under the considered basis, the charged +lepton mass matrix is also diagonal. We have expressed the light neutrino and heavy right-handed +neutrino mass matrix in terms of the modular forms. We have also studied briefly the contributions +of 0νββ in LRSM. The effective masses corresponding to standard light neutrino contribution, +right-handed contribution and scalar triplet contributions are determined in terms of (Y1, Y2, Y3) +and we have plotted the effective mass corresponding to the considered contributions against the +Yukawa couplings. To summarize our work, some results are stated as under, +• The absolute value of the modulus was found to be within the range 1.073 to 1.197, which +is greater than unity, that is the desired result. +• The Yukawa couplings, expressed in terms of modular forms ranges from 10−9 to 10−6. +• The sum of the neutrino masses for type-I dominance ranges from the order of 10−4 to 10−1 +27 + +INH +Excluded KamLAND-Zen region +0.100 +0.010 +(Aa) +0.001 +10-4 +Allowed 3o region +10-5 +1.×10-8 +5. × 10-8 1. ×107 +5.×10-71.×10-6 +5.×10-6 +[Y3].IH +Exciuded KamLAND-Zen region +0.100 +0.010 +effsclr +0.001 +10-4 +Allowed 3oregion +10-5 +5. ×10-8 1. × 10-7 +5. × 10-7 1. ×10-6 +5.×10-6 +[Y3]for both normal and inverted hierarchy. +• The sum of the neutrino masses for type-II dominance ranges from the order of 10−4 to 10−2 +for both normal and inverted hierarchy. +The effective masses for the 0νββ contributions are calculated and by determining their relations +with the modular forms, we have plotted the effective masses with the three Yukawa couplings and +it has been found that the values for the effective mass corresponding to each contribution is well +within the experimental bounds, which infact makes us clearly state that the building of the model +with modular symmetry is advantageous to that of flavor symmetries. In this model, we have not +used any extra particles and the analysis has been done taking into consideration the calculated +and computed values for the model parameters and the results are found to be satisfactory, so it +can be stated that the Left-Right Symmetric Model can be constructed with modular symmetry +while satisfying the experimental bounds on the desired parameters. +VIII. +APPENDIX A +Let us consider the Higgs potential of our model that has quadratic and quartic coupling terms +given by [36], +Vφ,∆L,∆R = −µ2 +ijTr[φ† +iφj] + λijklTr[φ† +iφj]Tr[φ† +kφl] + λ +′ +ijklTr[φ† +iφjφ† +kφl] − µ2 +ijTr[∆† +L∆L + ∆† +R∆R] +ρ1[(Tr[∆† +L∆L])2 + (Tr[∆† +L∆L])2] + ρ2(Tr[∆† +L∆L∆† +L∆L] + Tr[∆† +R∆R∆† +R∆R]) + ρ3Tr[∆† +L∆L∆† +R∆R]+ +αijTr[φ† +iφj](Tr[∆† +L∆L]+Tr[∆† +R∆R])+βij(Tr[∆† +L∆Lφiφ† +j]+Tr[∆† +R∆Rφiφ† +j])+γij(Tr[∆† +Lφi∆Rφ† +j]+h.c) +(8.1) +where, i,j,k,l runs from 1 to 2 with φ1 = φ and φ2 = ˜φ. As mentioned above after SSB, the scalar +sector obtains VEV. So after the substitution of the respective VEVs and determining the traces, +so after simplification the potential can be written as, +V = −µ2(v2 +L + v2 +R) + ρ +4(v4 +L + v4 +R) + ρ′ +2 + α +2 (v2 +L + v2 +R)k2 +1 + γvLvRk2 +(8.2) +where, we have used the approximation k′ << k, and ρ′ = 2ρ3. Our minimization conditions are, +δV +δvL = δV +δvR = δV +δk = δV +δk′ = 0 +28 + +Therefore, we get, +δV +δvL += −2µ2vL + ρv3 +L + ρ′vLk2 + γvRk2 +(8.3) +Here, it is evident that the Majorana mass of the left-handed neutrino MLL is dependent on the +vev vL as already defined above. Again, we have +δV +δvR += −2µ2vR + ρv3 +R + ρ′vRk2 + γvLk2 +(8.4) +So, the right handed Majorana mass MRR is dependent on the vev vR. Similarly, the calculations +for the same can be carried out and it can be found out the Dirac mass term MD can be expressed +in terms of the vev for the Higgs bidoublet as also defined previously. +Now, we are to determine a relation between the VEVs for the scalars and so after using the +minimization conditions and simplifying the equations, we come to a relation given by, +vLvR = γ +ξ k +(8.5) +where, ξ = ρ − ρ′. +The neutrino mass for LRSM is given as a summation of the type-I and type-II term as already +mentioned above. So, in the approximation that k′ << k, and if we consider that our Yukawa +coupling Y l corresponding to the neutrino masses is yD and the coupling � +Y l for the charged fermion +masses is denoted by yL, so considering yDk >> ylk′ we can write, +Mν = k2 +vR +yDf −1 +R yT +D + fLvL +(8.6) +Since, for due to left-right symmetry, we can consider fL = fR = f, so the above equation can be +written as, +Mν = k2 +vR +yDf −1yT +D + fvL +(8.7) +So, from this equation we can come to a relation given by, +Mν = (f γ +ξ + yDf −1yT +D) k2 +vR +(8.8) +Here, we can consider two situations, namely +• If f( γ +ξ ) << yDf −1yT +D, the light neutrino mass is given by the type-I term MDM −1 +RRM T +D. That +is, here type-I is dominant and the light neutrino mass is from the suppression of heavy νR. +• If f( γ +ξ ) >> yDf −1yT +D, the light neutrino mass is given by the type-II term fvL. That is, in +this case type-II mass term is dominant and the light neutrino mass is because of the tiny +value of νL. +29 + +IX. +APPENDIX B +Properties of A4 group +A4 is a non-abelian discrete symmetry group which represents even permuatations of four ob- +jects. It has four irreducible representations, three out of which are singlets (1, 1′, 1′′) and one +triplet 3 (3A represents the anti-symmetric part and 3S the symmetric part). Products of the +singlets and triplets are given by, +1 ⊗ 1 = 1 +1′ ⊗ 1′ = 1′′ +1′ ⊗ 1′′ = 1 +1′′ ⊗ 1′′ = 1′ +3 ⊗ 3 = 1 ⊕ 1′ ⊕ 1′′ ⊕ 3A ⊕ 3S +If we have two triplets under A4 say, (a1, a2, a3) and (b1, b2, b3) , then their multiplication rules are +given by, +1 ≈ a1b1 + a2b3 + a3b2 +1′ ≈ a3b3 + a1b2 + a2b1 +30 + +1′′ ≈ a2b2 + a3b1 + a1b2 +3S ≈ +� +� +� +� +2a1b1 − a2b3 − a3b2 +2a3b3 − a1b2 − a2b1 +2a2b2 − a1b3 − a3b1 +� +� +� +� +3A ≈ +� +� +� +� +a2b3 − a3b2 +a1b2 − a2b1 +a3b1 − a1b3 +� +� +� +� +X. +REFERENCES +[1] Justin Evans. 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JHEP, 04:046, 2016. +[54] Debasish Borah, Sudhanwa Patra, and Utpal Sarkar. TeV scale Left Right Symmetry with sponta- +neous D-parity breaking. Phys. Rev. D, 83:035007, 2011. +[55] Happy Borgohain and Mrinal Kumar Das. Lepton number violation, lepton flavor violation, and +baryogenesis in left-right symmetric model. Phys. Rev. D, 96(7):075021, 2017. +35 + diff --git a/ltFRT4oBgHgl3EQfZTcN/content/tmp_files/load_file.txt b/ltFRT4oBgHgl3EQfZTcN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fe3c4e635accc2899a48a0482c8fa74383f2fd3 --- /dev/null +++ b/ltFRT4oBgHgl3EQfZTcN/content/tmp_files/load_file.txt @@ -0,0 +1,1135 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf,len=1134 +page_content='Minimal Left-Right Symmetric Model with A4 modular symmetry Ankita Kakoti,1, ∗ Bichitra Bijay Boruah,1, † and Mrinal Kumar Das1, ‡ 1Department of Physics, Tezpur University, Tezpur 784028, India Abstract In this paper, we have realized the left-right symmetric model with modular symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have used Γ(3) modular group which is isomorphic to non-abelian discrete symmetry group A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The advantage of using modular symmetry is the non-requirement for the use of extra particles called ’flavons’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In this model, the Yukawa couplings are expressed in terms of modular forms (Y1, Y2, Y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In this work, we have studied minimal Left-Right Symmetric Model for both type-I and type-II dominances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The results obtained are well within the experimental limits for the desired values of sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions within Left-Right Symmetric Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ∗ ankitak@tezu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='ernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='in † bijay@tezu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='ernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='in ‡ mkdas@tezu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='ernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='13552v1 [hep-ph] 31 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' INTRODUCTION Despite the huge and continued success of the Standard Model (SM) of particle physics, it leaves some of the puzzles unanswered like the existence of neutrino masses, baryon asymmetry of the universe, existence of dark matter etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The discovery of neutrino oscillation by Sudbury neutrino observatory and Super-Kamiokande experiments was a milestone discovery in the area of neutrino physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The experiments like MINOS [1], T2K [2], Daya-Bay [3], Double-Chooz [4], RENO [5] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' provided evidence on the neutrinos being massive which is one of the most compelling rev- elation that we need to go beyond Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However inspite of the huge achievements in determining the neutrino oscillation parameters in solar, atmospheric , reactor and accelerator neutrino experiments, many questions related to neutrino still remain unsolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Among these lies the question regarding the absolute mass scale of neutrinos, exact nature of the particle (Dirac or Majorana), hierarchical pattern of the mass spectrum (Normal or Inverted) and leptonic CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The absolute mass scale of the neutrinos is not yet known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However experiments like Planck has given an upper bound on the sum of the light neutrino masses to be Σ|mνi| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='23eV in 2012 [6] and recently the bound has been constarined to Σ|mνi| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='11eV [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The most successful data pertaining to neutrino oscillation parameters is found in the 3σ global fit data [8] as shown in table (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Parameters Normal Ordering Inverted Ordering ∆ m2 21 (10−5eV 2) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='82 → 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='82 → 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='04 ∆ m2 3l (10−5eV 2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='435 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='598 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='581 → −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='414 sin2 θ12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='264 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='269 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='343 sin2 θ23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='415 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='419 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='617 sin2 θ13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='02032 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='02410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='02052 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='02428 TABLE I: Global fit 3σ values for neutrino oscillation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 2 We have used the definition, ∆m2 3l = ∆m2 31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ∆m2 31 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' NO (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) ∆m2 3l = ∆m2 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ∆m2 32 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' IO (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) The simplest way to look for neutrino masses is by the seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The mechanism may be of type I [9], [10],type II [11], [12],type III [13] and Inverse Seesaw [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' These are extensions of the SM where we incorporate extra particles like right-handed fermions,scalar fermion triplets, gauge singlet neutral fermions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The BSM physics also sheds light upon the phenomena like baryon asymmetry of the universe (BAU) [15], Lepton Number Violation (LNV) [16], Lepton Flavor violation (LFV) [17], existence of dark matter [18], [19] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' A BSM framework which has been successful in explaining the first three of the phenomenologies is the Left- Right Symmetric Model (LRSM) [20–24], an extension of the SM corresponding to the addition of SU(2)R group into the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The gauge group of LRSM is SU(3)C ⊗ SU(2)R ⊗ SU(2)L ⊗ U(1)B−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The type I and type II seesaw masses appear naturally in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The right-handed neutrinos are an essential part of the model, which acquires Majorana mass when SU(2)R symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' LRSM provides a natural framework to understand the spontaneous breaking of parity and origin of small neutrino masses by seesaw mechanism [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Another concerning aspect is the ambiguity regarding nature of neutrinos which has not been yet predicted by the SM of particle physics, that whether neutrinos are Dirac or Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' This problem is directly connected to the issue of lepton number conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' One of the process of fundamental importance which arises in almost any extension of the SM is Neutrinoless Double Beta Decay(NDBD) [26], [27] which when verified can assure that neutrinos are Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' NDBD is a slow, radiative process that transforms a nuclide of atomic number Z into its isobar with atomic number Z+2 [28], N(A, Z) → N(A, Z + 2) + e− + e− (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) The main aim in the search of NDBD (0νββ) is the measurement of effective Majorana neutrino mass, which is a combination of the neutrino mass eigenstates and neutrino mixing matrix terms [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, no experimental evidence regarding the decay has been in picture till date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In 3 addition to the determination of the effective masses, the half-life of the decay [29] combined with sufficient knowledge of the nuclear matrix elements (NME), we can set a constraint on the neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The experiments like KamLAND-Zen [30] and GERDA [31] which uses Xenon-136 and Germanium-76 respectively have improved the lower bound on the half-life of the decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, KamLAND-Zen imposes the best lower limit on the half life as T 0ν 1/2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='07 × 1026 yr at 90 % CL and the corresponding upper limit of the effective Majorana mass in the range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='061- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='165)eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' There are several contributions in LRSM that appear due to additional RH current interactions, giving rise to sizeable LFV rates for TeV scale RH neutrino that occur at rates accessible in current experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' It has been found that the most significant constraints has been provided by the decays, µ → 3e and µ → γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In the Standard Model, these LFV decays are suppressed by the tiny neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' No experiment has so far observed any flavor violating processes including charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, many experiments are currently going on to set strong limits on the most relevant LFV observables that will constrain the parameter space of many new models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The best bounds on the branching ratio for LFV decays of the form µ → γe comes from MEG experiment and it is set at BR(µ → γe) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2 × 10−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In case of the decay µ → 3e, the bound is set by the SINDRUM experiment at BR(µ → 3e) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0 × 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As mentioned LRSM is an important theory that incorporates the above mentioned phe- nomenologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=', the phenomenologies related to neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' There are many works where the authors make use of discrete symmetry groups like A4 [32],S4 [33],Z2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' [34] to analyze the prob- lem of flavor structure of fermions and to study various related phenomenologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our work, we have used A4 modular symmetry to study neutrino masses and mixings and hence study Neu- trinoless Double Beta Decay within the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The advantage of using modular symmetry over discrete flavor symmetries is that the study of the model using symmetries can be done without the introduction of extra particles called ’flavons’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Hence the model is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, in this work we have not done a very detailed analysis of the above mentioned phe- nomenologies, but only realized the left-right symmetric model with the help of A4 modular sym- metry and studied the variations of new physics contributions of neutrinoless double beta decay within LRSM with the range of values for Yukawa couplings, which in our model is expressed as modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In section (II), we have given a detailed explanation of the left-right symmetric model, the associated Lagrangian and the mass terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We begin section (III) by introducing 4 modular symmetry and then in section (IV), we incorporate modular symmetry into LRSM and determine the associated mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In section (V), we present a very brief discussion of neutrinoless double beta decay and its associated contributions and their relations with the modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In section (VI), the numerical analysis and results of this work has been discussed and the last section reads the conclusion for the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' MINIMAL LEFT-RIGHT SYMMETRIC MODEL The Left-Right Symmetric Model (LRSM) was first introduced around 1974 by Pati and Salam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Rabindra N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Mohapatra and Goran Senjanovic were also some pioneers of this very elegant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' LRSM is an extension of the Standard Model of particle physics, the gauge group being SU(3)C ⊗ SU(2)R ⊗ SU(2)L ⊗ U(1)B−L, which has been studied by several groups since 1970’s [25], [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The usual type-I and type-II seesaw neutrino masses arises naturally in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The seesaw scale is identified by the breaking of SU(2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Some other problems are also addressed in LRSM like parity, CP violation of weak interaction, massive neutrinos, hierarchy problems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' LRSM removes the disparity between the left and right-handed fields by considering the RH fields to be doublet under the additional SU(2)R keeping the right sector couplings same as the left-one by left- right symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In this model, the electric charge is given by Q = T3L +T3R + B−L 2 , where T3L and T3R are the generators of SU(2)L and SU(2)R respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' B − L refers to baryon number minus lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The particle content of the model along with their respective charge assignments are given in table(III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The matrix representation for the scalar sector is given by, φ = � �φ0 1 φ+ 1 φ− 2 φ0 2 � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) ∆L,R = � � δ+ L,R √ 2 δ++ L,R δ0 L,R − δ+ L,R √ 2 � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) In order for the fermions to attain mass, a Yukawa Lagrangian is necessary which couples to the bidoublet φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa Lagrangian incorporating the bidoublet is given by, LD = liL(Y l ijφ + � Y l ij �φ)ljR + QiL(Y q ijφ + � Y q ij �φ)QjR + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='c (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) 5 where, lL and lR are the left-handed and right-handed lepton fields, QL and QR are the left- handed and right-handed quark fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Y l being the Yukawa coupling corresponding to leptons and Y q being the Yukawa coupling for the quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa Lagrangian incorporating the scalar triplets which play a role in providing Majorana mass to the neutrinos is given by, LM = fL,ijΨL,i TCiσ2∆LΨL,j + fR,ijΨR,i TCiσ2∆RΨR,j + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='c (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) fL and fR are the Majorana Yukawa couplings and are equal subjected to discrete left-right sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The scalar potential in LRSM is a combination of interaction terms consisting the potential and after spontaneous symmetry breaking the scalars attain VEVs given by, < ∆L,R >= 1 √ 2 � � 0 0 vL,R 0 � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5) < φ >= � �k 0 0 eiθk′ � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) The magnitudes of the VEVs follows the relation, |vL|2 < |k2 + k′2| < |vR|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The breaking pattern of the LRSM gauge group takes place in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The LRSM gauge group is first broken down to the Standard Model gauge group by the vev of the scalar triplet ∆R, and then the Standard Model gauge group is broken down to the electromagnetic gauge group i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=', U(1)em by the vev of the bidoublet and a tiny vev of the scalar triplet ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Dirac mass terms for the leptons come from the Yukawa Lagrangian, which for the charged leptons and the neutrinos are given by, Ml = 1 √ 2(k′Yl + k ˜Yl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7) MD = 1 √ 2(kYl + k′ ˜Yl) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='8) The light neutrino mass after spontaneous symmetry breaking (SSB), generated within a type (I+II) seesaw can be written as, Mν = Mν I + Mν II, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='9) Mν = MDMRR −1MD T + MLL (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='10) 6 where, MLL = √ 2vLfL (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='11) and, MRR = √ 2vRfR (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='12) The first and second terms in corresponds to type-I seesaw and type-II seesaw masses respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' It is an interesting fact that in the context of LRSM both type-I and type-II terms can be equally dominant or either of the two terms can be dominant, but under certain conditions [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' It has been demonstrated in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In the context of LRSM however, both the type-I and type-II mass terms can be expressed in terms of the heavy right-handed Majorana mass matrix, so equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='10) will follow, Mν = MDM −1 RRM T D + γ � MW vR �2 MRR (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='13) where, γ is a dimensionless parameter which is a function of various couplings, appearing in the VEV of the triplet Higgs ∆L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=', vL = γ( v2 vR) and here, v = √ k2 + k′2, and γ = β1kk′ + β2k2 + β3k′2 (2ρ1 − ρ3)(k2 + k′2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='14) In our model, the dimensionless parameter γ has been fine tuned to γ ≈ 10−6 and vR is of the order of TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' MODULAR SYMMETRY Modular symmetry has gained much importance in aspects of model building [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' This is because it can minimize the extra particle called ’flavons’ while analyzing a model with respect to a particular symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' An element q of the modular group acts on a complex variable τ which belongs to the upper-half of the complex plane given as [38] [39] qτ = aτ + b cτ + d (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) where a, b, c, d are integers and ad − bc = 1, Imτ>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 7 The modular group is isomorphic to the projective special linear group PSL(2,Z) = SL(2,Z)/Z2 where, SL(2,Z) is the special linear group of integer 2 × 2 matrices having determinant unity and Z2 = (I, −I) is the centre, I being the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The modular group can be represented in terms of two generators S and T which satisfies S2 = (ST)3 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' S and T satisfies the following matrix representations: S = � � 0 1 −1 0 � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) T = � �1 1 0 1 � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) corresponding to the transformations, S : τ → −1 τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' T : τ → τ + 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) Finite modular groups (N ≤ 5) are isomorphic to non-abelian discrete groups, for example, Γ(3) ≈ A4, Γ(2) ≈ S3, Γ(4) ≈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' While using modular symmetry, the Yukawa couplings can be expressed in terms of modular forms, and the number of modular forms present depends upon the level and weight of the modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' For a modular form of level N and weight 2k, the table below shows the number of modular forms associated within and the non-abelian discrete symmetry group to which it is isomorphic [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' N No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' of modular forms Γ(N) 2 k + 1 S3 3 2k + 1 A4 4 4k + 1 S4 5 10k + 1 A5 6 12k 7 28k - 2 TABLE II: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' of modular forms corresponding to modular weight 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 8 In our work, we will be using modular form of level 3, that is, Γ(3) which is isomorphic to A4 discrete symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The weight of the modular form is taken to be 2, and hence it will have three modular forms (Y1, Y2, Y3) which can be expressed as expansions of q given by, Y1 = 1 + 12q + 36q2 + 12q3 + 84q4 + 72q5 + 36q6 + 96q7 + 180q8 + 12q9 + 216q10 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5) Y2 = −6q1/3(1 + 7q + 8q2 + 18q3 + 14q4 + 31q5 + 20q6 + 36q7 + 31q8 + 56q9) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) Y3 = −18q2/3(1 + 2q + 5q2 + 4q3 + 8q4 + 6q5 + 14q6 + 8q7 + 14q8 + 10q9) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7) where, q = exp(2πiτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' MINIMAL LRSM WITH A4 MODULAR SYMMETRY In particle physics, symmetries have always played a very crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The realization of LRSM with the help of discrete flavor symmetries have been done in earlier works [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our work we have incorporated A4 modular symmetry into LRSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The advantage of using modular symmetry rather than flavor symmetry is the minimal use of extra particles (flavons) and hence the model is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The model contains usual particle content of LRSM [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The lepton doublets transform as triplets under A4 and the bidoublet and scalar triplets transform as 1 under A4 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As we have considered modular symmetry, we assign modular weights to the particles, keeping in mind that matter multiplets corresponding to the model can have negative modular weights, but the modular forms cannot be assigned negative weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The assignment of these weights are done in such a way that in the Lagrangian the sum of the modular weights in each term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Modular weights corresponding to each particle is shown in table (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa Lagrangian for the leptonic and quark sector in LRSM is given by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) and with a reference to that we can write the Yukawa Lagrangian of our A4 modular symmetric LRSM, for the fermionic sector, by introducing Yukawa coupling in the form of modular forms Y is given as, LY = lLφlRY + lL ˜φlRY + QLφQRY + QL ˜φQRY + lR TCiτ2∆RlRY + lL TCiτ2∆LlLY (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) 9 Gauge group QL QR lL lR φ ∆L ∆R SU(3)C 3 3 1 1 1 1 1 SU(2)L 2 1 2 1 2 3 1 SU(2)R 1 2 1 2 2 1 3 U(1)B−L 1/3 1/3 -1 -1 0 2 2 A4 3 3 3 3 1 1 1 kI 0 2 0 -2 0 -2 2 TABLE III: Charge assignments for the particle content of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa couplings Y = (Y1, Y2, Y3) are expressed as modular forms of level 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Y (modular forms) A4 3 kI 2 TABLE IV: Charge assignment and modular weight for the corresponding modular Yukawa form for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our work, we are concerned with the mass of the neutrinos and as such, using A4 modular symmetry and using the multiplication rules for A4 group, we construct the Dirac and Majorana mass matrices as given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Dirac mass matrix is given by, MD = v � � � � 2Y1 −Y3 −Y2 −Y2 −Y1 2Y3 −Y3 2Y2 −Y1 � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) where, v is considered to be the VEV for the Higgs bidoublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The right-handed Majorana mass matrix is given by, MR = vR � � � � 2Y1 −Y3 −Y2 −Y3 2Y2 −Y1 −Y2 −Y1 2Y3 � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) 10 where, vR is the VEV for the scalar triplet ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As it is seen that the Majorana mass matrix for our model is found to be symmetric in nature as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Under these assumptions for modular symmetric LRSM and the basis that we have considered, our charged lepton mass matrix is also found to be diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The type-I seesaw mass is then given by, MνI = MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='MR −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='MD T (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) and, the type-II seesaw mass is given by, MνII = MLL (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5) As mentioned above, in LRSM type-II seesaw mass can also be expressed in terms of the right- handed mass MR as, MνII = γ � MW vR �2 MR (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Type-I dominanace In LRSM, the type-I seesaw mass dominates when the vev of the left-handed triplet is taken to be negligibly small and hence the type-II term is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In such a case the lightest neutrino mass can be given in terms of the type-I seesaw mass term given by, Mν = MDMR −1MD T (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7) and the heavy right-handed Majorana mass term can be given as, MR = fRvR (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='8) where, fR is the right-handed Majorana Yukawa coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In the approximation that k′ << k, and if we consider that our Yukawa coupling Y l corresponding to the neutrino masses is yD and the coupling � Y l for the charged fermion masses is denoted by yL, so considering yDk >> yLk′ we can write the type-I mass term as [44], Mν = k2 vR yDf −1 R yT D (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='9) 11 If we consider that UR is a unitary matrix that diagonalizes MR, so since the VEV vR is a constant the same matrix will also diagonalize the coupling matrix fR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Taking fR = fL = f, so f = URf diaU T R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='10) If we take inverse on both sides and taking into account the property of a unitary matrix (U −1 R = U T R), we get, f −1 = U T R(f dia)−1UR (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='11) Therefore, we get Mν = k2 vR yDU T R(f dia)−1URyT D (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='12) Multiplying both sides of the equation with U T R from the right and with UR from left, we finally arrive at the following equation, URMνU T R = (Mν)dia (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='13) where we have used URyDU T R = yD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So, the unitary matrix diagonalizing the matrix MR also diagonalizes the light neutrino mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So in this case it can be determined that if mi denotes the light neutrino mass and Mi denotes the heavy neutrino mass, then they are related as mi ∝ 1 Mi (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='14) For our model, the Yukawa couplings are modular forms expressed as expansions of q, and the mass matrices are expressed in terms of the modular forms (Y1, Y2, Y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So, the light neutrino mass matrix, Mν for type-I dominance is given by the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As already stated in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3), the Dirac and Majorana mass matrices are determined by the application of multiplication rules for the A4 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So, for type-I dominance, our light neutrino mass matrix will be given by, Mν = v2 vR � � � � 2Y1 −Y2 −Y3 −Y2 2Y3 −Y1 −Y3 −Y1 2Y2 � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='15) As mentioned previously, the value for vR is of the order of TeV and that for v is in GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have computed the values of the sum of the neutrino masses for type-I dominance and checked the correctness of our model by plotting it against the Yukawa couplings and the result was found to match the experimental bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 1: Variation of |Y1| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 2: Variation of |Y2| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 3: Variation of |Y3| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='050 F (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010F Em 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='005E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001E Allowed 3oregion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='10- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} 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+page_content='001 10-4 Allowed Bo region 10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y3]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Type-II dominance Type-II seesaw mass in LRSM dominates when the Dirac term connecting the right-handed and left-handed parts is negligible as compared to that of the type-II term [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In that case, our light neutrino mass mν will given by the type-II seesaw mass term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=', MνL = fLvL (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='16) And the heavy mass matrix is given by, MR = fRvR (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='17) Again if we consider that UL and UR diagonalizes MνL and MR respectively, so for the reason mentioned above the same matrices will also diagonalize fL and fR respectively and since in our model, fL = fR, so we can consider UL = UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In such a case, we arrive at an important result that mi ∝ Mi (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='18) Now using modular symmetry the light neutrino mass matrix for type-II dominance in our model is given by, mν = vL � � � � 2Y1 −Y3 −Y2 −Y3 2Y2 −Y1 −Y2 −Y1 2Y3 � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='19) where, vL is the vev for left-handed scalar triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The value of vL is taken to be of the order of eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The sum of the neutrino masses is computed for type-II dominance and plotting of the sum is done with the Yukawa couplings which are found to be as shown under, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 4: Variation of |Y1| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 14 HN Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': 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[YI]FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 5: Variation of |Y2| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 6: Variation of |Y3| with sum of neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' NEUTRINOLESS DOUBLE BETA DECAY (0νββ) IN MINIMAL LRSM Neutrinoless double beta decay is a lepton number violating process, which if proven to exist will directly imply the Majorana nature of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' N(A, Z) → N(A, Z + 2) + e− + e− (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) Many groups have however already done a lot of work on NDBD in the model , [21],[28, 45–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In LRSM [51], there are several contributions to NDBD in addition to the standard contribution via light Majorana neutrino exchange owing to the presence of several heavy additional scalar,vector 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Exciuded KamLAND-Zen region 0.' metadata={'source': 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-6 [Y2]NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed3oregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': 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+page_content='×10-6 [Y3]NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed3gregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-6 [Y2]and fermionic fields [52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Various contributions to NDBD transition rate in LRSM are discussed as follows : Standard Model contribution to NDBD where the intermediate particles are the WL bosons and light neutrinos, the process in which the amplitude depends upon the leptonic mixing matrix elements and light neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Heavy right-handed neutrino contribution in which the mediator particles are the WL bosons and the amplitude depends upon the mixing between light and heavy neutrinos as well as the mass of the heavy neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Light neutrino contribution to NDBD where the intermediate particles are WR bosons and the amplitude depends upon the mixing between light and heavy neutrinos as well as mass of the right-handed gauge boson WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Heavy right-handed neutrino contribution where the mediator particles are the WR bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The amplitude of this process is dependent on the elements of the right handed leptonic mixing matrix and mass of the right-handed gauge boson, WR as well as the mass of the heavy right handed Majorana neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Light neutrino contribution from the Feynman diagram mediated by both WL and WR, and the amplitude of the process depends upon the mixing between light and heavy neutri- nos,leptonic mixing matrix elements, light neutrino masses and the mass of the gauge bosons, WL and WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Heavy neutrino contribution from the Feynman diagram mediated by both WL and WR, and the amplitude of the process depends upon the right handed leptonic mixing matrix elements, mixing between the light and heavy neutrinos, also the mass of the gauge bosons, WL and WR and the mass of the heavy right handed neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Scalar triplet contribution (∆L) in which the mediator particles are WL bosons, and the amplitude for the process depends upon the masses of the WL bosons, left-handed triplet Higgs, as well as their coupling to leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 16 Right-handed scalar triplet contribution (∆R) contribution to NDBD in which the mediator particles are WR bosons, and the amplitude for the process depends upon the masses of the WR bosons, right-handed triplet Higgs, ∆R as well as their coupling to leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our work, where we have incorporated A4 modular symmetry to LRSM and in our present work we have considered three of the above mentioned contributions, one from the standard light neutrino contribution and the other two new physics contribution mediated by W − R and ∆R re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' For simple approximations, an assumption has been made in the mass scales of heavy particles, where, MR ≈ MWR ≈ M∆L++ ≈ M∆R++ ≈ TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Under these assumptions, the amplitude for the light-heavy mixing contribution which is pro- portional to mD2 MR remains very small, since mν ≈ mD2 MR ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1)eV, mD ≈ (105 − 106)eV which implies mD MR ≈ (10−7 − 10−6)eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Thus in our model, we ignore the contributions involving the light and heavy neutrino mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' When NDBD is done in the framework of LRSM, the standard light neutrino contribution is given by, meff v = U 2 Limi (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) where, ULi are the elements of the first row of the neutrino mixing matrix UPMNS, in which the elements are dependent on known mixing angles θ13 , θ12 and the Majorana phases κ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The UPMNS matrix is given by, UPMNS = � � � � c12c13 s12c13 s13e−iδ −c23s12 − s23s13c12eiδ −c23c12 − s23s12s13eiδ s23c13 s23s12 − c23s13c12eiδ −s23c12 − c23s13s12eiδ c23c13 � � � � P (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) where, P = diag(1, eiκ, eiη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So the effective mass can be parametrized in terms of the elements of the diagonalizing matrix and the eigenvalues as, meff v = m1c2 12c2 13 + m2s2 12c2 13e2iκ + m3s2 13e2iη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' NUMERICAL ANALYSIS AND RESULTS In our present work, we have modified left-right symmetric model by incorporating A4 modu- 17 lar symmetry for both type-I and type-II dominances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As we are using modular symmetry, the Yukawa couplings are expressed as expansions of q as shown in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our model, the value of q is found to be of the order of 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The aboslute value of the modulus should however be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' τ = Re(τ) + Im(τ) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) Re(τ) Im(τ) |τ| Range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='715,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='789] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='9] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='073,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='197] TABLE V: Range of values corresponding to real and imaginary parts of the modulus τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Yukawa couplings Normal Hierarchy Inverted hierarchy Y1(min) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='29155 × 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='32276 × 10−7 Y1(max) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='22986 × 10−7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='21382 × 10−7 Y2(min) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='02229 × 10−9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='42826 × 10−9 Y2(max) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='32006 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='45952 × 10−6 Y3(min) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='39287 × 10−9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='759 × 10−9 Y3(max) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3165 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='50082 × 10−6 TABLE VI: Range of Yukawa couplings (Y1, Y2, Y3) for both normal and inverted hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' From table (V), it is seen that the absolute value of the modulus is greater than unity, which is the expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The range of the Yukawa couplings for our model is shown in the table above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' It is seen from the table that the minimum value of the Yukawa coupling Y1 is in the scale of 10−10 for normal hierarchy while for inverted hierarchy it is in the scale of 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, for the maximum of Y1 both the orderings are in the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' For Y2 and Y3, the minimum and maximum values for both normal and inverted hierarchies are within the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have plotted the effective masses against the Yukawa couplings (Y1, Y2, Y3) and it was found to be well within the bound set by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As shown both for type-I and type-II dominances, we have plotted the absolute values of the 18 Yukawa couplings against the sum of the neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The range of the values for sum of neutrino masses for both the cases are given as under, � mν Normal Hierarchy Inverted hierarchy Type − I(min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='000980556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='000437758 Type − I(max) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='177296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='186377 Type − II(min) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='000219304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='000035 Type − II(max) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0200981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0203081 TABLE VII: Range of values for sum of neutrino masses for type-I and type-II dominances for both normal and inverted hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Standard Light Neutrino Contribution to 0νββ As mentioned above, in the standard light neutrino contribution to 0νββ, the intermediate particles are the WL bosons and light neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective mass for the contribution is given by equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Simplifying for the respective elements of ULi and mi, the value of the effective mass is obtained in terms of the modular forms (Y1, Y2, Y3) as, meff ν = meff 1 + meff 2 + meff 3 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) where, meff 1 = ν(Y2 − Y3)2(Y1 + Y2 + Y3) νR(Y1 − Y3)2 meff 2 = ν2(Y1 − Y2)2(Y1 + Y2 + Y3 − √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2νR(Y1 − Y3)2 meff 3 = ν2(Y1 + Y2 + Y3 − √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2νR for type-I dominance, and the plots are shown as, 19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 7: Variation of |Y1| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 8: Variation of |Y2| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 9: Variation of |Y3| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 20 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10~ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10- [Y1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='H Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 (Aa) meff 0T0:0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 Allowed 3o region 107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10-7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10-7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 [Y1]1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 (Aa) meff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 Allowed 3o region 107 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed3oregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-6 [Y3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0015 Allowed 3o region 10-4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 /Y3For type-II dominance, we have meff 1 = −νL(−Y2 + Y3)(Y1 + Y2 + Y3) Y1 − Y2 meff 2 = −νL(Y1 − Y3)(Y1 + Y2 + Y3 − √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2(Y1 − Y2) meff 3 = νL(Y1 − Y3)(Y1 + Y2 + Y3 + √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 10: Variation of |Y1| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 11: Variation of |Y2| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed3gregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (Aa) Meff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3g region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x107 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='× 10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' x10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 (Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 (A) O1O1O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 Allowed 3o region 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y2]FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 12: Variation of |Y3| with effective neutrino mass for standard light neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Heavy Right-Handed Neutrino contribution to 0νββ In our work, we have considered contributions of heavy right-handed neutrino and scalar Higgs triplet to NDBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective mass for heavy right-handed neutrino is given by, meff R = p2 � M 4 WL M 4 WR �� U ∗2 Rei Mi � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) where, p2 is the typical momentum exchange of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As it is known that TeV scale LRSM plays a very important role in the process of neutrinoless double beta decay (0νββ), we have considered the values as MWR = 10TeV , MWL = 80GeV , M∆R ≈ 3TeV and after calculation, the value for heavy right-handed neutrino is found to be in the scale of TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The allowed value of p is in the range (100 − 200)MeV and so we consider, p ≈ 180MeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Thus, we get, p2 � M 4 WL M 4 WR � = 1010eV 2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) where, URei refers to the first row elements of the diagonalizing matrix of the heavy Majorana mass matrix and Mi are its eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective mass corresponding to the heavy right-handed neutrino can be expressed in terms of the modular forms as, mR eff = 1010(mR1 eff + mR2 eff + mR3 eff) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5) where, mR1 eff = 2 νR(Y1 + Y2 + Y3 + √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 22 1 NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3oregion 10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 Allowed 3oregion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10~9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10-S 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10~7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x106 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10~6 (Y3)mR2 eff = 2(Y ∗ 1 − Y ∗ 3 )2 νR(Y1 + Y2 + Y3 − √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 )(Y ∗ 1 − Y ∗ 2 )2 mR3 eff = (−Y ∗ 2 + Y ∗ 3 )2 νR(Y1 + Y2 + Y3)(Y ∗ 1 − Y ∗ 2 )2 The total effective mass is also calculated for the standard light and right-handed heavy neutrino contribution, given by, |mefftotal ν | = |meff ν + mR eff| (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) which can be obtained in terms of the modular forms as a summation of the above mentioned terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 13: Variation of |Y1| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 14: Variation of |Y2| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 23 NH Excluded KamLAND-Zenregion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (10) total 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 105 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-$1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×107 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 /Y2]FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 15: Variation of |Y3| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The plots above are for type-I dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 16: Variation of |Y1| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 17: Variation of |Y2| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 24 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 /Y2]FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 18: Variation of |Y3| with total effective neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The figures above are the plots for type-II dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Scalar Triplet contribution to 0νββ The magnitude of ∆R contribution is controlled by the factor Mi M∆R [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' However, scalar triplet contribution is not included in the total contribution under the assumption Mi M∆R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' But, some the mixing parameters in the large part of the parameter space may result in a higher Mi M∆R ratio and in such cases we will have to include it in the total contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The impact of this contribution here is studied in the limit, M∆R ≊ Mheaviest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective mass for scalar triplet contribution is given as, |meff ∆ | = |p2 M 4 WL M 4 WR U 2 ReiMi M 2 ∆R | (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7) The value of the mass for the right-handed scalar triplet is taken as, M∆R = 3TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So, the value of the coefficient results as, p2 M 4 WL M 4 WR 1 M 2 ∆R = 1010 9 × 1024 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='8) In terms of modular forms, the effective scalar mass can be expressed as, m∆R eff = m∆R eff1 + m∆R eff2 + m∆R eff3 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='9) where, m∆R eff1 = νR(Y2 + Y3)2(Y1 + Y2Y3) (Y1 − Y2)2 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 ltotal( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3g region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y3]Exciuded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 Hltot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 Allowed 3o region 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x1081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 IY3Im∆R eff2 = νR(Y1 − Y3)2(Y1 + Y2 + Y3 − √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2(Y1 − Y2)2 m∆R eff3 = νR(Y1 + Y2 + Y3 + √ 3 � 3Y 2 1 − 2Y1Y2 + 3Y 2 2 − 2Y1Y3 − 2Y2Y3 + 3Y 2 3 ) 2 The plots are shown as under.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 19: Variation of |Y1| with effective neutrino mass for scalar triplet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 20: Variation of |Y2| with effective neutrino mass for scalar triplet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' NH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='x10-6 (Y1]·IH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 meffxh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='01 104F Allowed3o region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0-10-6 13:10-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0-10-5 25±10-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='10-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5x10-5 YINH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='0105 (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Exciuded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 effsclr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3oregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='× 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-6 [Y2]FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 21: Variation of |Y3| with effective neutrino mass for scalar triplet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' CONCLUSION The discovery of neutrino oscillations paved the gateway for physics beyond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In our paper, we have realized LRSM with the help of modular A4 symmetry for both type-I and type-II dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Using modular symmetry provides the advantage of using no extra particles called ’flavons’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa couplings are represented as modular forms expressed as expansions of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The values of the Yukawa couplings (Y1, Y2, Y3) are calculated using ’Mathematica’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The mass matrices are then determined using the multiplication rules for A4 group stated in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Majorana mass matrix is found to be symmetric and under the considered basis, the charged lepton mass matrix is also diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have expressed the light neutrino and heavy right-handed neutrino mass matrix in terms of the modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' We have also studied briefly the contributions of 0νββ in LRSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective masses corresponding to standard light neutrino contribution, right-handed contribution and scalar triplet contributions are determined in terms of (Y1, Y2, Y3) and we have plotted the effective mass corresponding to the considered contributions against the Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' To summarize our work, some results are stated as under, The absolute value of the modulus was found to be within the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='073 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='197, which is greater than unity, that is the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The Yukawa couplings, expressed in terms of modular forms ranges from 10−9 to 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The sum of the neutrino masses for type-I dominance ranges from the order of 10−4 to 10−1 27 INH Excluded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 (Aa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3o region 10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×107 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='IH Exciuded KamLAND-Zen region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='010 effsclr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='001 10-4 Allowed 3oregion 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' × 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' ×10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='×10-6 [Y3]for both normal and inverted hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The sum of the neutrino masses for type-II dominance ranges from the order of 10−4 to 10−2 for both normal and inverted hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The effective masses for the 0νββ contributions are calculated and by determining their relations with the modular forms, we have plotted the effective masses with the three Yukawa couplings and it has been found that the values for the effective mass corresponding to each contribution is well within the experimental bounds, which infact makes us clearly state that the building of the model with modular symmetry is advantageous to that of flavor symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' In this model, we have not used any extra particles and the analysis has been done taking into consideration the calculated and computed values for the model parameters and the results are found to be satisfactory, so it can be stated that the Left-Right Symmetric Model can be constructed with modular symmetry while satisfying the experimental bounds on the desired parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' APPENDIX A Let us consider the Higgs potential of our model that has quadratic and quartic coupling terms given by [36], Vφ,∆L,∆R = −µ2 ijTr[φ† iφj] + λijklTr[φ† iφj]Tr[φ† kφl] + λ ′ ijklTr[φ† iφjφ† kφl] − µ2 ijTr[∆† L∆L + ∆† R∆R] ρ1[(Tr[∆† L∆L])2 + (Tr[∆† L∆L])2] + ρ2(Tr[∆† L∆L∆† L∆L] + Tr[∆† R∆R∆† R∆R]) + ρ3Tr[∆† L∆L∆† R∆R]+ αijTr[φ† iφj](Tr[∆† L∆L]+Tr[∆† R∆R])+βij(Tr[∆† L∆Lφiφ† j]+Tr[∆† R∆Rφiφ† j])+γij(Tr[∆† Lφi∆Rφ† j]+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='c) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='1) where, i,j,k,l runs from 1 to 2 with φ1 = φ and φ2 = ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' As mentioned above after SSB, the scalar sector obtains VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So after the substitution of the respective VEVs and determining the traces, so after simplification the potential can be written as, V = −µ2(v2 L + v2 R) + ρ 4(v4 L + v4 R) + ρ′ 2 + α 2 (v2 L + v2 R)k2 1 + γvLvRk2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='2) where, we have used the approximation k′ << k, and ρ′ = 2ρ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Our minimization conditions are, δV δvL = δV δvR = δV δk = δV δk′ = 0 28 Therefore, we get, δV δvL = −2µ2vL + ρv3 L + ρ′vLk2 + γvRk2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='3) Here, it is evident that the Majorana mass of the left-handed neutrino MLL is dependent on the vev vL as already defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Again, we have δV δvR = −2µ2vR + ρv3 R + ρ′vRk2 + γvLk2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='4) So, the right handed Majorana mass MRR is dependent on the vev vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Similarly, the calculations for the same can be carried out and it can be found out the Dirac mass term MD can be expressed in terms of the vev for the Higgs bidoublet as also defined previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Now, we are to determine a relation between the VEVs for the scalars and so after using the minimization conditions and simplifying the equations, we come to a relation given by, vLvR = γ ξ k (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='5) where, ξ = ρ − ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The neutrino mass for LRSM is given as a summation of the type-I and type-II term as already mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' So, in the approximation that k′ << k, and if we consider that our Yukawa coupling Y l corresponding to the neutrino masses is yD and the coupling � Y l for the charged fermion masses is denoted by yL, so considering yDk >> ylk′ we can write, Mν = k2 vR yDf −1 R yT D + fLvL (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='6) Since, for due to left-right symmetry, we can consider fL = fR = f, so the above equation can be written as, Mν = k2 vR yDf −1yT D + fvL (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='7) So, from this equation we can come to a relation given by, Mν = (f γ ξ + yDf −1yT D) k2 vR (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content='8) Here, we can consider two situations, namely If f( γ ξ ) << yDf −1yT D, the light neutrino mass is given by the type-I term MDM −1 RRM T D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' That is, here type-I is dominant and the light neutrino mass is from the suppression of heavy νR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' If f( γ ξ ) >> yDf −1yT D, the light neutrino mass is given by the type-II term fvL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' That is, in this case type-II mass term is dominant and the light neutrino mass is because of the tiny value of νL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 29 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' APPENDIX B Properties of A4 group A4 is a non-abelian discrete symmetry group which represents even permuatations of four ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' It has four irreducible representations, three out of which are singlets (1, 1′, 1′′) and one triplet 3 (3A represents the anti-symmetric part and 3S the symmetric part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Products of the singlets and triplets are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 1 ⊗ 1 = 1 1′ ⊗ 1′ = 1′′ 1′ ⊗ 1′′ = 1 1′′ ⊗ 1′′ = 1′ 3 ⊗ 3 = 1 ⊕ 1′ ⊕ 1′′ ⊕ 3A ⊕ 3S If we have two triplets under A4 say,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' (a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' a3) and (b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' b3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' then their multiplication rules are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' 1 ≈ a1b1 + a2b3 + a3b2 1′ ≈ a3b3 + a1b2 + a2b1 30 1′′ ≈ a2b2 + a3b1 + a1b2 3S ≈ � � � � 2a1b1 − a2b3 − a3b2 2a3b3 − a1b2 − a2b1 2a2b2 − a1b3 − a3b1 � � � � 3A ≈ � � � � a2b3 − a3b2 a1b2 − a2b1 a3b1 − a1b3 � � � � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' REFERENCES [1] Justin Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' The MINOS Experiment: Results and Prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFRT4oBgHgl3EQfZTcN/content/2301.13552v1.pdf'} +page_content=' High Energy 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Thai3 + +1 Ph.D. Candidate, Department of Urban and Regional Planning and Florida Institute for Built Environment +Resilience, College of Design, Construction and Planning, University of Florida, 1480 Inner Road, +Gainesville, FL, 32601, U.S.; Email: gao.shangde@ufl.edu; ORCID: 0000-0003-2218-2872. +2*Assistant Professor, Department of Urban and Regional Planning and Florida Institute for Built +Environment Resilience, University of Florida, P.O. Box 115706, Gainesville, FL 32611, U.S. +(corresponding author); E-mail: yanw@ufl.edu; ORCID: 0000-0002-3946-9418. +3 Professor, Department of Computer & Information Science & Engineering and Warren B. Nelms Institute +for the Connected World, University of Florida, Gainesville, FL 32611, U.S.; E-mail: mythai@cise.ufl.edu; +ORCID: 0000-0003-0503-2012. + +Abstract: Although social norms’ effect on mitigating misinformation is identified, scant knowledge exists +about the patterns of social norm emergence, such as the patterns and variations of social tipping in online +communities with diverse characteristics. Accordingly, this study investigates the features of social tipping +in online communities and examines the correlations between the tipping features and characteristics of +online communities. Taking “the side effects of COVID-19 vaccination” as the case topic, we first track +the patterns of tipping features in 100 online communities, which are detected using Louvain Algorithm +from the aggregated communication network on Twitter between May 2020 and April 2021. Then, we use +multi-variant linear regression to explore the correlations between tipping features and communities’ +characteristics. We find that social tipping in online communities can sustain for two to four months and +lead to a 50% increase in populations who accept the normative belief in online communities. The +regression indicates that the duration of social tipping is positively related to the community populations +and original acceptance of social norms, while the correlation between the tipping duration and the degrees +among community members is negative. Additionally, the network modularity and original acceptance of +social norms have negative relationships with the extent of social tipping, while the users’ degree and +betweenness centrality can have significant positive relationships with the extent of tipping. Our findings +shed light on more precise normative interventions on misinformation in digital environments as it offers +preliminary evidence about the timing and mechanism of social norm emergence. + +1 Introduction +The extensive development of online platforms has fostered the spread of messages generated by +stakeholders at various levels, e.g., governmental agencies and individual users, during public events (Y. +Wang et al., 2021). A large proportion of user-generated online messages contain inaccurate and misleading +information, i.e., misinformation (Del Vicario et al., 2016; Wang et al., 2022). The wide diffusion of +misinformation has threatened human society from multiple perspectives, e.g., interfering with collective +decision-making on democratic, environmental, and public health issues (West & Bergstrom, 2021). There +is an emergent need for suppressing misinformation spreading and mitigating the negative consequences of +online misinformation on human society (West & Bergstrom, 2021). Existing studies (e.g., D. T. Nguyen +et al. (2012), N. P. Nguyen et al. (2012), Zhang, Alim, et al. (2015, 2016), Zhang et al. (2018), Zhang, +Kuhnle, et al. (2016), Zhang, Zhang, et al. (2015)) tend to suppress misinformation with (i) debunking, i.e., +correcting the misinformation after people are exposed to it, and (ii) prebunking, i.e., helping people + +Page 2/17 +recognize the false/misleading contents (U. K. H. Ecker et al., 2022; Lewandowsky & van der Linden, +2021). The debunking strategy is widely adopted to provide targeted countermeasures for misinformation +of specific topics (U. K. H. Ecker et al., 2022), e.g., provide messages with factual elaboration (Gao et al., +2021; van der Meer & Jin, 2020; Wang et al., 2022), fact-checking content (Humprecht, 2020), and +messages that stimulate the health-protective measures (Humprecht, 2020). The debunking strategy is not +always effective when the explanations that support the misinformation exist widely (Chan et al., 2017). +The effect of debunking messages tends to be short-term and washed out by future exposure to +misinformation (Mourali & Drake, 2022). Also, the debunking strategy can only be conducted after +people’s initial exposure to the misinformation (van der Meer & Jin, 2020), while the negative +consequences of misinformation may already exist and cause notable social costs. +On the contrary, the prebunking strategy is potentially an effective vehicle that overcomes the limitations +of the debunking strategy and confers large-scale resistance against misinformation among the public (van +der Linden et al., 2020). The prebunking strategy is based on the social psychological theory of +“inoculation”. If people are pre-warned and form the belief of rejecting misinformation, they might be +“immune” to misinformation (Lewandowsky & van der Linden, 2021). Compared to the debunking strategy, +the prebunking strategy focuses on influencing people’s beliefs on the topics of misinformation, posing +long-term effects on the public and reducing the occurrence of negative consequences of misinformation +(Basol et al., 2021). When being implemented at a large scale, the pre-bunking strategy is conducted with +social norm interventions, which aim to generate the social norms and consensus that support the factual +evidence and reject misinformation (Dow et al., 2022). +The basis of social norm interventions is people’s adherence to the surrounding social norms (Constantino +et al., 2022). Existing in both the digital and physical world (Gao et al., 2022), social norms, i.e., the shared +beliefs or acceptable behaviors in communities, have shown a significant relationship with people’s belief +in the content of misinformation (Andı & Akesson, 2021; Gimpel et al., 2021; Lapinski & Rimal, 2005). +Adhering to social norms can satisfy a desire to avoid sanctions, confer benefits by coordinating with others, +and provide a simple heuristic about what is accepted/wise in a particular context (Constantino et al., 2022). +Based on this psychological phenomenon, social norm interventions have been implemented to help form +the belief of supporting factual evidence and rejecting misinformation in both the physical and digital +realms (Andı & Akesson, 2021; Gimpel et al., 2021; Lapinski & Rimal, 2005), such as suppressing +misinformation about climate actions and health behaviors (Constantino et al., 2022; U. K. Ecker et al., +2022). Specifically, by showing individuals the text that describes the “common beliefs” (i.e., social norms) +towards the misinformation of a certain topic, individuals tend to modify their beliefs to match the “common +beliefs” and reduce the reliance on the misinformation (U. K. Ecker et al., 2022). In another case, by +showing individuals a message that “most responsible people think twice before sharing articles” (a social +norm), individuals are not likely to share social media articles that contain misleading or contested content +(Andı & Akesson, 2021). +Though the role of the social norm in suppressing misinformation has been identified (Dow et al., 2022; +Constantino et al., 2022; U. K. Ecker et al., 2022), scant empirical evidence has been provided to inform +the implementation of social norm interventions. Several knowledge gaps and challenges remain. First, +with the controlled experiments in physical worlds, recent works have identified that social norm emergence +in their artificially designed communities tended to have a tipping process, i.e., social tipping (Berger, 2021; +Centola et al., 2018; Ehret et al., 2022). Social tipping is a process that when the “tipping point” is reached, +a small change in an individual community can create abrupt, nonlinear change in the acceptance of the +normative beliefs across the community (Berger, 2021). By predicting the occurrence and extent of social +tipping, policymakers can improve the effectiveness of the social norm interventions by adjusting the timing +and efforts of implementing the interventions (Andreoni et al., 2021; Ehret et al., 2022). However, due to +the lack of analysis of the online communities, it is unclear whether social tipping also exists in online +communities and follows certain patterns regarding the tipping features, e.g., the duration and extent of +social tipping. Little knowledge exists to guide the practices of social norm interventions regarding the + +Page 3/17 +timing and efforts that are needed to promote the tipping process of norm emergence. Second, experiments +in existing studies have identified some evidence regarding the potential relationships between community +characteristics and the diffusion of normative beliefs (Hu & Leung, 2017; Savarimuthu & Cranefield, 2011; +Sen & Sen, 2010; Yu et al., 2014). However, these experiments were generally based on artificially +designed communities in real-world or virtual scenarios, and the experiment findings may not be applicable +in the communities of the online environment. Also, how the social tipping process varies in the community +characteristics has not been disclosed in the existing studies. There is a need for empirical studies that +explore the relationships between community characteristics and social tipping based on real-world +communities, providing a reference for the design of social norm interventions. +To fill this research gap, this study aims to answer the following research questions (RQ): +• +RQ1: Does social tipping exist during the social norm emergence of online communities? If so, what +are the characteristics and patterns of social tipping? +• +RQ2: Do the features of social tipping correlate with different network characteristics of individual +communities? +This study takes the case of the norms on Twitter regarding the side effects of COVID-19 vaccines. The +diffusion of vaccine-related misinformation has led to severe consequences during the pandemic (Loomba +et al., 2021). A survey in 2020 showed that more than 55% of U.S. adult participants became hesitant in +obtaining COVID-19 vaccines because they believed in the misinformation about the side effects, political +issues, and safety issues of the vaccines (Graham et al., 2020). When exposed to misinformation about +COVID-19 vaccines, people can become hesitant to take the COVID-19 vaccines, exacerbating their risks +to be infected (Loomba et al., 2021). There is an emergent need for suppressing misinformation spreading +and mitigating the negative consequences of online misinformation on human society. We utilize Louvain +Algorithm (Blondel et al., 2008) to extract the communication communities between Twitter accounts from +the tweets containing the topics of COVID-19 vaccines. We adopt the definition of “beliefs” from existing +psychological studies (Camina et al., 2021; Durando et al., 2016; Herzog et al., 2013; Ritchie et al., 2021) +and focus on if a user thinks the manipulated “side effects” of COVID-19 vaccines exist and accepts/rejects +the COVID-19 vaccination. Regarding this case, “supporting COVID-19 vaccination” is our desired online +social norm and we investigate the social tipping of the expressed normative belief across communities. +We further examine how the dynamics of norm emergences vary across community characteristics, such as +modularity and betweenness centrality (Winkelmann et al., 2022). The study contributes to disclosing the +temporal patterns and mechanisms of social norm emergence in the online environment. Our findings can +facilitate the strategic design of normative interventions for precisely mitigating the dissemination of +misinformation in the online environment. + +2 Data and Methods +2.1 Overview +As shown in Fig. 1, this study starts by collecting real-time tweets regarding the COVID-19 vaccines and +related misinformation using Twitter Streaming API (Twitter, 2022). We define communities in the online +environment based on Newman (2003), i.e., groups of vertices that have a high density of edges within +them, with a lower density of edges between other groups. Specifically for this study, we detect +communities from the “retweeting” and “mentioning” networks among Twitter users in the whole study +period. For example, if one Twitter user retweets/mentions another user within the whole study period, one +edge will exist between these two users. Among the identified individual communities, we select those with +a relatively large population (i.e., more than ten users) and long periods of existence (i.e., more than ten +days). With these communities, we track the temporal change of the community population that follows the +normative belief (i.e., tracking norm emergence) and extract the community characteristics (e.g., modularity, +average degree). After preparation, we first answer RQ1 by observing if social tipping can be identified in + +Page 4/17 +the temporal trend of social norm emergence in our detected individual communities. If tipping exists, we +capture the patterns of the features of social tipping, which include the tipping extent and duration in this +study. Based on the tipping features and community characteristics, we answer RQ2 and explore if +significant correlations exist between social tipping and community characteristics. + +Figure 1 Research procedure + +2.2 Data Preparation +The basic dataset is collected with Twitter Streaming API between May 1, 2020, and April 30, 2021, +regarding COVID-19 vaccines. Specifically, we use keywords of COVID-19 vaccinations to filter out the +tweets that are related to COVID-19 vaccines, including the keywords of “vaccine,” “vax,” “vaccination,” +and brands of COVID-19 vaccines, e.g., “Pfizer”. We extract the online communities based on the +communication networks such as “mentioning/replying” messages (i.e., “@username”) and retweeting +messages (i.e., “RT @username”) for multiple reasons. First, retweeting/replying behaviors tend to happen +between the users who have following relationships and represent the active social ties between online users +(Ozer et al., 2016; B. Wang et al., 2021; Weitzeil et al., 2012). Especially, a study of retweets about COVID- +19 (B. Wang et al., 2021) indicated that more than 50% of the retweets about COVID-19 information were +generated between users with follower/following relationships. Second, retweeting/replying behaviors can +well reflect the social influence of social media users, as the users who tend to retweet or reply to the +messages from others if they are influenced by the tweet content (Evkoski et al., 2021; Yuan and Crooks, +2018). We can potentially capture how a certain belief diffuses among social media users based on the +interactions between the users (e.g., retweeting/replying to tweets) (Evkoski et al., 2021). +Based on the summary of COVID-19 vaccine-related misinformation from Skafle et al. (2022), we focus +on the “side effect” topic of COVID-19 vaccines from the collected tweets, which generally discuss: (a) +whether COVID-19 vaccines have side effects that can heavily threaten human health, (b) whether COVID- +19 vaccines can make people killed, and (c) whether COVID-19 vaccines have not passed trials and are +poisonous. We use keywords (Table 1) of these three topics to identify the related tweets in our collected +dataset. The keywords in the pattern of “word A + word B”, represent the queries that a tweet is regarded +as relevant to the topics if both “word A” and “word B” can be identified in the main text of the tweet. The +nodes in the online individual communities are the users of the tweets in the basic dataset. We only keep +the users whose tweets mentioned other users in the basic dataset, or the users who have been mentioned +by other users in the basic dataset. The news bot accounts are also removed. We finally extract 19,839,188 +tweets containing the keywords about the three topics of misinformation that were posted by 5,462,900 +distinct users (see Table 1). We furtherly detect individual communities and analyze the norm emergence +with this dataset. +Table 1 Keywords of misinformation related to the side effects of COVID-19 vaccines +Topics +Keywords + +Research Preparation +Research Questions +Tracking Norm +RQ1: Existence and Patterns of +Emergence in +Data Preparation +Social Tipping +Communities +Extracting +RQ2: Relationship between +Community +Tipping Features and Community +Characteristics +Characteristics (Hypotheses Test)Page 5/17 +COVID-19 vaccines have +side effects +"side effect", "autism", "autistic", "mental+illness", +"psychological+illness", "mental issue", "psychological issue", +"infertility", +COVID-19 vaccines can +make people killed +"children+die", "children+died", "children+dying", "soldier+die", +"soldier+died", "soldier+dying", "old+die", " old +died", " old+dying", +COVID-19 vaccines have +not passed trials and are +poisonous +"skip+trail", "poison", "not tested", "doesn't be tested", "isn't tested", +"aren't tested", "didn't be tested", "wasn't tested", "weren't tested", +"haven’t been tested" + +2.3 Community Detection +In the retrieved communication network, the edges between users are formed when users reply to or retweet +from other users. The weights of the edges are the frequencies of one user mentioning the other user within +one day. We detect individual communities from social networks using Louvain Algorithms (Blondel et al., +2008). Louvain Algorithm is a combinational optimization algorithm that aims to maximize the modularity +among the detected individual communities. The algorithm has a process that first assigns every node to be +in its community and then for each node it tries to find the maximum positive modularity gain by moving +each node to all its neighbor communities. If no positive gain is achieved the node remains in its original +community (Blondel et al., 2008). Compared to other algorithms, Louvain Algorithm can efficiently capture +the individual communities from a large-scale network, such as a social media network with millions of +users. To better reveal the social tipping in large communities instead of small groups (e.g., a small group +with less than ten members), we select the 100 communities with the largest populations among our detected +communities for the following analysis. +2.4 Classifying Individual Users’ Expressed beliefs towards Misinformation about COVID-19 Vaccines +and Tracking Norm Emergence in Communities +Based on the user’s tweets, we classify the expressed beliefs of individuals at a certain period regarding the +side effect of COVID-19 vaccines. We first classify the expressed beliefs in the tweets of individual users. +We train a Long Short-Term Memory (LSTM) model with 2,000 tweets related to COVID-19 vaccination +and use this model to estimate if tweets from specific users with expressed beliefs that support or reject +misinformation about the side effects of the COVID-19 vaccination. LSTM has a good performance in +existing studies regarding text classification because it captures phrase-level and sentence-level feature +patterns in the tweet text (Zhou et al., 2018). The validated accuracy and loss of the LSTM classifier during +training are shown in Fig. 2, which reach 0.8892 and 0.2292 separately after training, and the RMSE of the +classification outcomes are 0.3719. These metrics indicate that our LSTM classifier has an acceptable +performance in classifying the expressed beliefs of individual users. +After classifying the expressed beliefs delivered in the tweets, we obtain the overall expressed belief of +each user on each day based on their tweets on that day. Specifically, we calculate the proportion of tweets +that one user generates in one day that rejects the misinformation about COVID-19 vaccines. Specifically, +if more than 50% of the tweets are supporting the COVID-19 vaccination, we regard the user accept the +COVID-19 vaccination on that day. If only one tweet is generated by one user on one day, we regard the +expressed belief in the tweet as the expressed belief of that user on a specific date. +We then aggregate the individuals’ expressed beliefs to the community level and track the norm emergence +in our sample communities. We regard the normative belief as “rejecting the misinformation about COVID- +19 vaccines regarding side effects”, and the emergence of norms within a community is tracked by the +temporal trend of the proportion of community members who hold the normative belief. From the temporal +trends, we may identify the tipping points where the acceptance increased rapidly. + +Page 6/17 + +Figure 2 The accuracy and loss of the LSTM classifier during training +2.5 Characterizing the Emergence of Social Norm +Based on the temporal trends of norm emergence in the sample communities, we first observe the trends +and detect if social tipping exists in the communities (RQ1). We detect the existence of social tipping +according to the tipping’s definition, i.e., the increase of community members adopting the norms in +specific periods is relatively more rapid than in the past periods (Berger, 2021). We calculate the daily +increase in the proportion of community members adopting the normative belief, observing if the increase +in a certain period is relatively more rapid than the previous periods. If so, we will regard the social tipping +as existing during the norm emergence of our sample communities. If social tipping does exist in the sample +communities, we adopt the measurements of social tipping in existing studies (Andrighetto & Vriens, 2022), +including the duration and the extent of the social tipping (illustrated in Fig. 3). The duration represents the +number of time steps that the social tipping exists. The extent of social tipping is measured as the change +in the proportion of community members adopting the normative belief before and after social tipping. + +Figure 3 Illustration of duration and extent of social tipping +2.6 Investigating Relationships Between Community Characteristics and Tipping Features +Some characteristics of online individual communities (Table 2) may influence the duration of social +tipping by increasing or decreasing the rapidness of the tipping process. Some community characteristics + +Validation Accuracy +1.0 +ValidationLoss +0.8 +Accuracy +0.6 +0.4 +0.2 +0 +5 +10 +15 +20 +25 +30 +Epoch1.0 +Extent of Tipping +NormativeBelief +0.5 +0.0 +Duration of Tipping +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +DatesPage 7/17 +may also influence the extent of social tipping by causing large-scale social norm acceptance within the +individual community, e.g., the modularity of online individual communities. This study investigates the +statistical correlations between the characteristics of online communities (Table 2) and the duration and +extent of social tipping regarding the proportion of community members accepting the social norms (RQ2). +Table 2 Characteristics of online communities +Characteristics +Reference +Modularity +(Winkelmann et al., 2022) +Messaging frequency +(Centola et al., 2018) +Network size +(Sabarwal & Higgins, 2021) +Original acceptance levels of social norms +(Berger, 2021) +Degree and betweenness centrality of community members +(Winkelmann et al., 2022) + +We specify the influence of each community's characteristics on the duration and extent of social tipping +when examining each hypothesis. We specifically test the following hypotheses that are designed for each +of the community characteristics in Table 2. +• +H1: The modularity of a community has a positive relationship with the duration and extent of social +tipping. +• +H2: The average messaging frequency among members in an online community has a positive +relationship with the duration and extent of social tipping. +• +H3: The size, i.e., the number of members, of a community has a negative relationship with the duration +and extent of social tipping. +• +H4: The original proportion of community members who accept the normative belief has a negative +relationship with the duration and extent of social tipping. +• +H5.1: The average degree of network communities has a positive relationship with the duration and +extent of social tipping +• +H5.2: The average betweenness centrality of network communities has a positive relationship with the +duration and extent of social tipping +Before hypothesis testing, we check the statistical distributions of all the considered community +characteristics and the features of social tipping. In this way, we can identify if the data of hypothesis testing +has an obvious bias. As shown in Fig. 4, most communities have modularity that is lower than 0.1. The +network size of most communities is smaller than 200 users, and the messaging frequency among the +community users tends to be lower than 10 messages a day. For the original acceptance of social norms, +most communities have an acceptance level of lower than 40% when the communities emerge. But still, +more than twenty communities have the original acceptance that is higher than 80% when the communities +emerge. Additionally, the average degree and betweenness centrality of communities tend to evenly +distribute in a small range, e.g., 1.8 to 2.0 for the average degree, and 0 to 0.12 for the betweenness centrality. + +Page 8/17 + +Figure 4 Distributions of community characteristics +We examine all the hypotheses mentioned above with multi-variant linear regression (Eq. 1 and 2). Based +on the identified communities, we examine our proposed hypotheses based on the statistical significance +(i.e., ������������ − ������������������������������������������������������������) and whether the coefficients of community characteristics are positive or negative. For +example, to examine hypothesis H1 regarding the duration of social tipping, if the ������������ − ������������������������������������������������������������ for the +variable ������������������������������������������������������������������������������������������������������������������������ is low and the coefficient for this variable is positive, we can state that modularity +has a significantly positive relationship with the duration of social tipping in an online community. + +Duration~������������������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������ ������������������������������������������������ + ������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ ++ ������������������������������������������������������������������������������������ ������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ +(1) + +������������������������������������������������������������������������~������������������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������ ������������������������������������������������ + ������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ ++ ������������������������������������������������������������������������������������ ������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ +(2) + +3 Results +3.1 Trends and Patterns of Social Norm Emergence in the Sample Communities +To answer RQ1, we first check the temporal trends of social norm emergence, i.e., the change of norm +acceptance among sample communities, aiming to identify if “social tipping” can be identified. Specifically, +we determine that social tipping happened within a certain period (e.g., between two specific dates) if the +daily change of the proportion of the population who adopt the normative belief (i.e., rejecting +misinformation) in the community is much higher than in the past periods. From the temporal trends of +social norm emergence in the largest ten sample communities (Fig. 5), we find that social tipping does exist, +and the social tipping of different communities occurred nearly spontaneously between December 2020 +(when the U.S. FDA first issued emergency usage of COVID-19 vaccines (HHS, 2022)) to April 2021. +Especially at the end of December 2020, the daily increase of the population who adopt the norms exceeded +10%, which was much higher than the past daily increase (which tended to be lower than 4%). After tipping +in these communities, the populations that hold the normative belief towards COVID-19 vaccination in +each community generally reached 65% after three months of social tipping. + +Page 9/17 + +Figure 5 Temporal trends (a) and daily change (b) of social norm emergence in the ten largest sample +communities +Based on the social tipping we identified, we further check the statistical distributions of features of social +tipping among our detected communities, shown in Fig. 6. The histograms of the tipping durations and +extent indicate that, for norms related to the misinformation about COVID-19 vaccines, the social tipping +in online communities tends to be relatively long-term and intense. Specifically, the average duration of +tipping is 83.26 days, the median tipping duration is 96.5 days, and 95% of the sample communities have +a tipping duration between 59 days and 103 days. 17% of the communities have durations that are shorter +than one month, and the duration of 8% of the community is longer than four months. For the extent of +tipping, we can identify that the increase of population who adopt the normative belief in 86% of the sample +communities exceeds 40%, and the tipping in 56% of the sample communities even has the extent that +exceeds 50%. Overall, social tipping in online communities regarding the norms of rejecting +misinformation tends to exist for two to four months, and the tipping extent in more than half of the online +communities may exceed 50%. + +Figure 6 Distributions of features of social tipping among detected communities +3.2 Relationships Between Community Characteristics and Tipping Features +Before conducting the regression, we first check the dependence of the community characteristics, and the +outcome is shown in Fig. 7. Specifically, the absolute value of the correlation between each pair of + +0.8 +0.30 +'Adopting Normative Beliefs +Community 0 +Tipping +Community 0 +TippingPeriod +Period +Community1 +0.25 +Community 1 +0.6 +Community2 +Community2 +0.20 +Community3 +Community3 +Community 4 +Community 4 +0.4 +0.15 +Community 5 +Community 5 +Community 6 +0.10 +Community6 +0.2 +Community 7 +Community 7 +Community 8 +0.05 +Community8 +0.0 +0.00 +M +28 +02 +Date (m-d-y) +Date (m-d-y)Page 10/17 +community characteristics is lower than 0.2. The test outcomes indicate that the considered community +characteristics in this study are relatively independent of other characteristics and can be included in the +multi-variant linear regression models. + +Figure 7 Dependence test of community characteristics (whiter colors represent lower correlations) +Relationships between community characteristics and the duration of social tipping. The table of the +regression outcomes is shown in Table 3, and the estimated coefficients and 95% confidence intervals (CI) +for each community's characteristics are shown in Fig. 8. Among our selected characteristics of the detected +communities, the network sizes (H3), original acceptance levels of social norms (H4), and the average +degree (H5.1) among users have significantly positive impacts on the duration of social tipping. Specifically, +although not significant, modularity and communication frequency among community members have a +negative relationship with the duration of social tipping (H1, H2). The high-level betweenness centrality in +online communities has a positive relationship with the duration of social tipping, but this relationship is +not significant (H5b). Based on the outcomes of this regression outcomes, we identify that social tipping is +highly related to the context and interactions among the community members. Specifically, the high-level +average degree indicates that each community member can communicate with a large number of peers +within the community. The original proportion of community members adopting the normative belief +indicates the context literacy of the community members regarding the topics of misinformation. Our results +indicate that social norms can spread more easily if the individuals are exposed to the information and +interact with more peers than the communities with few interactions. Also, the community members who +originally do not reject the misinformation may not easily change their belief if they can expose to many +interactions with their peers that originally reject the misinformation (i.e., high-level original acceptance). +Additionally, the speed of norm emergence may not increase in the large-scale communities, making the +duration of tipping longer in the large-scale communities than in the small communities. + +Page 11/17 + +Figure 8 Estimated values and 95% CI of coefficients in regression for the duration of social tipping +(significant variables are within red boxes) +Table 3 Outcomes of multi-variant linear regression for the duration of social tipping (Adjusted ������������2: +0.648) +Variables +Coefficient +Standard Error +t Value +P>|t| +[0.025 +0.975] +Modularity +-0.3648 +0.427 +-0.855 +0.394 +-1.207 +0.477 +Network Size +0.0012 +0.001 +2.212 +0.028* +0 +0.002 +Messaging Frequency +-0.0011 +0.002 +-0.545 +0.587 +-0.005 +0.003 +Original Accept Level +0.6879 +0.305 +2.258 +0.025* +0.087 +1.289 +Average Degree of Users 0.4688 +0.08 +5.879 +< 0.001*** +0.311 +0.626 +Average Betweenness +Centrality of Users +2.3743 +2.324 +1.022 +0.308 +-2.212 +6.961 +Significance Levels: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 + +Relationships between community characteristics and the extent of social tipping. The table of the +regression outcomes is shown in Table 4, and estimated coefficients and 95% confidence intervals (95% +CI) for each variable of community characteristics are shown in Fig. 9. Among our selected characteristics +of the detected communities, the average degree (H5a) and betweenness centrality (H5b) among users have +a significantly positive impact on the duration of social tipping. Meanwhile, the modularity (H1) and +original acceptance (H4) of social norms can have a significantly negative relationship with the extent of +social tipping. Different from the average degree and betweenness centrality, the significance of the +relationships between other community characteristics and the extent of social tipping is not high. +Specifically, network size and communication frequency among community members have an insignificant +relationship with the extent of social tipping (H2, H3). Based on the regression outcomes, we identify that +the extent of social tipping is also highly related to the context and interactions among the community +members. Both the betweenness centrality and degree are related to how closely the community members +are connected, and the original acceptance of the normative belief is related to the literacy of community +members regarding the topics of misinformation. The positive and high-level influence of average degree +and betweenness centrality on the tipping extent indicates that more community members will finally turn +to the normative belief if they are exposed to heavy interactions with other community peers. Also, similar + +Value of Coefficients and 95% Cl +Average +Betweenness Centrality +Average +Degree +Original +Acceptance +Messaging +Frequency +Network Size +Modularity +-2 +-1 +0 +1 +2Page 12/17 +to the regression outcomes of tipping duration, the community members who originally do not reject the +misinformation may not easily change their expressed belief if they can expose to many interactions with +the peers that originally reject the misinformation (i.e., high-level original acceptance). + +Figure 9 Estimated values and 95% CI of coefficients in regression for the extent of social tipping +(significant variables are within red boxes) +Table 4 Outcomes of multi-variant linear regression for the extent of social tipping (Adjusted ������������2: 0.972) +Variables +Coefficient +Standard Error +t Value P>|t| +[0.025 +0.975] +Modularity +-0.1635 +0.073 +-2.256 +0.025* +-0.307 +-0.02 +Network Size +-0.0001 +0.0000897 +-1.271 +0.205 +0 +6E-05 +Messaging Frequency +0.0001 +0.000 +0.272 +0.786 +-0.001 +0.001 +Original Accept Level +-0.1127 +0.052 +-2.176 +0.031* +-0.215 +-0.01 +Average Degree of Users 0.4725 +0.014 +34.87 +< 0.001*** +0.446 +0.499 +Average Betweenness +Centrality of Users +1.2297 +0.395 +3.113 +0.002** +0.45 +2.009 +Significance Levels: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 + +4 Discussion +Social norm interventions can potentially mitigate the spread of misinformation, while insufficient +knowledge exists regarding the existence and patterns of social tipping in the online environment, as well +as how the tipping features vary in communities with different network characteristics. This study +investigates the existence of social tipping in the emergence process of the norms and focuses on rejecting +the misinformation about COVID-19 vaccines’ side effects. Also, our regression outcomes indicate that the +duration of tipping is more correlated to the size, average degree, and original acceptance of the normative +belief among the community members. The extent of social tipping (i.e., the increase of community +members adopting the normative belief) is more related to the average degree, average betweenness +centrality, modularity, and the original acceptance of the normative belief among the community members. +This study advances existing knowledge bodies from several perspectives. First, existing studies focused +more on the physical world or artificially designed communities (Berger, 2021; Centola et al., 2018; Ehret +et al., 2022), lacking exploration of the existence and patterns of social tipping in the online digital + +Value of Coefficients and 95% Cl +Average +Betweenness Centrality +Average +Degree +Original +Acceptance +Messaging +Frequency +Network Size +Modularity +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0Page 13/17 +environment. As there can be a difference between the social norm emergence in digital and other +environments, existing knowledge of social tipping may not be fully applicable to the online social norm +intervention. To fill this gap, we conduct empirical studies with the online communication dataset from +Twitter and investigate the social norm emergence in 100 sample communities. To a certain extent, our +study helps identify the statistical distributions of the duration and extent of social tipping in online +communities. The large datasets and sample communities with various characteristics in this study make it +possible to disclose the general patterns of social tipping in online environments. +Second, the existing knowledge body (e.g., Hu & Leung 2017, Savarimuthu & Cranefield 2011, and Sen & +Sen 2010) rarely analyzed the relationships between the patterns of social tipping and the network +characteristics of online communities, e.g., the modularity of the communities or the degree of community +members. To fill this gap, our hypothesis testing with 100 sample communities can help identify the +characteristics of online communities that are significantly correlated with the tipping duration and extent. +We highlight the significant correlation between the features of social tipping and the modularity, +community size, average degree, average betweenness centrality, and original acceptance of the normative +belief. Our findings can contribute to disclosing the general relationships between social tipping and +community characteristics, supporting the future designing of online social norm intervention strategies. +Limitations still exist in this study and open opportunities for our further studies. First, there are still some +external factors that can influence the individuals’ expressed belief (changes), such as governmental +policies, while this study does not include these factors. Our future studies will include the factors of +physical communities to capture the relationships more accurately between social tipping and online +community characteristics. Second, this study focuses on the topics of COVID-19 vaccine-related +misinformation, of which the community characteristics and social tipping may follow distinct temporal +patterns than other topics. To generate more generalizable findings regarding social tipping in online +communities, our future studies will study multiple topics of online communications, e.g., other prevention +measures for COVID-19. Third, this study regards each individual community as relatively isolated from +its neighboring communities, while it is possible that the norm emergence in the neighboring communities +also contributes to the social tipping of the individual communities. Our future studies will investigate the +norm emergence and social tipping in the circumstance of multi-community social networks and explore +the relationships between social tipping in different communities. Fourth, we regard the characteristics of +communities as relatively stable in the study period, while community characteristics may be temporally +dynamic and have different levels of influence on the norm emergence over different periods. Our future +studies will capture the dynamics of online communities and investigate the temporal interactions between +the network characteristics of individual communities and the trend of norm emergence. + +5 Conclusion +Exploring the patterns of social tipping and the relationship between social tipping and community +characteristics is critical for tailoring social norm interventions for mitigating online misinformation. Our +study contributes to the knowledge regarding the heterogeneous temporal patterns and mechanisms of social +tipping in online communities. Our findings can guide public health authorities, emergency responders, and +other crisis managers regarding suppressing online misinformation, such as actively disseminating and +endorsing messages delivering benign normative beliefs on online platforms. With tailored intervention +strategies, crisis managers can motivate the online populations to conduct appropriate prevention measures +(e.g., taking COVID-19 vaccines) as well as mitigate the adverse impacts caused by ineffective prevention +behaviors (e.g., rejecting vaccinations arbitrarily). With the probunking interventions with social norms, +individuals can potentially form positive attitudes towards the public health campaign and proactively reject +and suppress the spread of online misinformation. + + +Page 14/17 +Reference +Andı, S., & Akesson, J. (2021). Nudging Away False News: Evidence from a Social Norms Experiment. +Digital Journalism, 9(1), 106–125. https://doi.org/10.1080/21670811.2020.1847674 +Andreoni, J., Nikiforakis, N., & Siegenthaler, S. (2021). Predicting social tipping and norm change in +controlled experiments. 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Neural Networks, 108, 240–247. +https://doi.org/10.1016/j.neunet.2018.08.016 + diff --git a/ntAyT4oBgHgl3EQflfh4/content/tmp_files/load_file.txt b/ntAyT4oBgHgl3EQflfh4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ba2f0f5fbb450e6b5b367f99cb05ddfd6bae177 --- /dev/null +++ b/ntAyT4oBgHgl3EQflfh4/content/tmp_files/load_file.txt @@ -0,0 +1,1289 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf,len=1288 +page_content='Page 1/17 Investigating the Dynamics of Social Norm Emergence within Online Communities Shangde Gao1, Yan Wang2*, My T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Thai3 1 Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Candidate, Department of Urban and Regional Planning and Florida Institute for Built Environment Resilience, College of Design, Construction and Planning, University of Florida, 1480 Inner Road, Gainesville, FL, 32601, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Email: gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='shangde@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' ORCID: 0000-0003-2218-2872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 2*Assistant Professor, Department of Urban and Regional Planning and Florida Institute for Built Environment Resilience, University of Florida, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Box 115706, Gainesville, FL 32611, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (corresponding author);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' E-mail: yanw@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' ORCID: 0000-0002-3946-9418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 3 Professor, Department of Computer & Information Science & Engineering and Warren B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Nelms Institute for the Connected World, University of Florida, Gainesville, FL 32611, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' E-mail: mythai@cise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' ORCID: 0000-0003-0503-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Abstract: Although social norms’ effect on mitigating misinformation is identified, scant knowledge exists about the patterns of social norm emergence, such as the patterns and variations of social tipping in online communities with diverse characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Accordingly, this study investigates the features of social tipping in online communities and examines the correlations between the tipping features and characteristics of online communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Taking “the side effects of COVID-19 vaccination” as the case topic, we first track the patterns of tipping features in 100 online communities, which are detected using Louvain Algorithm from the aggregated communication network on Twitter between May 2020 and April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Then, we use multi-variant linear regression to explore the correlations between tipping features and communities’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We find that social tipping in online communities can sustain for two to four months and lead to a 50% increase in populations who accept the normative belief in online communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The regression indicates that the duration of social tipping is positively related to the community populations and original acceptance of social norms, while the correlation between the tipping duration and the degrees among community members is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Additionally, the network modularity and original acceptance of social norms have negative relationships with the extent of social tipping, while the users’ degree and betweenness centrality can have significant positive relationships with the extent of tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our findings shed light on more precise normative interventions on misinformation in digital environments as it offers preliminary evidence about the timing and mechanism of social norm emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 1 Introduction The extensive development of online platforms has fostered the spread of messages generated by stakeholders at various levels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', governmental agencies and individual users, during public events (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' A large proportion of user-generated online messages contain inaccurate and misleading information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', misinformation (Del Vicario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The wide diffusion of misinformation has threatened human society from multiple perspectives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', interfering with collective decision-making on democratic, environmental, and public health issues (West & Bergstrom, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' There is an emergent need for suppressing misinformation spreading and mitigating the negative consequences of online misinformation on human society (West & Bergstrom, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Existing studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2012), N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2012), Zhang, Alim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2015, 2016), Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2018), Zhang, Kuhnle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2016), Zhang, Zhang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2015)) tend to suppress misinformation with (i) debunking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', correcting the misinformation after people are exposed to it, and (ii) prebunking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', helping people Page 2/17 recognize the false/misleading contents (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Lewandowsky & van der Linden, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The debunking strategy is widely adopted to provide targeted countermeasures for misinformation of specific topics (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', provide messages with factual elaboration (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' van der Meer & Jin, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022), fact-checking content (Humprecht, 2020), and messages that stimulate the health-protective measures (Humprecht, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The debunking strategy is not always effective when the explanations that support the misinformation exist widely (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The effect of debunking messages tends to be short-term and washed out by future exposure to misinformation (Mourali & Drake, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Also, the debunking strategy can only be conducted after people’s initial exposure to the misinformation (van der Meer & Jin, 2020), while the negative consequences of misinformation may already exist and cause notable social costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' On the contrary, the prebunking strategy is potentially an effective vehicle that overcomes the limitations of the debunking strategy and confers large-scale resistance against misinformation among the public (van der Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The prebunking strategy is based on the social psychological theory of “inoculation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If people are pre-warned and form the belief of rejecting misinformation, they might be “immune” to misinformation (Lewandowsky & van der Linden, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Compared to the debunking strategy, the prebunking strategy focuses on influencing people’s beliefs on the topics of misinformation, posing long-term effects on the public and reducing the occurrence of negative consequences of misinformation (Basol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' When being implemented at a large scale, the pre-bunking strategy is conducted with social norm interventions, which aim to generate the social norms and consensus that support the factual evidence and reject misinformation (Dow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The basis of social norm interventions is people’s adherence to the surrounding social norms (Constantino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Existing in both the digital and physical world (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022), social norms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the shared beliefs or acceptable behaviors in communities, have shown a significant relationship with people’s belief in the content of misinformation (Andı & Akesson, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Gimpel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Lapinski & Rimal, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Adhering to social norms can satisfy a desire to avoid sanctions, confer benefits by coordinating with others, and provide a simple heuristic about what is accepted/wise in a particular context (Constantino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on this psychological phenomenon, social norm interventions have been implemented to help form the belief of supporting factual evidence and rejecting misinformation in both the physical and digital realms (Andı & Akesson, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Gimpel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Lapinski & Rimal, 2005), such as suppressing misinformation about climate actions and health behaviors (Constantino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, by showing individuals the text that describes the “common beliefs” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', social norms) towards the misinformation of a certain topic, individuals tend to modify their beliefs to match the “common beliefs” and reduce the reliance on the misinformation (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' In another case, by showing individuals a message that “most responsible people think twice before sharing articles” (a social norm), individuals are not likely to share social media articles that contain misleading or contested content (Andı & Akesson, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Though the role of the social norm in suppressing misinformation has been identified (Dow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Constantino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022), scant empirical evidence has been provided to inform the implementation of social norm interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Several knowledge gaps and challenges remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' First, with the controlled experiments in physical worlds, recent works have identified that social norm emergence in their artificially designed communities tended to have a tipping process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', social tipping (Berger, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Centola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ehret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Social tipping is a process that when the “tipping point” is reached, a small change in an individual community can create abrupt, nonlinear change in the acceptance of the normative beliefs across the community (Berger, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' By predicting the occurrence and extent of social tipping, policymakers can improve the effectiveness of the social norm interventions by adjusting the timing and efforts of implementing the interventions (Andreoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ehret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' However, due to the lack of analysis of the online communities, it is unclear whether social tipping also exists in online communities and follows certain patterns regarding the tipping features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the duration and extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Little knowledge exists to guide the practices of social norm interventions regarding the Page 3/17 timing and efforts that are needed to promote the tipping process of norm emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Second, experiments in existing studies have identified some evidence regarding the potential relationships between community characteristics and the diffusion of normative beliefs (Hu & Leung, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Savarimuthu & Cranefield, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Sen & Sen, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' However, these experiments were generally based on artificially designed communities in real-world or virtual scenarios, and the experiment findings may not be applicable in the communities of the online environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Also, how the social tipping process varies in the community characteristics has not been disclosed in the existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' There is a need for empirical studies that explore the relationships between community characteristics and social tipping based on real-world communities, providing a reference for the design of social norm interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To fill this research gap, this study aims to answer the following research questions (RQ): RQ1: Does social tipping exist during the social norm emergence of online communities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If so, what are the characteristics and patterns of social tipping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' RQ2: Do the features of social tipping correlate with different network characteristics of individual communities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' This study takes the case of the norms on Twitter regarding the side effects of COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The diffusion of vaccine-related misinformation has led to severe consequences during the pandemic (Loomba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' A survey in 2020 showed that more than 55% of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' adult participants became hesitant in obtaining COVID-19 vaccines because they believed in the misinformation about the side effects, political issues, and safety issues of the vaccines (Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' When exposed to misinformation about COVID-19 vaccines, people can become hesitant to take the COVID-19 vaccines, exacerbating their risks to be infected (Loomba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' There is an emergent need for suppressing misinformation spreading and mitigating the negative consequences of online misinformation on human society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We utilize Louvain Algorithm (Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2008) to extract the communication communities between Twitter accounts from the tweets containing the topics of COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We adopt the definition of “beliefs” from existing psychological studies (Camina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Durando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Herzog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ritchie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021) and focus on if a user thinks the manipulated “side effects” of COVID-19 vaccines exist and accepts/rejects the COVID-19 vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Regarding this case, “supporting COVID-19 vaccination” is our desired online social norm and we investigate the social tipping of the expressed normative belief across communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We further examine how the dynamics of norm emergences vary across community characteristics, such as modularity and betweenness centrality (Winkelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The study contributes to disclosing the temporal patterns and mechanisms of social norm emergence in the online environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our findings can facilitate the strategic design of normative interventions for precisely mitigating the dissemination of misinformation in the online environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 2 Data and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1 Overview As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 1, this study starts by collecting real-time tweets regarding the COVID-19 vaccines and related misinformation using Twitter Streaming API (Twitter, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We define communities in the online environment based on Newman (2003), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', groups of vertices that have a high density of edges within them, with a lower density of edges between other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically for this study, we detect communities from the “retweeting” and “mentioning” networks among Twitter users in the whole study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' For example, if one Twitter user retweets/mentions another user within the whole study period, one edge will exist between these two users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Among the identified individual communities, we select those with a relatively large population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', more than ten users) and long periods of existence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', more than ten days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' With these communities, we track the temporal change of the community population that follows the normative belief (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', tracking norm emergence) and extract the community characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', modularity, average degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' After preparation, we first answer RQ1 by observing if social tipping can be identified in Page 4/17 the temporal trend of social norm emergence in our detected individual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If tipping exists, we capture the patterns of the features of social tipping, which include the tipping extent and duration in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on the tipping features and community characteristics, we answer RQ2 and explore if significant correlations exist between social tipping and community characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Figure 1 Research procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2 Data Preparation The basic dataset is collected with Twitter Streaming API between May 1, 2020, and April 30, 2021, regarding COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, we use keywords of COVID-19 vaccinations to filter out the tweets that are related to COVID-19 vaccines, including the keywords of “vaccine,” “vax,” “vaccination,” and brands of COVID-19 vaccines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', “Pfizer”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We extract the online communities based on the communication networks such as “mentioning/replying” messages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', “@username”) and retweeting messages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', “RT @username”) for multiple reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' First, retweeting/replying behaviors tend to happen between the users who have following relationships and represent the active social ties between online users (Ozer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Weitzeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Especially, a study of retweets about COVID- 19 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021) indicated that more than 50% of the retweets about COVID-19 information were generated between users with follower/following relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Second, retweeting/replying behaviors can well reflect the social influence of social media users, as the users who tend to retweet or reply to the messages from others if they are influenced by the tweet content (Evkoski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Yuan and Crooks, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We can potentially capture how a certain belief diffuses among social media users based on the interactions between the users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', retweeting/replying to tweets) (Evkoski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on the summary of COVID-19 vaccine-related misinformation from Skafle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2022), we focus on the “side effect” topic of COVID-19 vaccines from the collected tweets, which generally discuss: (a) whether COVID-19 vaccines have side effects that can heavily threaten human health, (b) whether COVID- 19 vaccines can make people killed, and (c) whether COVID-19 vaccines have not passed trials and are poisonous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We use keywords (Table 1) of these three topics to identify the related tweets in our collected dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The keywords in the pattern of “word A + word B”, represent the queries that a tweet is regarded as relevant to the topics if both “word A” and “word B” can be identified in the main text of the tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The nodes in the online individual communities are the users of the tweets in the basic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We only keep the users whose tweets mentioned other users in the basic dataset, or the users who have been mentioned by other users in the basic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The news bot accounts are also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We finally extract 19,839,188 tweets containing the keywords about the three topics of misinformation that were posted by 5,462,900 distinct users (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We furtherly detect individual communities and analyze the norm emergence with this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Table 1 Keywords of misinformation related to the side effects of COVID-19 vaccines Topics Keywords Research Preparation Research Questions Tracking Norm RQ1: Existence and Patterns of Emergence in Data Preparation Social Tipping Communities Extracting RQ2: Relationship between Community Tipping Features and Community Characteristics Characteristics (Hypotheses Test)Page 5/17 COVID-19 vaccines have side effects "side effect",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "autism",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "autistic",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "mental+illness",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "psychological+illness",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "mental issue",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "psychological issue",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "infertility",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' COVID-19 vaccines can make people killed "children+die",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "children+died",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "children+dying",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "soldier+die",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "soldier+died",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "soldier+dying",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "old+die",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' " old +died",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' " old+dying",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' COVID-19 vaccines have not passed trials and are poisonous "skip+trail",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "poison",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "not tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "doesn\'t be tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "isn\'t tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "aren\'t tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "didn\'t be tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "wasn\'t tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "weren\'t tested",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' "haven’t been tested" 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3 Community Detection In the retrieved communication network, the edges between users are formed when users reply to or retweet from other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The weights of the edges are the frequencies of one user mentioning the other user within one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We detect individual communities from social networks using Louvain Algorithms (Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Louvain Algorithm is a combinational optimization algorithm that aims to maximize the modularity among the detected individual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The algorithm has a process that first assigns every node to be in its community and then for each node it tries to find the maximum positive modularity gain by moving each node to all its neighbor communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If no positive gain is achieved the node remains in its original community (Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Compared to other algorithms, Louvain Algorithm can efficiently capture the individual communities from a large-scale network, such as a social media network with millions of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To better reveal the social tipping in large communities instead of small groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', a small group with less than ten members), we select the 100 communities with the largest populations among our detected communities for the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='4 Classifying Individual Users’ Expressed beliefs towards Misinformation about COVID-19 Vaccines and Tracking Norm Emergence in Communities Based on the user’s tweets, we classify the expressed beliefs of individuals at a certain period regarding the side effect of COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We first classify the expressed beliefs in the tweets of individual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We train a Long Short-Term Memory (LSTM) model with 2,000 tweets related to COVID-19 vaccination and use this model to estimate if tweets from specific users with expressed beliefs that support or reject misinformation about the side effects of the COVID-19 vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' LSTM has a good performance in existing studies regarding text classification because it captures phrase-level and sentence-level feature patterns in the tweet text (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The validated accuracy and loss of the LSTM classifier during training are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 2, which reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='8892 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2292 separately after training, and the RMSE of the classification outcomes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' These metrics indicate that our LSTM classifier has an acceptable performance in classifying the expressed beliefs of individual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' After classifying the expressed beliefs delivered in the tweets, we obtain the overall expressed belief of each user on each day based on their tweets on that day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, we calculate the proportion of tweets that one user generates in one day that rejects the misinformation about COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, if more than 50% of the tweets are supporting the COVID-19 vaccination, we regard the user accept the COVID-19 vaccination on that day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If only one tweet is generated by one user on one day, we regard the expressed belief in the tweet as the expressed belief of that user on a specific date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We then aggregate the individuals’ expressed beliefs to the community level and track the norm emergence in our sample communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We regard the normative belief as “rejecting the misinformation about COVID- 19 vaccines regarding side effects”, and the emergence of norms within a community is tracked by the temporal trend of the proportion of community members who hold the normative belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' From the temporal trends, we may identify the tipping points where the acceptance increased rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Page 6/17 Figure 2 The accuracy and loss of the LSTM classifier during training 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 Characterizing the Emergence of Social Norm Based on the temporal trends of norm emergence in the sample communities, we first observe the trends and detect if social tipping exists in the communities (RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We detect the existence of social tipping according to the tipping’s definition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the increase of community members adopting the norms in specific periods is relatively more rapid than in the past periods (Berger, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We calculate the daily increase in the proportion of community members adopting the normative belief, observing if the increase in a certain period is relatively more rapid than the previous periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If so, we will regard the social tipping as existing during the norm emergence of our sample communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' If social tipping does exist in the sample communities, we adopt the measurements of social tipping in existing studies (Andrighetto & Vriens, 2022), including the duration and the extent of the social tipping (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The duration represents the number of time steps that the social tipping exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The extent of social tipping is measured as the change in the proportion of community members adopting the normative belief before and after social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Figure 3 Illustration of duration and extent of social tipping 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='6 Investigating Relationships Between Community Characteristics and Tipping Features Some characteristics of online individual communities (Table 2) may influence the duration of social tipping by increasing or decreasing the rapidness of the tipping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Some community characteristics Validation Accuracy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 ValidationLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2 0 5 10 15 20 25 30 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 Extent of Tipping NormativeBelief 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 Duration of Tipping 2 3 4 5 6 7 8 9 10 11 12 13 14 DatesPage 7/17 may also influence the extent of social tipping by causing large-scale social norm acceptance within the individual community, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the modularity of online individual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' This study investigates the statistical correlations between the characteristics of online communities (Table 2) and the duration and extent of social tipping regarding the proportion of community members accepting the social norms (RQ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Table 2 Characteristics of online communities Characteristics Reference Modularity (Winkelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022) Messaging frequency (Centola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2018) Network size (Sabarwal & Higgins, 2021) Original acceptance levels of social norms (Berger, 2021) Degree and betweenness centrality of community members (Winkelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=", 2022) We specify the influence of each community's characteristics on the duration and extent of social tipping when examining each hypothesis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We specifically test the following hypotheses that are designed for each of the community characteristics in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H1: The modularity of a community has a positive relationship with the duration and extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H2: The average messaging frequency among members in an online community has a positive relationship with the duration and extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H3: The size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the number of members, of a community has a negative relationship with the duration and extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H4: The original proportion of community members who accept the normative belief has a negative relationship with the duration and extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' H5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1: The average degree of network communities has a positive relationship with the duration and extent of social tipping H5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2: The average betweenness centrality of network communities has a positive relationship with the duration and extent of social tipping Before hypothesis testing, we check the statistical distributions of all the considered community characteristics and the features of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' In this way, we can identify if the data of hypothesis testing has an obvious bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 4, most communities have modularity that is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The network size of most communities is smaller than 200 users, and the messaging frequency among the community users tends to be lower than 10 messages a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' For the original acceptance of social norms, most communities have an acceptance level of lower than 40% when the communities emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' But still, more than twenty communities have the original acceptance that is higher than 80% when the communities emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Additionally, the average degree and betweenness centrality of communities tend to evenly distribute in a small range, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 for the average degree, and 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='12 for the betweenness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Page 8/17 Figure 4 Distributions of community characteristics We examine all the hypotheses mentioned above with multi-variant linear regression (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on the identified communities, we examine our proposed hypotheses based on the statistical significance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', ������������ − ������������������������������������������������������������) and whether the coefficients of community characteristics are positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' For example, to examine hypothesis H1 regarding the duration of social tipping, if the ������������ − ������������������������������������������������������������ for the variable ������������������������������������������������������������������������������������������������������������������������ is low and the coefficient for this variable is positive, we can state that modularity has a significantly positive relationship with the duration of social tipping in an online community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='Duration~������������������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������ + ������������������������������������������������������������������������ ������������������������������������������������ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������~������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������ ������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='������������������������������������������������������������������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3 Results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1 Trends and Patterns of Social Norm Emergence in the Sample Communities To answer RQ1, we first check the temporal trends of social norm emergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the change of norm acceptance among sample communities, aiming to identify if “social tipping” can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, we determine that social tipping happened within a certain period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', between two specific dates) if the daily change of the proportion of the population who adopt the normative belief (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', rejecting misinformation) in the community is much higher than in the past periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' From the temporal trends of social norm emergence in the largest ten sample communities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 5), we find that social tipping does exist, and the social tipping of different communities occurred nearly spontaneously between December 2020 (when the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' FDA first issued emergency usage of COVID-19 vaccines (HHS, 2022)) to April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Especially at the end of December 2020, the daily increase of the population who adopt the norms exceeded 10%, which was much higher than the past daily increase (which tended to be lower than 4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' After tipping in these communities, the populations that hold the normative belief towards COVID-19 vaccination in each community generally reached 65% after three months of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Page 9/17 Figure 5 Temporal trends (a) and daily change (b) of social norm emergence in the ten largest sample communities Based on the social tipping we identified, we further check the statistical distributions of features of social tipping among our detected communities, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The histograms of the tipping durations and extent indicate that, for norms related to the misinformation about COVID-19 vaccines, the social tipping in online communities tends to be relatively long-term and intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, the average duration of tipping is 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='26 days, the median tipping duration is 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 days, and 95% of the sample communities have a tipping duration between 59 days and 103 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 17% of the communities have durations that are shorter than one month, and the duration of 8% of the community is longer than four months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' For the extent of tipping, we can identify that the increase of population who adopt the normative belief in 86% of the sample communities exceeds 40%, and the tipping in 56% of the sample communities even has the extent that exceeds 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Overall, social tipping in online communities regarding the norms of rejecting misinformation tends to exist for two to four months, and the tipping extent in more than half of the online communities may exceed 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Figure 6 Distributions of features of social tipping among detected communities 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2 Relationships Between Community Characteristics and Tipping Features Before conducting the regression, we first check the dependence of the community characteristics, and the outcome is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, the absolute value of the correlation between each pair of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content="30 'Adopting Normative Beliefs Community 0 Tipping Community 0 TippingPeriod Period Community1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='25 Community 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='6 Community2 Community2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='20 Community3 Community3 Community 4 Community 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='15 Community 5 Community 5 Community 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='10 Community6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2 Community 7 Community 7 Community 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='05 Community8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='00 M 28 02 Date (m-d-y) Date (m-d-y)Page 10/17 community characteristics is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The test outcomes indicate that the considered community characteristics in this study are relatively independent of other characteristics and can be included in the multi-variant linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Figure 7 Dependence test of community characteristics (whiter colors represent lower correlations) Relationships between community characteristics and the duration of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=" The table of the regression outcomes is shown in Table 3, and the estimated coefficients and 95% confidence intervals (CI) for each community's characteristics are shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Among our selected characteristics of the detected communities, the network sizes (H3), original acceptance levels of social norms (H4), and the average degree (H5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1) among users have significantly positive impacts on the duration of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, although not significant, modularity and communication frequency among community members have a negative relationship with the duration of social tipping (H1, H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The high-level betweenness centrality in online communities has a positive relationship with the duration of social tipping, but this relationship is not significant (H5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on the outcomes of this regression outcomes, we identify that social tipping is highly related to the context and interactions among the community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, the high-level average degree indicates that each community member can communicate with a large number of peers within the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The original proportion of community members adopting the normative belief indicates the context literacy of the community members regarding the topics of misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our results indicate that social norms can spread more easily if the individuals are exposed to the information and interact with more peers than the communities with few interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Also, the community members who originally do not reject the misinformation may not easily change their belief if they can expose to many interactions with their peers that originally reject the misinformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', high-level original acceptance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Additionally, the speed of norm emergence may not increase in the large-scale communities, making the duration of tipping longer in the large-scale communities than in the small communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Page 11/17 Figure 8 Estimated values and 95% CI of coefficients in regression for the duration of social tipping (significant variables are within red boxes) Table 3 Outcomes of multi-variant linear regression for the duration of social tipping (Adjusted ������������2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='648) Variables Coefficient Standard Error t Value P>|t| [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='975] Modularity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='394 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='477 Network Size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='028* 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='002 Messaging Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='003 Original Accept Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='6879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='305 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='025* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='087 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='289 Average Degree of Users 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='4688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='879 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='626 Average Betweenness Centrality of Users 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='3743 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='324 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='308 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='212 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='961 Significance Levels: 0 ‘***’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001 ‘**’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='01 ‘*’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='05 ‘.’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1 ‘ ’ 1 Relationships between community characteristics and the extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The table of the regression outcomes is shown in Table 4, and estimated coefficients and 95% confidence intervals (95% CI) for each variable of community characteristics are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Among our selected characteristics of the detected communities, the average degree (H5a) and betweenness centrality (H5b) among users have a significantly positive impact on the duration of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Meanwhile, the modularity (H1) and original acceptance (H4) of social norms can have a significantly negative relationship with the extent of social tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Different from the average degree and betweenness centrality, the significance of the relationships between other community characteristics and the extent of social tipping is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Specifically, network size and communication frequency among community members have an insignificant relationship with the extent of social tipping (H2, H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Based on the regression outcomes, we identify that the extent of social tipping is also highly related to the context and interactions among the community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Both the betweenness centrality and degree are related to how closely the community members are connected, and the original acceptance of the normative belief is related to the literacy of community members regarding the topics of misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The positive and high-level influence of average degree and betweenness centrality on the tipping extent indicates that more community members will finally turn to the normative belief if they are exposed to heavy interactions with other community peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Also, similar Value of Coefficients and 95% Cl Average Betweenness Centrality Average Degree Original Acceptance Messaging Frequency Network Size Modularity 2 1 0 1 2Page 12/17 to the regression outcomes of tipping duration, the community members who originally do not reject the misinformation may not easily change their expressed belief if they can expose to many interactions with the peers that originally reject the misinformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', high-level original acceptance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Figure 9 Estimated values and 95% CI of coefficients in regression for the extent of social tipping (significant variables are within red boxes) Table 4 Outcomes of multi-variant linear regression for the extent of social tipping (Adjusted ������������2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='972) Variables Coefficient Standard Error t Value P>|t| [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='975] Modularity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='073 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='025* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='02 Network Size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0000897 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='205 0 6E-05 Messaging Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001 Original Accept Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='052 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='031* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='01 Average Degree of Users 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='4725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='014 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='87 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='499 Average Betweenness Centrality of Users 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='2297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='395 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='002** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='009 Significance Levels: 0 ‘***’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='001 ‘**’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='01 ‘*’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='05 ‘.’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='1 ‘ ’ 1 4 Discussion Social norm interventions can potentially mitigate the spread of misinformation, while insufficient knowledge exists regarding the existence and patterns of social tipping in the online environment, as well as how the tipping features vary in communities with different network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' This study investigates the existence of social tipping in the emergence process of the norms and focuses on rejecting the misinformation about COVID-19 vaccines’ side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Also, our regression outcomes indicate that the duration of tipping is more correlated to the size, average degree, and original acceptance of the normative belief among the community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The extent of social tipping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the increase of community members adopting the normative belief) is more related to the average degree, average betweenness centrality, modularity, and the original acceptance of the normative belief among the community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' This study advances existing knowledge bodies from several perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' First, existing studies focused more on the physical world or artificially designed communities (Berger, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Centola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Ehret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', 2022), lacking exploration of the existence and patterns of social tipping in the online digital Value of Coefficients and 95% Cl Average Betweenness Centrality Average Degree Original Acceptance Messaging Frequency Network Size Modularity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='0Page 13/17 environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' As there can be a difference between the social norm emergence in digital and other environments, existing knowledge of social tipping may not be fully applicable to the online social norm intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To fill this gap, we conduct empirical studies with the online communication dataset from Twitter and investigate the social norm emergence in 100 sample communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To a certain extent, our study helps identify the statistical distributions of the duration and extent of social tipping in online communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' The large datasets and sample communities with various characteristics in this study make it possible to disclose the general patterns of social tipping in online environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Second, the existing knowledge body (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', Hu & Leung 2017, Savarimuthu & Cranefield 2011, and Sen & Sen 2010) rarely analyzed the relationships between the patterns of social tipping and the network characteristics of online communities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', the modularity of the communities or the degree of community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To fill this gap, our hypothesis testing with 100 sample communities can help identify the characteristics of online communities that are significantly correlated with the tipping duration and extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' We highlight the significant correlation between the features of social tipping and the modularity, community size, average degree, average betweenness centrality, and original acceptance of the normative belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our findings can contribute to disclosing the general relationships between social tipping and community characteristics, supporting the future designing of online social norm intervention strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Limitations still exist in this study and open opportunities for our further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' First, there are still some external factors that can influence the individuals’ expressed belief (changes), such as governmental policies, while this study does not include these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our future studies will include the factors of physical communities to capture the relationships more accurately between social tipping and online community characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Second, this study focuses on the topics of COVID-19 vaccine-related misinformation, of which the community characteristics and social tipping may follow distinct temporal patterns than other topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' To generate more generalizable findings regarding social tipping in online communities, our future studies will study multiple topics of online communications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', other prevention measures for COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Third, this study regards each individual community as relatively isolated from its neighboring communities, while it is possible that the norm emergence in the neighboring communities also contributes to the social tipping of the individual communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our future studies will investigate the norm emergence and social tipping in the circumstance of multi-community social networks and explore the relationships between social tipping in different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Fourth, we regard the characteristics of communities as relatively stable in the study period, while community characteristics may be temporally dynamic and have different levels of influence on the norm emergence over different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our future studies will capture the dynamics of online communities and investigate the temporal interactions between the network characteristics of individual communities and the trend of norm emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' 5 Conclusion Exploring the patterns of social tipping and the relationship between social tipping and community characteristics is critical for tailoring social norm interventions for mitigating online misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our study contributes to the knowledge regarding the heterogeneous temporal patterns and mechanisms of social tipping in online communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Our findings can guide public health authorities, emergency responders, and other crisis managers regarding suppressing online misinformation, such as actively disseminating and endorsing messages delivering benign normative beliefs on online platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' With tailored intervention strategies, crisis managers can motivate the online populations to conduct appropriate prevention measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', taking COVID-19 vaccines) as well as mitigate the adverse impacts caused by ineffective prevention behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', rejecting vaccinations arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' With the probunking interventions with social norms, individuals can potentially form positive attitudes towards the public health campaign and proactively reject and suppress the spread of online misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' Page 14/17 Reference Andı, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=', & Akesson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content=' (2021).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} +page_content='016' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntAyT4oBgHgl3EQflfh4/content/2301.00453v1.pdf'} diff --git a/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/2301.00696v1.pdf.txt b/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/2301.00696v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a2224651f3e9ca452a04b34be0aabef0487b248 --- /dev/null +++ b/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/2301.00696v1.pdf.txt @@ -0,0 +1,1042 @@ +arXiv:2301.00696v1 [math.GN] 2 Jan 2023 +ON CERTAIN GENERALIZED NOTIONS USING +I-CONVERGENCE IN TOPOLOGICAL SPACES +PRATULANANDA DAS∗, UPASANA SAMANTA∗, SHOU LIN† +Abstract. In this paper, we consider certain topological properties along +with certain types of mappings on these spaces defined by the notion of +ideal convergence. +In order to do that, we primarily follow in the foot- +steps of the earlier studies of ideal convergence done by using functions +(from an infinite set S to X) in [8, 9, 29], as that is the most general per- +spective and use functions instead of sequences/nets/double sequences etc. +This functional approach automatically provides the most general settings +for such studies and consequently extends and unifies the proofs of sev- +eral old and recent results in the literature about spaces like sequential, +Fr´echet-Uryshon spaces and sequential, quotient and covering maps. +In +particular, we introduce and investigate the notions of I-functional spaces, +I-functional continuous, quotient and covering mappings and finally I- +functional Fr´echet-Uryshon spaces. In doing so, we take help of certain set +theoretic and other properties of ideals. +Key words and phrases: Ideal, ideal convergence of functions, I-functional +space, I-functional continuous, quotient and covering mappings, I-functional +Fr´echet-Uryshon space. +1. Introduction +. The idea of statistical convergence of sequence was introduced in [12,33] as an +extension of the usual notion of convergence. Apart from a lot of investigations +in the fields of summability theory, measure theory, functional analysis etc., this +idea has led to various investigations in the settings of topological spaces (for ex- +ample see [4,5,10,24,25,34–36]). The most important generalization of almost +all types of convergence including statistical convergence had been proposed +by Kostyrko et al. [20] who had introduced the concepts of I-convergence and +I∗-convergence in metric spaces using ideals of the set of all natural numbers. +Following the line of Kostyrko et al., the same has been studied for sequences +2010 Mathematics Subject Classification. Primary: +54A20, 54B15, 54C08 Secondary: +40A05, 26A03 . +The +first +author +is +thankful +to +NBHM +for +granting +the +project +(sanction +no. +02011/9/2022/NBHM(RP)/RD II/10378) during the tenure of which this work was done. +1 + +2 +P. DAS, U. SAMANTA, S. LIN +in general topological spaces [22], for nets in topological and uniform spaces [23] +(subsequently studied in [6, 7]) and for functions in topological spaces [9, 29], +uniform spaces [8] for example, where other references can be found. +On the other hand, it is a well known fact that the topology of a topological +space, in general, can not be determined by convergent sequences, unlike metric +spaces where sequences play a much more important role in characterizing several +notions. From the beginning, it has been a very rich and challenging topic of +investigations as to, in which topological spaces sequences play a better role. +The first countable, Fr´echet-Uryshon and sequential spaces are examples of some +such spaces that are determined by convergence of sequences [11, 28]. Instead +of usual convergence of sequences, first in [32,34] the authors have worked with +statistical convergence to define statistical counterparts of Fr´echet-Uryshon and +sequential spaces. Subsequently the more general idea of ideal convergence of +sequences has been widely used to introduce these notions as also several other +new ideas in topological settings (for example one can see [3,31,32,35,36]). +In particular in [36], Zhou and his co-authors defined I-continuous, I-quotient +and I-covering mappings and checked how they interact with I-sequential, +I-Fr´echet spaces. +As a natural consequence, in this paper, we further generalize the whole set- +ting of such investigations by considering ideals of an arbitrary infinite set S, and +as a natural replacement, instead of sequences in X we take functions from S to +X. This approach unifies the two directions mentioned above and provides the +most general type of results. Primarily we use the idea of I-convergence of func- +tions to introduce I-functional open sets, I-functional closed sets, I-functional +spaces and I-functional Fr´echet-Uryshon spaces and establish several proper- +ties. +We also proceed in the same way to extend the ideas of I-continuous, +I-quotient and I-covering mappings and subsequently investigate their coun- +terparts, namely, I-functional continuous, quotient and covering mappings and +their effects on I-functional spaces, I-functional Fr´echet-Uryshon spaces. In or- +der to clear ambiguity and for the sake of continuity, we call all mappings with +domain S “functions” (continuing the nomenclature of [8,9,29]) and mappings +from one topological space to another as just “mappings”. +As a consequence, not only the results of [3,31,32,35,36] become special cases +of our results, also the whole treatment seems much more simplified, at the same +time underscoring the focal point that, several topological concepts can actually +be studied without restricting the domain set. + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES3 +2. Preliminaries +Let N denote the set of all natural numbers and let K ⊂ N. Recall that the nat- +ural or asymptotic density of K is defined by d(K) = +lim +n−→∞ +1 +n|{k ∈ N : k ≤ n}| +if the limit exists. If X is a topological space then a sequence (xn : n ∈ N) +in X is statistically convergent to x ∈ X if for each neighbourhood U of x in +X, d({n ∈ N : xn ̸∈ U}) = 0 [10]. +The notion of statistical convergence has subsequently been extended to the +notion of I-convergence, which is based on the notion of ideal of subsets of N. +Let Y be a non-empty set and let P(Y ) be the family of all subsets of Y. A +family I(⊂ P(Y )) of subsets of a non-empty set Y is said to be an ideal of Y +if (i) A, B ∈ I imply A ∪ B ∈ I (ii) A ∈ I, B ⊂ A imply B ∈ I, while an +admissible ideal I of Y covers Y . Such ideals are also called free ideals. If I +is a proper non-trivial ideal of Y (i.e. Y /∈ I, I ̸= ∅), then the family of sets +F(I) = {M ⊂ Y : Y \ M ∈ I} is a filter (called the dual filter) of Y whereas +the coideal of I is I+ = {A ⊂ Y : A ̸∈ I}. We denote the ideal consisting of all +finite subsets and density zero subsets of N by Ifin and Id respectively. +If I is a maximal ideal then for any A ⊂ S, we have either A ∈ I or S \A ∈ I. +For each ideal I of S, the set of all maximal ideals J of S such that I ⊂ J is +denoted by Θ(I). It is known that I = +� +J ∈Θ(I) +J [17]. Recall that B ⊂ N is said +to be a pseudounion of a family A ⊂ P(N) if N \ B is infinite and A \ B is finite +for each A ∈ A [3]. +A sequence (xn : n ∈ N) in a topological space X is said to be I-convergent +to x ∈ X provided for each neighbourhood U of x, the set {n ∈ N : xn ̸∈ U} +belongs to I [22]. I-convergence of sequence coincides with ordinary convergence +of sequence if we take I = Ifin and with the statistical convergence if I = Id. +The concept of I∗-convergence of real sequence arises from a result of statis- +tical convergence that: a real sequence (xn : n ∈ N) is statistically convergent to +x if and only if there exists a set M = {mk : k ∈ N} with m1 < m2 < · · · mk · · · +such that d(M) = 1 and +lim +k−→∞ xmk = x. This idea has been extended to +I∗-convergence of a sequence in a topological space as a sequence (xn : n ∈ N) +in X is I∗-convergent to x ∈ X if and only if there exists a set M ∈ F(I) where +m1 < m2 < · · · < mk < · · · such that +lim +k−→∞ xmk = x [22]. +Throughout the paper X stands for a topological space, S an infinite set and +I, an admissible ideal of S unless otherwise stated. Further by a “space” we will +always mean a topological space. Our topological terminology and notation are +as in the book [11]. + +4 +P. DAS, U. SAMANTA, S. LIN +3. I-functional open sets, I-functional closed sets and I-functional +space +Before we proceed to introduce our main concepts of this section, we present +certain basic observations about convergence of functions which happen to be +the main tool behind these generalizations. +Definition 3.1. For x ∈ X, we say that a function f : S → X +• is convergent to x, whenever for every open set U containing x, the set +f −1(U) is co-finite. +• is I-convergent to x, whenever for every open set U containing x, the +set f −1(U) is in F(I) [29]. +• is I∗-convergent to x, whenever there is a set M ∈ F(I) such that g +defined by “g(s) = f(s) if s ∈ M and g(s) = x if s /∈ M” is convergent +to x [29]. +Suppose g : S −→ X, is I-convergent to x. Let S′ be an infinite subset of S +with |S′| = |S|. Let h : S −→ S′ be a bijective function and let Φ = g|S′. Now +Φ is said to be I-convergent to x if (Φ ◦ h)(s) = g(s), ∀ s ∈ S is I-convergent +to x. +Further if f : S −→ X is convergent to x ∈ X, then for any infinite S′ ⊂ +S, f|S′ is convergent to x. In a Hausdorff space I-limit of a function is unique. +For two ideals I ⊂ J of S, if f : S −→ X is I-convergent to x then f : S −→ X +is J -convergent to x. +Following [3] we can say that an ideal I of S has a pseudounion if there exists +an infinite set A ⊂ S with |S| = |S \ A| such that I \ A is finite for each I ∈ I. +Lemma 3.1. If I has a pseudounion and f : S −→ X is I-convergent to x then +there exists a function from S to X which is convergent to x. +Proof. Let f : S −→ X be I-convergent to x. Since I has a pseudounion, there +exists an infinite set A ⊂ S with S \ A ∈ I+ such that I ∩ (S \ A) is finite +for each I ∈ I. As f is I-convergent to x, for every open set O containing +x, AO = {s ∈ S : f(s) ̸∈ O} ∈ I. Thus AO ∩ (S \ A) is finite. Then Φ : S −→ X +defined by Φ(s) = f(s) if s ∈ S \A and Φ(s) = x if s ∈ A, is convergent to x. +□ +Proposition 3.1. Let g : S → X be given. Then g is I-convergent to x if and +only if g is J -convergent to x for each J ∈ Θ(I). +Example 3.1. Let I be an ideal of S. Take ∞ ̸∈ S. We define a topology on +S ∪ {∞} by considering each s ∈ S isolated and each basic open neighbourhood +U of ∞ as (S \ I) ∪ {∞} for some I ∈ I. This space is denoted by � +S(I). +Clearly the inclusion mapping i : S −→ � +S(I) is I-convergent to ∞. Note that + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES5 +if I ̸= Ifin, then I contains an infinite set I ‘say’. Then it readily follows that +the inclusion function is not convergent to ∞ in the usual sense. +Let us now look back at the history as to how the notion of closed sets in topo- +logical spaces have been generalized using sequences. Recall that a subset F ⊂ X +is called sequentially closed if for each sequence (xn : n ∈ N) in F converging to +x ∈ X, we have x ∈ F. X is called a sequential space [13] if each sequentially +closed subset of X is closed. A subset U ⊂ X is called sequentially open if X \U +is sequentially closed. Di Maio and Koˇcinac introduced statistical version of +sequential space in [10] while Pal [31] further extended it to I-sequential spaces. +Very recently Zhou et al. revisited the notion of I-sequential space in [36] where +following notions were introduced. A subset F ⊂ X is called I-closed if for +each sequence (xn : n ∈ N) in F, I-convergent to x ∈ X, we have x ∈ F. A +subset U ⊂ X is called I-open if X \ U is I-closed. X is called an I-sequential +space if each I-closed subset of X is closed. Motivated by the generalization +of I-sequential spaces from the idea of sequential spaces, we now introduce the +main concept of this section. +Definition 3.2. Let X be a topological space. (i) F ⊂ X is said to be I-functional +closed if for each function g : S → F that is I-convergent to x ∈ X we have +x ∈ F. +(ii) U ⊂ X is said to be I-functional open if X \ U is I-functional closed. +(iii) X is called an I-functional space if each I-functional closed subset of X is +closed. +If we consider “usual” convergence of functions (see Definition 3.1) instead +of I-convergence, we call I-functional closed sets, I-functional open sets and +I-functional spaces as functional closed, functional open and functional spaces +respectively. Clearly, every I-functional closed set is functional closed but the +following example shows that the converse is not generally true. +Example 3.2. Let I be a maximal ideal of S. We consider the space � +S(I) as +in Example 3.1. Then S is a functional closed set in � +S(I) but not I-functional +closed. +As an immediate consequence of Lemma 3.1 we can see that +Proposition 3.2. A ⊂ X is I-functional closed if and only if A is functional +closed provided I has a pseudounion. Therefore X is an I-functional space if +and only if X is a functional space provided I has a pseudounion. +We can modify Definition 3.2(ii) in the following way. +Lemma 3.2. A subset O of X is I-functional open if and only if no function +h : S −→ X \ O is I-convergent to a point in O. + +6 +P. DAS, U. SAMANTA, S. LIN +Proof. Sufficiency directly follows from Definition 3.2(i), (ii). As O is I-functional +open so X \ O is I-functional closed. Hence for every function h : S −→ X \ O +which is I-convergent to x, we must have x ∈ X \ O. +□ +It is evident that every open set (and so every closed set) is I-functional open +(I-functional closed). Following example establishes the existence of a space +which is not I-functional. +Example 3.3. Consider the Cartesian product S × S. For a ∈ S, we call the +subset S × {a} as the a-th row of S × S. Let I have a pseudounion and let ∞ +be an element outside S × S. Let X = (S × S) ∪ {∞}. We define a topology on +X as follows. Let τ1 = P(S × S) and let τ2 be the collection of those subsets A +of X so that ∞ ∈ A and {a ∈ S : ({s ∈ S : (s, a) ∈ A} ∈ F(I))} ∈ F(I). Take +τ = τ1 ∪ τ2. Then it can be verified that τ is a topology on X. +No function from S to X can be I-convergent to ∞. If g : S −→ X is +I-convergent to ∞ then by Lemma 3.1 there is a function f : S −→ X which +is convergent to ∞. Note that each row contains at most finitely many elements +of the form f(s). Excluding these terms from each row, we obtain an open set +containing ∞ which contains no terms of the form f(s). Also no function from S +to S × S can be I-convergent to a point of S × S unless it is eventually constant. +But ∞ is a limit point of S ×S. Hence S ×S is I-functional closed but not closed +and therefore X is not I-functional. +However there exists an ideal for which every sequential space is I-functional. +Proposition 3.3. Let S = � +i∈N Si such that Si ∩ Sj = ∅ for different i, j and +let I0 = {A ⊂ S : A ∩ Si ̸= ∅ for finitely many i}. Then every sequential space +X is an I0-functional space. +Proof. Let O ⊂ X be I-functional open. If O is not open then there is a sequence +xn ∈ X \ O converging to x ∈ O. Define a function g : S −→ X \ O by g(s) = xi +if s ∈ Si. Then g is I-convergent to x ∈ O. Hence X \ O is not I-functional +closed, which is a contradiction. +□ +Proposition 3.4. The following are equivalent for any A ⊂ X. +(i) A ⊂ X is I-functional open +(ii) For any function g : S −→ X which is I-convergent to x ∈ A, we have +{s ∈ S : g(s) ∈ A} ∈ I+. +(iii) |{s ∈ S : g(s) ∈ A}| ≥ ω for each function g : S −→ X which is +I-convergent to x ∈ A. +Proof. (i) =⇒ (ii) Let A ⊂ X be I-functional open and let g : S −→ X be +I-convergent to x ∈ A. If possible let C = {s ∈ S : g(s) ∈ A} ∈ I. Fix +an element a ∈ X \ A. Define a function h : S −→ X \ A by h(s) = g(s) +for s ∈ S \ C and h(s) = a if s ∈ C. Let U be a neighbourhood of x. Then + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES7 +{s ∈ S : g(s) ∈ U} ∩S \ C ⊂ {s ∈ S : h(s) ∈ U} ∈ F(I). Thus h : S −→ X \ A is +I-convergent to x, this contradicts that A is I-functional open. Thus (ii) holds. +(ii) =⇒ (iii) As I is an admissible ideal, thus (iii) holds. +(iii) =⇒ (i) If possible let A ⊂ X be not I-functional open. So X \ A is not +I-functional closed. Therefore there is a function g : S −→ X \ A which is +I-convergent to x ∈ A and evidently {s ∈ S : g(s) ∈ A} = ∅ which contradicts +(iii). +□ +Example 3.4. Let I be a maximal ideal of S and let g : S −→ X be I-convergent +to x. Let Y = {g(s) : s ∈ S} ∪ {x}. Endow {g(s) : s ∈ S} ⊂ Y with the discrete +topology and let a basic neighbourhood of x be of the form {x}∪{g(s) : s ∈ A} for +some A ∈ F(I). Y endowed with this topology is an I-functional space. To prove +that, let U be I-functional open in Y. Without any loss of generality assume that +x ∈ U. As I is maximal, by Proposition 3.4(ii), we have {s ∈ S : g(s) ∈ U} ∈ +F(I). Hence {x} ∪ {g(s) ∈ U} ⊂ U which implies that U is open in Y. +Lemma 3.3. Let X = +� +i∈Λ +Xi have the product topology. Then a function f : +S −→ X is I-convergent to x = (xi) if and only if πi ◦ f is I-convergent to xi +for each i ∈ Λ. +Proof. Let πi ◦ f be I-convergent to xi for each i ∈ Λ. Let O = +� +i∈Λ +Oi be a basic +open set in X containing x. Let Oi = Ui for i = m1, m2, · · · , mk and Oi = Xi +otherwise. Then {s ∈ S : (πi ◦ f)(s) ∈ Ui} ∈ F(I) for each i = m1, m2, · · · , mk. +Now +� +i∈{m1,m2,··· ,mk} +{s ∈ S : (πi ◦ f)(s) ∈ Ui} ∈ F(I). Consequently the result +follows. Clearly the converse holds. +□ +Proposition 3.5. Let X = +� +i∈Λ +Xi have the product topology and let O be +I-functional open in X. Then πi(O) is I-functional open in Xi for each i ∈ Λ. +Proof. If possible let πi(O) be not I-functional open in Xi. Then there exists a +function g : S −→ Xi which is I-convergent to x ∈ πi(O) and {s ∈ S : g(s) ∈ +πi(O)} ∈ I. Now fix some aj ∈ πj(O) for j ̸= i. Define a function h : S −→ X +by +(πj ◦ h)(s) = +� +aj +if +j ̸= i +g(s) +if +j = i. +Let y = (yi) be defined as follows. +yj = +� +aj +if +j ̸= i, +x +if +j = i. + +8 +P. DAS, U. SAMANTA, S. LIN +Then h : S −→ X is I-convergent to y (by Lemma 3.3). Also {s ∈ S : g(s) ∈ +πi(O)} = {s ∈ S : h(s) ∈ O} ∈ I, contradicts that O is I-functional open. +□ +We now state certain basic results regarding I-functional spaces without +proofs. +(i) Let I ⊂ J be two ideals of S and let X be a space. +If U ⊂ X is +J -functional open then it is I-functional open. +(ii) Let I ⊂ J be two ideals of S. If X is I-functional then it is J -functional. +(iii) Suppose that {Iα : α ∈ A} is a collection of ideals of S. If X is a space +and U ⊂ X is Iα-functional open for some α ∈ A, then U is +� +α∈A +Iα-functional +open. +Lemma 3.4. Let I be a maximal ideal of S. If U, V are two I-functional open +subsets of X then U ∩ V is also I-functional open. +Proof. Let g : S −→ X be I-convergent to x ∈ U ∩ V. So, {s ∈ S : g(s) ∈ U} ∈ +F(I) and {s ∈ S : g(s) ∈ V } ∈ F(I) (by Proposition 3.4). Now {s ∈ S : g(s) ∈ +U} ∩ {s ∈ S : g(s) ∈ V } = {s ∈ S : g(s) ∈ U ∩ V } ∈ F(I) and therefore U ∩ V +is I-functional open. +□ +The I-functional coreflection of a space X is the set X endowed with the +topology generated by I-functional open subsets of X as a subbase and the +topology is denoted by I-fX. Clearly for a space X, I-fX is finer than the +topology of X. Further If I is a maximal ideal of S, then the collection of all +I-functional open sets itself forms a topology on X. +Definition 3.3. Let I be an ideal of S and A ⊂ X. A function f : S −→ X +is said to be I-eventually in A if there is a E ∈ I such that f(s) ∈ A for all +s ∈ S \ E. +Proposition 3.6. Let I be a maximal ideal of S. Then A ⊂ X is I-functional +open if and only if for each function which is I-convergent to a point of A, it is +I-eventually in A. +Proof. The result follows from Proposition 3.4. +□ +Theorem 3.1. Every I-functional space is hereditary with respect to I-functional +open (I-functional closed) subspaces. +Proof. Let X be an I-functional space. Suppose that Y is an I-functional open +set in X. Then Y is open in X. Let U(⫋ Y ) be I-functional open in Y. We +have to show that U is I-functional open in X. Suppose that g : S −→ X is + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES9 +I-convergent to x ∈ U ⊂ Y. Since Y is open, {s ∈ S : g(s) ∈ Y } ∈ F(I). +Let y ∈ Y \ U. Define a function h : S −→ Y by h(s) = g(s) if g(s) ∈ Y +and h(s) = y if g(s) ̸∈ Y. Therefore, h : S −→ X is I-convergent to x. Since +|{s ∈ S : g(s) ̸∈ U}| = |{s ∈ S : h(s) ̸∈ U}|, by Proposition 3.4, it follows that +U is I-functional open in X. As X is I-functional space, so U is open in X and +so open in Y. +Let Y be an I-functional closed subset of X. Then Y is closed in X. Let +F(⫋ Y ) be an I-functional closed subset of Y. We have to show that F is +I-functional closed subset of X. Suppose that g : S −→ F is I-convergent to +x. So x ∈ Y as Y is closed. Therefore x ∈ F since F is an I-functional closed +subset of Y. Thus F is an I-functional closed subset of X, so F is a closed subset +of X and hence a closed subset of Y. +□ +Theorem 3.2. I-functional spaces are preserved by topological sums. +Theorem 3.3. Any quotient space of an I-functional space is an I-functional +space. +Proof. Let X be an I-functional space and let f : X −→ Y be a quotient +mapping. Let F ⊂ Y be I-functional closed. If F is not closed, f −1(F) is not +closed (as f is a quotient mapping) and so f −1(F) is not I-functional closed. +Then there exists a function g : S −→ f −1(F) which is I-convergent to x ̸∈ +f −1(F). Since F is I-functional closed and f is continuous, we obtain that f ◦g : +S −→ F is I-convergent to f(x) ∈ F. This contradicts that x ̸∈ f −1(F). +□ +Theorem 3.4. Every I-functional space is a quotient of some metric space +provided I = I0, the ideal defined in Proposition 3.3. +Proof. Let X be an I-functional space and let tn = +1 +n + 1, n ∈ N. Define a +function f : S −→ R by f(s) = tn if s ∈ Sn. Then f is I-convergent to 0. +Take Y = { +1 +n + 1 : n ∈ N} ∪ {0}. The topology of Y is induced from the usual +metric topology of R. Clearly O ⊂ Y is open if and only if either 0 ̸∈ O or if +0 ∈ O then f(s) ∈ O if s ∈ A for some A ∈ F(I). Let S = {g : S −→ X : g +is I-convergent to some g0 ∈ g(S)}. Writing {(g(s) : s ∈ S)} = Z, let d be a +metric on Z × Y = {(Z, y) : y ∈ Y } defined by d(Z, a), (Z, b)) = |a − b|. +Now consider the topological sum +L = +� +Z∈S +Z × Y . +We observe that A ⊂ L is open if and only if {y ∈ Y : (Z, y) ∈ A} is open in Y. +Consider the mapping Φ : L −→ X defined by Φ(Z, 0) = g0 and Φ(Z, f(s)) = +g(s). Clearly Φ is onto. Now we show that Φ is a quotient mapping. + +10 +P. DAS, U. SAMANTA, S. LIN +Let U ⊂ X be open. +Then for every g : S −→ X, I-convergent to a ∈ +U, {s ∈ S : g(s) ∈ U} ∈ F(I). If (Z, 0) ∈ Φ−1(U) then g0 ∈ U, also {s ∈ +S : g(s) ∈ U} ∈ F(I). Write E = {s ∈ S : g(s) ∈ U}. By the definition of +Φ, Φ(Z, f(s)) = g(s) ∈ U for each s ∈ E. Therefore, Φ−1(U) is open in L. +Again if U is not open in X, then there exists a function g : S −→ (X \ U) +which is I-convergent to g0 ∈ U. Consequently {y ∈ Y : (Z, y) ∈ Φ−1(U)} = {0}, +which is not open in Y. Hence Φ−1(U) is not open in L. +□ +4. I-functional continuity +In this section our main object of investigation is the notion of I-functional +continuity. +Recall that a mapping f from a space X to another space Y is +called sequentially continuous [2] provided for any sequentially open set U in Y, +f −1(U) is sequentially open in X. It is proved in [2] that a mapping f : X −→ Y +is sequentially continuous if and only if f preserves the convergence of sequences, +i.e., for each sequence (xn : n ∈ N) in X converging to x, the sequence (f(xn) : +n ∈ N) converges to f(x). In [36], authors introduced the notion of I-continuity +in terms of I-open sets. Extending this notion in the language of functions, we +introduce following definitions. +Definition 4.1. Let I be an ideal of S and f : X −→ Y be a mapping. Then +(i) f is called an I-functional convergence preserving mapping provided for a +function g : S −→ X, I-convergent to x, f ◦ g is I-convergent to f(x). +(ii) f is called I-functional continuous provided for any I-functional open set U +in Y , f −1(U) is I-functional open in X. +We call f simply functional continuous if we take functional open set instead +of I-functional open set in Definition 4.1. +Lemma 4.1. Let Y ⊂ X and let U be I-functional open in X. Then U ∩ Y is +I-functional open in Y. +Proof. Let g : S −→ Y be I-convergent to y ∈ U ∩Y. Then {s ∈ S : g(s) ∈ U} ∈ +I+ (by Proposition 3.4) and therefore {s ∈ S : g(s) ∈ U ∩ Y } ∈ I+. +□ +Lemma 4.2. Let I be a maximal ideal of S and let U ⊂ Y ⊂ X. Suppose +that U is I-functional open in Y and Y is I-functional open in X. Then U is +I-functional open in X. +Proof. If U is not I-functional open in X then there exits a mapping f : S −→ +X, I-converging to some a ∈ U and f −1(U) ∈ I. As Y is I-functional open in X, +and I is maximal, f −1(Y ) ∈ F(I). Therefore f −1(Y \ U) = f −1(Y ) \ f −1(U) ∈ + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +11 +F(I). Let F = f −1(Y \ U). Define a mapping φ : S −→ Y by +φ(s) = +� +f(s) +if +s ∈ F +a +if +s ̸∈ F. +Evidently φ is I-convergent to a. Also φ−1(U) ∈ I, contradicts that U is +I-functional open in Y. +□ +Theorem 4.1. Let X be a space and let U be a cover of X by I-functional +open sets. Then a mapping f : X −→ Y is I-functional continuous if and only +if for each U ∈ U the restriction f|U is I-functional continuous provided I is +maximal. +Proof. Let f : X −→ Y be I-functional continuous and let U ∈ U. Suppose that +V ⊂ Y is I-functional open. Then (f|U)−1(V ) = f −1(V ) ∩ U is I-functional +open in U. Conversely let the condition hold. +Then by Lemma 4.2 for any +I-functional open set V ⊂ Y, f −1(V ) ∩ U is I-functional open in X. As X = +� U, f −1(V ) = +� +U∈U +(f|U)−1(V ) and is I-functional open as each is so. +□ +In [36, Theorem 4.2], it was shown that every continuous mapping preserves +I-convergence of sequences and if a mapping preserves I-convergence of se- +quences then the mapping is I-continuous. Here also, similar kind of results +hold. +Proposition 4.1. Let X, Y be two spaces and f : X −→ Y be a mapping. +(i) If f is continuous then f preserves I-functional convergence. +(ii) If f preserves I-functional convergence then f is I-functional continu- +ous. +The examples given below, show that the converses of preceding Proposition +are not generally true. +Example 4.1. Let I be a maximal ideal. Take X = � +S(I) as in Example 3.1 +and let Y = X be endowed with the discrete topology. Let f : X −→ Y be the +identity mapping. +Clearly i : S −→ X, the inclusion function is not convergent to ∞. Let S′ ⊂ S +and i|S′. If S′ ∈ I then i can not converge to ∞. +Otherwise take an infinite S′′ ⊂ S′ satisfying |S′ \ S′′| = |S′|. If S′′ ∈ I then +(S \ S′′) ∪ {∞} is a neighbourhood of ∞. So i|S′ again can not be convergent to +∞. +Finally if S′′ ∈ F(I) then S′′ ∪ {∞} is a neighbourhood of ∞ but it does not +contain all but finitely many terms of i|S′. So i|S′ is not convergent to ∞. +Therefore there is no convergent function from S to X except for eventual +constant mappings. So f preserves Ifin-functional convergence trivially. Thus + +12 +P. DAS, U. SAMANTA, S. LIN +by Proposition 4.1, f is also functional continuous. But evidently f is not con- +tinuous. +Example 4.2. Let X = � +S(I) and let Y = {1, 0} be endowed with discrete +topology. Also let I be a non-maximal ideal of S. Then there is A ⊂ S for which +both A ∈ I+ and S \ A ∈ I+. Define a mapping g : X −→ Y by g(x) = 1 if +x ∈ A and g(x) = 0 otherwise. As (S \ A) ∪ {∞} is I-functional open, so g is +I-functional continuous. But g does not preserve I-functional convergence. +Let f : X −→ Y be an I-functional continuous mapping and let g : S −→ X +be I-convergent to x. If V ⊂ Y is an I-functional open set containing f(x) then +by Proposition 3.4, {s ∈ S : g(s) ∈ f −1(V )} ∈ I+ and thus {s ∈ S : (f ◦ g)(s) ∈ +V } ∈ I+. This observation leads to the following result immediately. +Theorem 4.2. Let I be a maximal ideal of S. Then a mapping f : X −→ Y is +I-functional continuous if and only if it preserves I-functional convergence. +For the next result we recall the following definition. I is called a P-ideal if +for any (An)n∈ω from F(I) there is A ∈ F such that A \ An is finite for each +n [29]. +Theorem 4.3. Let I be a P-ideal and X be a first-countable space. +Then +f : X −→ Y is I-functional continuous if and only if it preserves I-functional +convergence. +Proof. From [29], it follows that I-convergence implies I∗-convergence of func- +tions, as I is a P-ideal. Let f : X −→ Y be I-functional continuous and let +g : S −→ X be I-convergent to x. Then there is A ∈ I such that g ⇃S\A−→ X +is convergent to x. Let U be an open neighbourhood of f(x). Then f −1(U) is +I-functional open in X, and so is a functional open set containing x. Conse- +quently g(s) ∈ f −1(U) for all s ∈ S \(A∪F) (for a suitable finite subset F of S) +and hence (f ◦ g)(s) ∈ U for all s ∈ S \ (A ∪ F). By admissibility of I it follows +that {s ∈ S : (f ◦ g)(s) ∈ U} ∈ F(I). +The converse result follows directly from Proposition 4.1. +□ +Next we investigate the interrelationships between the notions of continuity +and I-f-continuity. +Theorem 4.4. Let f be a mapping from an I-functional space X to another +space Y. Then f is continuous if and only if f is I-functional continuous. +Proof. Let f : X −→ Y be continuous. Then by Proposition 4.1, f is I-functional +continuous. +Since every open set is I-functional open and X is an I-functional space the +converse follows immediately. +□ + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +13 +Corollary 4.1. Let f be a mapping from a functional space X to another space +Y. Then following are equivalent. +(1) f is continuous. +(2) f preserves I-functional convergence. +(3) f is I-functional continuous. +(4) f is functional continuous. +Proof. (1) =⇒ (2) follows from the Proposition 4.1(i). (2) =⇒ (3) follows di- +rectly from Proposition 4.1(ii). As each functional space is I-functional space +and continuity implies functional continuity, preceding theorem establishes that +(3) =⇒ (4). Finally (4) =⇒ (1) since X is a functional space. +□ +Theorem 4.5. Let X be a sequential space and I be as defined in Proposition +3.3. Then for a mapping f : X −→ Y, f is continuous if and only if f preserves +I-functional convergence. +Proof. Let V ⊂ Y be an open set. If f −1(V ) is not open then f −1(V ) is not +I-functional open (by Proposition 3.3). Then there is a function g : S −→ X +which is I-convergent to x ∈ f −1(V ) such that {s ∈ S : g(s) ∈ f −1(V )} ∈ I. So +f ◦ g : S −→ Y is I-convergent to f(x). But {s ∈ S : g(s) ∈ f −1(V )} ∈ I =⇒ +{s ∈ S : (f ◦ g)(s) ∈ V } ∈ I, which contradicts that V is I-functional open. +Converse is obvious. +□ +Proposition 4.2. If f : X −→ Y preserves J -functional convergence for each +J ∈ Θ(I) then f preserves I-functional convergence. +Corollary 4.2. If f : X −→ Y is J -functional continuous for each J ∈ Θ(I) +then f is I-functional continuous. +Proof. The result follows from Proposition 4.1 and 4.2. +□ +Example 4.3. There exists a mapping which preserves Ifin-functional conver- +gence but is not J -functional continuous for J ∈ Θ(Ifin). Let X = � +S(J ) +and let Y = S, endowed with discrete topology. Define a mapping f : X −→ Y +by f(s) = s if s ∈ S and f(∞) = a for some particular a ∈ S. There is no +function from S to X which is convergent. Hence f preserves Ifin-functional +convergence but is not J -functional continuous since S \ {a} is J -functional +closed and f −1(S \ {a}) = S is not J -functional closed in X. +Lemma 4.3. Let X = +� +i∈Λ +Xi have the product topology. Then πi : X −→ Xi is +I-functional continuous for each i ∈ Λ. +Proof. Follows from Lemma 3.3. +□ + +14 +P. DAS, U. SAMANTA, S. LIN +Proposition 4.3. Let X = +� +i∈Λ +Xi have the product topology and let Y be a +space. Then a mapping f : Y −→ X is I-functional continuous if and only if +πi ◦ f is so for each i ∈ Λ provided I is maximal. +Proof. Let πi ◦ f be I-functional continuous for each i ∈ Λ. Let g : S −→ Y +be I-convergent to y. Then πi ◦ f ◦ g : S −→ Xi is I-convergent to (πi ◦ f)(y) +for each i ∈ Λ (by Theorem 4.2). Using Lemma 3.3, it follows that f ◦ g is +I-convergent to f(y). Therefore f is I-functional continuous (by Theorem 4.2). +Conversely let f be I-functional continuous. Let Uα ⊂ Xi be I-functional +open in Xi for some i ∈ Λ. Now (πi ◦ f)−1(Uα) = f −1(π−1 +i +(Uα)) where π−1 +i +(Uα) +is I-functional open in X (by Lemma 4.3). +Consequently f −1(π−1 +i +(Uα)) is +I-functional open in Y as f is I-functional continuous +□ +5. I-functional quotient and I-functional covering mappings +In the literature (see the papers [2,26–28]), the notions of quotient, sequen- +tially quotient and sequence covering mappings play an important role in study- +ing sequential spaces. These notions have been extended using ideal convergence +of sequences to I-quotient and I-covering mappings in [36]. In this section we +intend to further extend these concepts by defining them in terms of functions +over an arbitrary set S. +Let X, Y be two spaces. Recall that an onto mapping f : X −→ Y is said to +be a quotient mapping provided U is open in Y if and only if f −1(U) is open in +X; f is said to be sequentially quotient [2] provided U is sequentially open in Y +if and only if f −1(U) is sequentially open in X; f is said to be I-quotient [36] +provided U is I-open in Y if f −1(U) is I-open in X; f is said to be sequence +covering [2] if whenever (yn : n ∈ N) is a sequence in Y converging to y ∈ Y, +there exists a sequence (xn : n ∈ N) in X satisfying xn ∈ f −1(yn) for all n ∈ N +and x ∈ f −1(y) such that xn converges to x; f is said to be I-covering [36] if +whenever (yn : n ∈ N) is a sequence in Y, I-converging to y ∈ Y, there exists a +sequence (xn : n ∈ N) in X satisfying xn ∈ f −1(yn) for all n ∈ N and x ∈ f −1(y) +such that (xn : n ∈ N) is I-convergent to x. +Our next definitions are introduced following this line. +Definition 5.1. Let f be a mapping from a space X onto another space Y. +(i) f is said to be I-functional quotient provided U is I-functional open in Y if +and only if f −1(U) is I-functional open in X. +(ii) f is said to be I-functional covering if for any g : S −→ Y , I-converging +to y ∈ Y, there exists a function h : S −→ X satisfying (f ◦ h)(s) = g(s) for all +s ∈ S and x ∈ f −1(y) such that h is I-convergent to x. + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +15 +We call f simply functional quotient if we take functional open set instead of +I-functional open set in Definition 5.1. +Theorem 5.1. Let X be a space and Y be a non-empty set. Further let f : +X −→ Y be an onto mapping and I be a maximal ideal of S. There exists a +strongest topology on Y w.r.t which f is I-functional continuous. +Proof. Let J = {V ⊂ Y : f −1(V ) is I-functional open in X}. Then J is a +topology on Y w.r.t which f is I-functional continuous. Next let J ′ be any +other topology on Y w.r.t which f is I-functional continuous. Then for every +J ′-functional open set V ⊂ Y, f −1(V ) is I-functional open in X. So for each +V ∈ J ′, V ∈ J and hence J ′ ⊂ J as required. +□ +In the above Theorem f is an I-functional quotient mapping. +Definition 5.2. A mapping f : X −→ Y is said to be I-functional open provided +f(U) is I-functional open in Y whenever U is I-functional open in X. +Proposition 5.1. Every I-functional continuous, I-functional open onto map- +ping is I-functional quotient. +Proposition 5.2. Let f : X −→ Y and g : Y −→ Z be two mappings. Then +the following results hold. +(i) If f and g are I-functional quotient mappings then g ◦ f is an I-functional +quotient mapping. +(ii) If f and g ◦ f are I-functional quotient mappings then g is an I-functional +quotient mapping. +Proof. (i) If g, f are I-functional continuous then g ◦ f is also I-functional con- +tinuous. Again for any V ⊂ Z, (g ◦ f)−1(V ) = (f −1(g−1(V ))) and therefore +g ◦ f is an I-functional quotient mapping. +(ii) Let V ⊂ Z such that g−1(V ) is I-functional open in Y. So (f −1(g−1(V ))) +is I-functional open in X as f is an I-functional quotient mapping. Now (g ◦ +f)−1(V ) = (f −1(g−1(V ))) and g ◦ f being I-functional quotient, together imply +that V is I-functional open in Z. Next let V be I-functional open in Z. Then as +g ◦ f is I-functional continuous and f is I-functional quotient, we have g−1(V ) +is I-functional open in Y. +□ +Proposition 5.3. I-functional quotient mappings are preserved by finite prod- +ucts provided I is a maximal ideal of S. +Proof. Let fi : Xi −→ Yi be an I-functional quotient mapping for i = 1, 2, · · · , N. +We define a mapping f : +N +� +i=1 +Xi −→ +N +� +i=1 +Yi by +f(x1, x2, · · · , xN) = (f1(x1), f2(x2), · · · , fN(xN)). + +16 +P. DAS, U. SAMANTA, S. LIN +By Proposition 4.3, f is I-functional continuous. It is also onto. +Next let U ⊂ +N +� +i=1 +Yi be such that f −1(U) is I-functional open. Then there +exists a function g : S −→ +N +� +i=1 +Yi, I-convergent to y ∈ U and {s ∈ S : g(s) ∈ +U} ∈ I. Consequently πi◦g : S −→ Yi is I-convergent to πi(y) for i = 1, 2, · · · , N +(by Lemma 3.3). If {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ F(I) for each i = 1, 2, · · · , N +then {s ∈ S : g(s) ∈ U} = +N +� +i=1 +{s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ F(I), which is a +contradiction. So there exists i such that {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ̸∈ F(I). +Maximality of I implies that {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ I. Consequently +πi(U) is not I-functional open in Yi. Thus f −1 +i +(πi(U)) is not I-functional open as +fi is I-functional quotient. But (πi ◦ f −1)(U) = f −1 +i +(πi(U)), which contradicts +the fact that f −1(U) is I-functional open (by Proposition 3.5). +□ +The interrelationships results among I-quotient, quotient and I-covering map- +pings that have been studied in [36] can be further generalized as below. +Proposition 5.4. Let f be a mapping from a space X onto a space Y. +(i) If f is I-functional continuous and an I-functional covering mapping then +f is an I-functional quotient mapping. +(ii) If f is one-to-one and an I-functional quotient mapping then f is an I-functional +covering provided I is a maximal ideal. +Proof. (1) Let f : X −→ Y be an I-functional continuous and I-functional +covering mapping. Suppose U ⊂ Y is such that f −1(U) is I-functional open. If +U is not I-functional open there exists a function g : S −→ Y, I-converging to +y ∈ U for which {s ∈ S : g(s) ∈ U} ∈ I. As f is I-functional covering there is a +function h : S −→ X, I-convergent to x ∈ X such that f ◦ h = g and f(x) = y. +Also {s ∈ S : g(s) ∈ U} = {s ∈ S : (f ◦ h)(s) ∈ U} = {s ∈ S : h(s) ∈ f −1(U)} +which implies that f −1(U) is not I-functional open in X. +(ii) Let f : X −→ Y be an one-to-one and I-functional quotient mapping. +Let g : S −→ Y be I-convergent to y. As f is one-to-one and onto, for each +s ∈ S there exists an unique xs ∈ X such that f(xs) = g(s). Define a function +h : S −→ X by h(s) = xs and let f(x) = y. If h is not I-convergent to x, there +exists an open set O containing x such that {s ∈ S : h(s) ∈ O} ̸∈ F(I). Since +I is maximal, {s ∈ S : h(s) ∈ O} ∈ I, so {s ∈ S : (f ◦ h)(s) ∈ f(O)} ∈ I (as f +is one-to-one) i.e. {s ∈ S : g(s) ∈ f(O)} ∈ I (because f ◦ h = g). Now f(O) is + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +17 +I-functional open in Y as f −1(f(O)) = O is I-functional open in X, and f is +I-functional quotient. This contradicts Proposition 3.4. +□ +The next result establishes when an I-functional quotient mapping becomes +a quotient mapping and conversely. +Theorem 5.2. Let f be a continuous mapping from an I-functional space X +to another space Y. Then f is a quotient map if and only if f is I-functional +quotient and Y is an I-functional space. +Proof. First let f : X −→ Y be a quotient mapping and let F ⊂ Y be not closed. +Then f −1(F) is not closed in X as f is a quotient mapping. So f −1(F) is not +I-functional closed since X is an I-functional space. Hence by Theorem 3.4 there +exists a function g : S −→ f −1(F) which is I-convergent to x ∈ X \ f −1(F). +Therefore, f ◦ g : S −→ F is I-convergent to f(x) ∈ Y \ F and consequently F +is not I-functional closed. Thus Y is an I-functional space. Let F ⊂ Y be such +that f −1(F) is I-functional closed. Now if F is not I-functional closed, F is not +closed. Thus f −1(F) is not closed (as f is a quotient mapping). Since X is an +I-functional space, f −1(F) is not I-functional closed. +Conversely let f be an I-functional quotient mapping and let Y be an I-functional +space. Take F ⊂ Y such that f −1(F) is closed in X and so I-functional closed. +Since f is an I-functional quotient mapping F is I-functional closed. Y is an +I-functional space, thus F is closed. This concludes that f is a quotient map- +ping. +□ +Generally an I-functional quotient mapping is not quotient and vice-versa. +The next result characterises I-functional spaces in terms of the interrelations of +I-functional quotient and quotient mappings, which can be proved in the same +way as that of [36, Theorem 5.6]. +Theorem 5.3. Let X be a space and I be a maximal ideal of S. Then X is an +I-functional space if and only if each I-functional quotient mapping onto X is +quotient. +Theorem 5.4. Let I be a maximal ideal. An onto mapping p : X −→ Y is +I-functional quotient if and only if it has the property that for any space W and +a mapping f : Y −→ W, I-functional continuity of f ◦ p implies that of f. +Proof. First let p : X −→ Y be I-functional quotient and let f be a mapping +from Y to some space W. Take f ◦ p, an I-functional continuous mapping. Let +F ⊂ W be I-functional closed. Then (f ◦p)−1(F) = p−1(f −1(F)) is I-functional +closed in X. Since p : X −→ Y is I-functional quotient, f −1(F) is I-functional +closed in X. +Conversely let the condition hold. +Consider F ⊂ Y such that p−1(F) is +I-functional closed. Let W = {0, 1}. Define a mapping f : Y −→ W by f(y) = 1 + +18 +P. DAS, U. SAMANTA, S. LIN +if y ∈ F and f(y) = 0 if y ∈ Y \ F. So (f ◦ p)(x) = 1 if x ∈ p−1(F) and +(f ◦p)(x) = 0 if x ̸∈ p−1(F). As in Theorem 5.1, the topology on W is induced by +f ◦p. As (f ◦p)−1({1}) = p−1(F) is I-functional closed in X, {1} is I-functional +closed in W. Therefore by I-functional continuity of f, F is I-functional closed +in Y. Consequently p : X −→ Y is I-functional quotient. +□ +If I has a pseudounion, then the next theorem readily follows from Proposition +3.2. +Theorem 5.5. If f is a mapping from a space X onto Y and if I has a pseu- +dounion then following results hold. +(1) f is I-functional continuous if and only if f is functional continuous. +(2) f is I-functional quotient if and only if f is functional quotient. +Theorem 5.6. Let f : X −→ Y be a mapping and let I have a pseudounion. +Then f is I-functional quotient if and only if for any function g : S −→ +Y, I-converging to p there exists a function h : S −→ X which is I-convergent +to some x ∈ X so that (f ◦ h)(S) ⊂ g(S) and f(x) = p. +Proof. Let f be a I-functional quotient mapping. Then f is functional quo- +tient (by Theorem 5.5). Let g : S −→ Y be I-convergent to p. Since I has +pseudounion, as in Lemma 3.1, we obtain a function g′ : S −→ Y which is +convergent to p. Now f −1(g′(S) \ {p}) is not functional closed, and so we get a +function k : S −→ f −1(g′(S) \ {p}), converging to a ̸∈ f −1(g′(S) \ {p}). Now +(f ◦ k) : S −→ g′(S) is convergent to f(a) = p. +Conversely let the condition hold and let F ⊂ Y so that f −1(F) is an +I-functional closed set in X. Let g : S −→ F be I-convergent to y ∈ Y. Then +there exists an infinite set S′ ⊂ S and g|S′ is convergent to y. Let Φ = g|S′ and +h : S −→ S′ be an onto mapping satisfying (Φ ◦ h)(s) = g(s) for every s ∈ S. +Then (Φ◦ h) is convergent to y. So, there is a function k : S −→ f −1((Φ◦ h)(S)) +that is I-convergent to x ∈ f −1(y). As f −1(F) is I-functional closed, x ∈ f −1(F) +so y ∈ F. +□ +6. I-functional Fr´echet-Uryshon space +Recall that a space X is Fr´echet-Uryshon [13] (resp., statistically Fr´echet- +Uryshon [10], I-Fr´echet-Uryshon [36]) if for each A ⊂ X with x ∈ cl(A) +there exists a sequence in A which is convergent (resp. statistically convergent, +I-convergent) to x. We can extend these notions to I-functional Fr´echet-Uryshon +space in the following way. +Definition 6.1. A space X is called an I-functional Fr´echet-Uryshon space +if for each A ⊂ X and each x ∈ cl(A) there exists a function f : S −→ +A, I-convergent to x. + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +19 +Proposition 6.1. Every I-functional Fr´echet-Uryshon space is an I-functional +space. +We modify [34, Example 3.1] to show that the converse of the above Propo- +sition does not hold for a maximal ideal. +Example 6.1. Let I be a maximal ideal of S and let X be a non-empty set. +For every a ∈ S, let ga be a function from S to X and take Ga = ga(S). For +fixed a ∈ S assume that ga(x) ̸= ga(y) for x ̸= y ∈ S. Take G = {xa : a ∈ S} +with xa ̸= xb for a ̸= b ∈ S. Consider a point ∞ outside �{Ga : a ∈ S} � G and +take X = �{Ga : a ∈ S} � G �{∞}. +The topology on X is defined in the following way: +(1) Each point ga(s) is isolated; +(2) For each a ∈ S, an open neighbourhood of xa is taken as a set of the form +{xa} ∪ Ma where Ma = {ga(s) : s ∈ F} for some F ∈ F(I); +(3) Each open neighbourhood of ∞ is a set of the form {∞}∪M ∪{Ma : a ∈ M} +where M = {xa : a ∈ F} for some F ∈ F(I). +First to show that X is an I-functional space, take Y ⊂ X which is I-functional +closed. Also let y ∈ cl(Y ). Now one of the three cases may occur. +Case 1: If y ∈ �{Ga : a ∈ S} then {y} is an open neighbourhood of y and +y ∈ cl(Y ) implies that y ∈ Y. +Case 2: If y ∈ G, then y = xa for some a ∈ S. If possible let y ̸∈ Y. Now if +{s ∈ S : ga(s) ∈ Y } ∈ I then U = (Ga\Y )∪{xa} is an open neighbourhood of xa. +Also U ∩Y = ∅ contradicts that y ∈ cl(Y ). We define a function h : S −→ Y ∩Ga +by h(s) = ga(s) if ga(s) ∈ Y and h(s) = ga(s0) for some fixed s0 ∈ S. Then h +is I-convergent to xa. Because Y is I-functional closed, xa = y ∈ Y, which is a +contradiction. Therefore y ∈ Y. +Case 3: Consider the case when y = ∞. If A = {s ∈ S : xs ∈ Y } ∈ I +put V = X \ �{Ga : a ∈ A}. Then V is an open neighbourhood of ∞ and +V ∩ Y = ∅, which contradicts ∞ ∈ cl(Y ). Hence as in Case 2, we obtain a +function f : S −→ Y, I-convergent to ∞. As Y is I-functional closed, ∞ ∈ Y. +To prove that X is not an I-functional Fr´echet-Uryshon space, take ∞ ∈ +cl(X \ ({xa : a ∈ S} ∪ {∞})). Let h : S −→ X \ ({xa : a ∈ S} ∪ {∞}) be +I-convergent to ∞. Define Aa = h(S) ∩ Ga, a ∈ S. If for each a ∈ S, {s ∈ S : +ga(s) ∈ h(S)} ∈ I then V = {∞} ∪ {Ga \ Aa : a ∈ S} is an open set containing +∞. But V ∩ h(S) = ∅ and this contradicts that h is I-convergent to ∞. So there +is a ∈ S such that {s ∈ S : ga(s) ∈ h(S)} ∈ F(I). As in case 2, a function +k : S −→ h(S) ∩ Ga can be constructed which is I-convergent to xa. +Next if {s ∈ S : h(s) ∈ Ga} ∈ I then k : S −→ h(S) ∩ Ga can be made +to I-converge to a point different from xa. This contradicts Lemma ??. Now if +{s ∈ S : h(s) ∈ Ga} ∈ F(I) then k can be made to be I-convergent to ∞ ̸= xa, + +20 +P. DAS, U. SAMANTA, S. LIN +which again is a contradiction as X is a Hausdorff space. +Considering functions over S instead of functions over N, we restate [36, +Theorem 6.3] as follows. +Theorem 6.1. A space X is an I-functional Fr´echet-Uryshon space if and only +if each subset of X is an I-functional space. +Proposition 6.2. Subspaces of an I-functional Fr´echet-Uryshon space are I-functional +Fr´echet-Uryshon space. +Theorem 6.2. I-functional Fr´echet-Uryshon spaces are preserved by topological +sums. +Proof. Let {Xa : a ∈ Λ} be a disjoint family of I-functional Fr´echet-Uryshon +spaces and let X = +� +a∈Λ +Xa be its topological sum. From Proposition 6.1 and +Theorem 3.2, X is an I-functional space. +For every Y ⊂ X and for every +a ∈ Λ, Y ∩Xa is an I-functional Fr´echet-Uryshon space in Xa, and consequently +is an I-functional space in Xa. Hence the topological sum +� +a∈Λ +Y ∩ Xa becomes +an I-functional space. As Y is an I-functional space, therefore by Theorem 6.1, +X is an I-functional Fr´echet-Uryshon space. +□ +Although the line of proof of the next result is analogous with that of [36, +Lemma 6.9], it has its own significance. +Theorem 6.3. Every space is a continuous and I-functional covering image of +an I-functional Fr´echet-Uryshon space provided I is a maximal ideal of S. +In general product of two I-functional Fr´echet-Uryshon spaces may not be an +I-functional Fr´echet-Uryshon space. This follows from a modification of an Ex- +ample from [32]. Actually product of two I-functional Fr´echet-Uryshon spaces +need not be an I-functional space either. +A mapping f : X −→ Y is called pseudo-open [1] if for each y ∈ Y and an +open subset U in X with f −1(y) ⊂ U, f(U) is a neighbourhood of y in Y. +Theorem 6.4. Let f be a pseudo-open mapping from an I-functional Fr´echet- +Uryshon space X onto a space Y. If f preserves I-functional convergence then +Y is an I-functional Fr´echet-Uryshon space. +Proof. Let f preserve I-functional convergence. Let A ⊂ Y and choose y ∈ +cl(A). If f −1({y}) ∩ (cl(f −1(A))) = ∅ then f −1(y) ∈ X \ (cl(f −1(A))). Since +f is pseudo-open, y ∈ int(f(X \ cl(f −1(A))) = int(f(int(X \ f −1(A)))) ⊂ +int(f(X \ f −1(A))) = int(Y \ A) = Y \ cl(A). So, y ∈ Y \ cl(A), a contradiction. + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +21 +Therefore, there exists x ∈ f −1({y}) ∩ (cl(f −1(A))). Since X is an I-functional +Fr´echet-Uryshon space, there is a function g : S −→ f −1(A), I-convergent to x. +Hence f ◦ g : S −→ A is I-convergent to f(x) = y, which consequently implies +that Y is an I-functional Fr´echet-Uryshon space. +□ +Corollary 6.1. I-functional Fr´echet-Uryshon spaces are preserved by continu- +ous pseudo-open mappings. +Theorem 6.5. A space Y is an I-functional Fr´echet-Uryshon space if every +mapping onto Y that preserves I-functional convergence is pseudo-open provided +I is a maximal ideal of S. +Proof. Let Y be a space and let g : S −→ Y be I-convergent to yg. Consider +Sg = g(S)∪{yg} and S is the family of all functions on S which are I-convergent +in Y. A topology on Sg is defined as in Example 3.4 and is denoted by SI +g . Clearly +g is I-convergent to yg in SI +g . In view of Example 3.4, Theorem 3.1 and Theorem +6.1, one can conclude that SI +g is an I-functional Fr´echet-Uryshon space since +every subset of Sg is open or closed in SI +g . Let Z = +� +g∈S +SI +g be the topological +sum of {SI +g }. By Theorem 6.2, Z is an I-functional Fr´echet-Uryshon space. We +define a mapping f : Z −→ Y such that f|Sg : SI +g −→ (Sg, τSg) is the identity +mapping. Then f preserves I-functional convergence and therefore by Theorem +6.4 Y is an I-functional Fr´echet-Uryshon space. +□ +But suitably modifying [36, Theorem 6.7], we can obtain the next theorem. +Theorem 6.6. Let Y be an I-functional Fr´echet-Uryshon space. Then every +I-functional covering mapping from a space onto Y is pseudo-open. +We end the section with the following interesting observation. +Theorem 6.7. A space X is an I-functional Fr´echet-Uryshon space if and +only if every continuous I-functional covering mapping onto X is pseudo-open +provided I is a maximal ideal of S. +Proof. This follows from Theorem 6.6, Theorem 6.3 and corollary 6.1. +□ +References +[1] A.V. Arhangel’skiˇi, Mappings and spaces, Russian Math. Surveys, 21 (1966), 115-162. +[2] J.R. 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Yan, Sequence-covering maps of metric spaces, Topology Appl., 109 (2001), +301-314. +[27] S. Lin, A note on sequence-covering mappings, Acta Math. Hungar., 107 (3) (2005), +187-191. +[28] S. Lin and Z.Q. Yun, Generalized Metric Spaces and Mappings, Atlantis Studies in Math- +ematics, vol. 6, Atlantis Press, Paris (2016). +[29] M. Macˆaj, M. Sleziak, IK-convergence, Real Anal. Exchange, 36 (1) (2010-2011), 177–194. +[30] A.E. Michael, A quintuple quotient quest, Gen. Topol. Appl., 2 (1972), 91-138. +[31] S.K. Pal, I-sequential topological spaces, Appl. Math. E Not. 14 (2014), 236-241. +[32] V. Renukadevi and B. Prakash, I-Fr´echet-Uryshon spaces, Math. Morav., 20 (2) (2016), +87-97. +[33] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., +2 (1951), 73-74. +[34] Z. Tang and F. Lin, Statistical versions of sequential and Fr´echet-Urysohn spaces, Adv. +Math. (China), 44 (2015), 945-954. +[35] X.G. Zhou and M. Zhang, More about the kernel convergence and the ideal convergence, +Acta Math. Sin. Engl. Ser. 29 (2013), 2367-2372. + +ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES +23 +[36] X. Zhou, L. Liu and S. Lin, On topological spaces defined by I-convergence, Bull. Iranian +Math. Soc., (Accepted), July 2019. +* Department of Mathematics, Jadavpur University, Kolkata-32, West Bengal, In- +dia +Email address: pratulananda@yahoo.co.in, samantaupasana@gmail.com +† Department of Mathematics, Ningde Normal University, Ningde, 352100, Fujian, +People’s Republic of China +Email address: shoulin60@163.com + diff --git a/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/load_file.txt b/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0caa959525371f6bb31bf8a8a69817c3e459f135 --- /dev/null +++ b/oNAyT4oBgHgl3EQfy_n2/content/tmp_files/load_file.txt @@ -0,0 +1,1046 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf,len=1045 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='00696v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='GN] 2 Jan 2023 ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES PRATULANANDA DAS∗, UPASANA SAMANTA∗, SHOU LIN† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In order to do that, we primarily follow in the foot- steps of the earlier studies of ideal convergence done by using functions (from an infinite set S to X) in [8, 9, 29], as that is the most general per- spective and use functions instead of sequences/nets/double sequences etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This functional approach automatically provides the most general settings for such studies and consequently extends and unifies the proofs of sev- eral old and recent results in the literature about spaces like sequential, Fr´echet-Uryshon spaces and sequential, quotient and covering maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In particular, we introduce and investigate the notions of I-functional spaces, I-functional continuous, quotient and covering mappings and finally I- functional Fr´echet-Uryshon spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In doing so, we take help of certain set theoretic and other properties of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Key words and phrases: Ideal, ideal convergence of functions, I-functional space, I-functional continuous, quotient and covering mappings, I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The idea of statistical convergence of sequence was introduced in [12,33] as an extension of the usual notion of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Apart from a lot of investigations in the fields of summability theory, measure theory, functional analysis etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=', this idea has led to various investigations in the settings of topological spaces (for ex- ample see [4,5,10,24,25,34–36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The most important generalization of almost all types of convergence including statistical convergence had been proposed by Kostyrko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' [20] who had introduced the concepts of I-convergence and I∗-convergence in metric spaces using ideals of the set of all natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Following the line of Kostyrko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=', the same has been studied for sequences 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Primary: 54A20, 54B15, 54C08 Secondary: 40A05, 26A03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The first author is thankful to NBHM for granting the project (sanction no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 02011/9/2022/NBHM(RP)/RD II/10378) during the tenure of which this work was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN in general topological spaces [22], for nets in topological and uniform spaces [23] (subsequently studied in [6, 7]) and for functions in topological spaces [9, 29], uniform spaces [8] for example, where other references can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' On the other hand, it is a well known fact that the topology of a topological space, in general, can not be determined by convergent sequences, unlike metric spaces where sequences play a much more important role in characterizing several notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' From the beginning, it has been a very rich and challenging topic of investigations as to, in which topological spaces sequences play a better role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The first countable, Fr´echet-Uryshon and sequential spaces are examples of some such spaces that are determined by convergence of sequences [11, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Instead of usual convergence of sequences, first in [32,34] the authors have worked with statistical convergence to define statistical counterparts of Fr´echet-Uryshon and sequential spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Subsequently the more general idea of ideal convergence of sequences has been widely used to introduce these notions as also several other new ideas in topological settings (for example one can see [3,31,32,35,36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In particular in [36], Zhou and his co-authors defined I-continuous, I-quotient and I-covering mappings and checked how they interact with I-sequential, I-Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As a natural consequence, in this paper, we further generalize the whole set- ting of such investigations by considering ideals of an arbitrary infinite set S, and as a natural replacement, instead of sequences in X we take functions from S to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This approach unifies the two directions mentioned above and provides the most general type of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Primarily we use the idea of I-convergence of func- tions to introduce I-functional open sets, I-functional closed sets, I-functional spaces and I-functional Fr´echet-Uryshon spaces and establish several proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We also proceed in the same way to extend the ideas of I-continuous, I-quotient and I-covering mappings and subsequently investigate their coun- terparts, namely, I-functional continuous, quotient and covering mappings and their effects on I-functional spaces, I-functional Fr´echet-Uryshon spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In or- der to clear ambiguity and for the sake of continuity, we call all mappings with domain S “functions” (continuing the nomenclature of [8,9,29]) and mappings from one topological space to another as just “mappings”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As a consequence, not only the results of [3,31,32,35,36] become special cases of our results, also the whole treatment seems much more simplified, at the same time underscoring the focal point that, several topological concepts can actually be studied without restricting the domain set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Preliminaries Let N denote the set of all natural numbers and let K ⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Recall that the nat- ural or asymptotic density of K is defined by d(K) = lim n−→∞ 1 n|{k ∈ N : k ≤ n}| if the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If X is a topological space then a sequence (xn : n ∈ N) in X is statistically convergent to x ∈ X if for each neighbourhood U of x in X, d({n ∈ N : xn ̸∈ U}) = 0 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The notion of statistical convergence has subsequently been extended to the notion of I-convergence, which is based on the notion of ideal of subsets of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y be a non-empty set and let P(Y ) be the family of all subsets of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A family I(⊂ P(Y )) of subsets of a non-empty set Y is said to be an ideal of Y if (i) A, B ∈ I imply A ∪ B ∈ I (ii) A ∈ I, B ⊂ A imply B ∈ I, while an admissible ideal I of Y covers Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Such ideals are also called free ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If I is a proper non-trivial ideal of Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Y /∈ I, I ̸= ∅), then the family of sets F(I) = {M ⊂ Y : Y \\ M ∈ I} is a filter (called the dual filter) of Y whereas the coideal of I is I+ = {A ⊂ Y : A ̸∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We denote the ideal consisting of all finite subsets and density zero subsets of N by Ifin and Id respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If I is a maximal ideal then for any A ⊂ S, we have either A ∈ I or S \\A ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For each ideal I of S, the set of all maximal ideals J of S such that I ⊂ J is denoted by Θ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' It is known that I = � J ∈Θ(I) J [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Recall that B ⊂ N is said to be a pseudounion of a family A ⊂ P(N) if N \\ B is infinite and A \\ B is finite for each A ∈ A [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A sequence (xn : n ∈ N) in a topological space X is said to be I-convergent to x ∈ X provided for each neighbourhood U of x, the set {n ∈ N : xn ̸∈ U} belongs to I [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-convergence of sequence coincides with ordinary convergence of sequence if we take I = Ifin and with the statistical convergence if I = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The concept of I∗-convergence of real sequence arises from a result of statis- tical convergence that: a real sequence (xn : n ∈ N) is statistically convergent to x if and only if there exists a set M = {mk : k ∈ N} with m1 < m2 < · · · mk · · · such that d(M) = 1 and lim k−→∞ xmk = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This idea has been extended to I∗-convergence of a sequence in a topological space as a sequence (xn : n ∈ N) in X is I∗-convergent to x ∈ X if and only if there exists a set M ∈ F(I) where m1 < m2 < · · · < mk < · · · such that lim k−→∞ xmk = x [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Throughout the paper X stands for a topological space, S an infinite set and I, an admissible ideal of S unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Further by a “space” we will always mean a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Our topological terminology and notation are as in the book [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional open sets, I-functional closed sets and I-functional space Before we proceed to introduce our main concepts of this section, we present certain basic observations about convergence of functions which happen to be the main tool behind these generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For x ∈ X, we say that a function f : S → X is convergent to x, whenever for every open set U containing x, the set f −1(U) is co-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' is I-convergent to x, whenever for every open set U containing x, the set f −1(U) is in F(I) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' is I∗-convergent to x, whenever there is a set M ∈ F(I) such that g defined by “g(s) = f(s) if s ∈ M and g(s) = x if s /∈ M” is convergent to x [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose g : S −→ X, is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let S′ be an infinite subset of S with |S′| = |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let h : S −→ S′ be a bijective function and let Φ = g|S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now Φ is said to be I-convergent to x if (Φ ◦ h)(s) = g(s), ∀ s ∈ S is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Further if f : S −→ X is convergent to x ∈ X, then for any infinite S′ ⊂ S, f|S′ is convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In a Hausdorff space I-limit of a function is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For two ideals I ⊂ J of S, if f : S −→ X is I-convergent to x then f : S −→ X is J -convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Following [3] we can say that an ideal I of S has a pseudounion if there exists an infinite set A ⊂ S with |S| = |S \\ A| such that I \\ A is finite for each I ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If I has a pseudounion and f : S −→ X is I-convergent to x then there exists a function from S to X which is convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : S −→ X be I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since I has a pseudounion, there exists an infinite set A ⊂ S with S \\ A ∈ I+ such that I ∩ (S \\ A) is finite for each I ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As f is I-convergent to x, for every open set O containing x, AO = {s ∈ S : f(s) ̸∈ O} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus AO ∩ (S \\ A) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then Φ : S −→ X defined by Φ(s) = f(s) if s ∈ S \\A and Φ(s) = x if s ∈ A, is convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S → X be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then g is I-convergent to x if and only if g is J -convergent to x for each J ∈ Θ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be an ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take ∞ ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We define a topology on S ∪ {∞} by considering each s ∈ S isolated and each basic open neighbourhood U of ∞ as (S \\ I) ∪ {∞} for some I ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This space is denoted by � S(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly the inclusion mapping i : S −→ � S(I) is I-convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Note that ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES5 if I ̸= Ifin, then I contains an infinite set I ‘say’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then it readily follows that the inclusion function is not convergent to ∞ in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let us now look back at the history as to how the notion of closed sets in topo- logical spaces have been generalized using sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Recall that a subset F ⊂ X is called sequentially closed if for each sequence (xn : n ∈ N) in F converging to x ∈ X, we have x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' X is called a sequential space [13] if each sequentially closed subset of X is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A subset U ⊂ X is called sequentially open if X \\U is sequentially closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Di Maio and Koˇcinac introduced statistical version of sequential space in [10] while Pal [31] further extended it to I-sequential spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Very recently Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' revisited the notion of I-sequential space in [36] where following notions were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A subset F ⊂ X is called I-closed if for each sequence (xn : n ∈ N) in F, I-convergent to x ∈ X, we have x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A subset U ⊂ X is called I-open if X \\ U is I-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' X is called an I-sequential space if each I-closed subset of X is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Motivated by the generalization of I-sequential spaces from the idea of sequential spaces, we now introduce the main concept of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) F ⊂ X is said to be I-functional closed if for each function g : S → F that is I-convergent to x ∈ X we have x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) U ⊂ X is said to be I-functional open if X \\ U is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (iii) X is called an I-functional space if each I-functional closed subset of X is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If we consider “usual” convergence of functions (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1) instead of I-convergence, we call I-functional closed sets, I-functional open sets and I-functional spaces as functional closed, functional open and functional spaces respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly, every I-functional closed set is functional closed but the following example shows that the converse is not generally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We consider the space � S(I) as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then S is a functional closed set in � S(I) but not I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As an immediate consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 we can see that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A ⊂ X is I-functional closed if and only if A is functional closed provided I has a pseudounion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore X is an I-functional space if and only if X is a functional space provided I has a pseudounion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We can modify Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2(ii) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A subset O of X is I-functional open if and only if no function h : S −→ X \\ O is I-convergent to a point in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Sufficiency directly follows from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2(i), (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As O is I-functional open so X \\ O is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence for every function h : S −→ X \\ O which is I-convergent to x, we must have x ∈ X \\ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ It is evident that every open set (and so every closed set) is I-functional open (I-functional closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Following example establishes the existence of a space which is not I-functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consider the Cartesian product S × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For a ∈ S, we call the subset S × {a} as the a-th row of S × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I have a pseudounion and let ∞ be an element outside S × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = (S × S) ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We define a topology on X as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let τ1 = P(S × S) and let τ2 be the collection of those subsets A of X so that ∞ ∈ A and {a ∈ S : ({s ∈ S : (s, a) ∈ A} ∈ F(I))} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take τ = τ1 ∪ τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then it can be verified that τ is a topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' No function from S to X can be I-convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If g : S −→ X is I-convergent to ∞ then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 there is a function f : S −→ X which is convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Note that each row contains at most finitely many elements of the form f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Excluding these terms from each row, we obtain an open set containing ∞ which contains no terms of the form f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also no function from S to S × S can be I-convergent to a point of S × S unless it is eventually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But ∞ is a limit point of S ×S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence S ×S is I-functional closed but not closed and therefore X is not I-functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' However there exists an ideal for which every sequential space is I-functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let S = � i∈N Si such that Si ∩ Sj = ∅ for different i, j and let I0 = {A ⊂ S : A ∩ Si ̸= ∅ for finitely many i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then every sequential space X is an I0-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let O ⊂ X be I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If O is not open then there is a sequence xn ∈ X \\ O converging to x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function g : S −→ X \\ O by g(s) = xi if s ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then g is I-convergent to x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence X \\ O is not I-functional closed, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The following are equivalent for any A ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) A ⊂ X is I-functional open (ii) For any function g : S −→ X which is I-convergent to x ∈ A, we have {s ∈ S : g(s) ∈ A} ∈ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (iii) |{s ∈ S : g(s) ∈ A}| ≥ ω for each function g : S −→ X which is I-convergent to x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) =⇒ (ii) Let A ⊂ X be I-functional open and let g : S −→ X be I-convergent to x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If possible let C = {s ∈ S : g(s) ∈ A} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Fix an element a ∈ X \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function h : S −→ X \\ A by h(s) = g(s) for s ∈ S \\ C and h(s) = a if s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let U be a neighbourhood of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES7 {s ∈ S : g(s) ∈ U} ∩S \\ C ⊂ {s ∈ S : h(s) ∈ U} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus h : S −→ X \\ A is I-convergent to x, this contradicts that A is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) =⇒ (iii) As I is an admissible ideal, thus (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (iii) =⇒ (i) If possible let A ⊂ X be not I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So X \\ A is not I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore there is a function g : S −→ X \\ A which is I-convergent to x ∈ A and evidently {s ∈ S : g(s) ∈ A} = ∅ which contradicts (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S and let g : S −→ X be I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y = {g(s) : s ∈ S} ∪ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Endow {g(s) : s ∈ S} ⊂ Y with the discrete topology and let a basic neighbourhood of x be of the form {x}∪{g(s) : s ∈ A} for some A ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Y endowed with this topology is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' To prove that, let U be I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Without any loss of generality assume that x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As I is maximal, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4(ii), we have {s ∈ S : g(s) ∈ U} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence {x} ∪ {g(s) ∈ U} ⊂ U which implies that U is open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � i∈Λ Xi have the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then a function f : S −→ X is I-convergent to x = (xi) if and only if πi ◦ f is I-convergent to xi for each i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let πi ◦ f be I-convergent to xi for each i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let O = � i∈Λ Oi be a basic open set in X containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Oi = Ui for i = m1, m2, · · · , mk and Oi = Xi otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then {s ∈ S : (πi ◦ f)(s) ∈ Ui} ∈ F(I) for each i = m1, m2, · · · , mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now � i∈{m1,m2,··· ,mk} {s ∈ S : (πi ◦ f)(s) ∈ Ui} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly the converse holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � i∈Λ Xi have the product topology and let O be I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then πi(O) is I-functional open in Xi for each i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If possible let πi(O) be not I-functional open in Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there exists a function g : S −→ Xi which is I-convergent to x ∈ πi(O) and {s ∈ S : g(s) ∈ πi(O)} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now fix some aj ∈ πj(O) for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function h : S −→ X by (πj ◦ h)(s) = � aj if j ̸= i g(s) if j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let y = (yi) be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' yj = � aj if j ̸= i, x if j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN Then h : S −→ X is I-convergent to y (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also {s ∈ S : g(s) ∈ πi(O)} = {s ∈ S : h(s) ∈ O} ∈ I, contradicts that O is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ We now state certain basic results regarding I-functional spaces without proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) Let I ⊂ J be two ideals of S and let X be a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If U ⊂ X is J -functional open then it is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) Let I ⊂ J be two ideals of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If X is I-functional then it is J -functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (iii) Suppose that {Iα : α ∈ A} is a collection of ideals of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If X is a space and U ⊂ X is Iα-functional open for some α ∈ A, then U is � α∈A Iα-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If U, V are two I-functional open subsets of X then U ∩ V is also I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ X be I-convergent to x ∈ U ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So, {s ∈ S : g(s) ∈ U} ∈ F(I) and {s ∈ S : g(s) ∈ V } ∈ F(I) (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now {s ∈ S : g(s) ∈ U} ∩ {s ∈ S : g(s) ∈ V } = {s ∈ S : g(s) ∈ U ∩ V } ∈ F(I) and therefore U ∩ V is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ The I-functional coreflection of a space X is the set X endowed with the topology generated by I-functional open subsets of X as a subbase and the topology is denoted by I-fX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly for a space X, I-fX is finer than the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Further If I is a maximal ideal of S, then the collection of all I-functional open sets itself forms a topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be an ideal of S and A ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A function f : S −→ X is said to be I-eventually in A if there is a E ∈ I such that f(s) ∈ A for all s ∈ S \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then A ⊂ X is I-functional open if and only if for each function which is I-convergent to a point of A, it is I-eventually in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The result follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Every I-functional space is hereditary with respect to I-functional open (I-functional closed) subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose that Y is an I-functional open set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then Y is open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let U(⫋ Y ) be I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We have to show that U is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose that g : S −→ X is ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES9 I-convergent to x ∈ U ⊂ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since Y is open, {s ∈ S : g(s) ∈ Y } ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let y ∈ Y \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function h : S −→ Y by h(s) = g(s) if g(s) ∈ Y and h(s) = y if g(s) ̸∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore, h : S −→ X is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since |{s ∈ S : g(s) ̸∈ U}| = |{s ∈ S : h(s) ̸∈ U}|, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4, it follows that U is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As X is I-functional space, so U is open in X and so open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y be an I-functional closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then Y is closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let F(⫋ Y ) be an I-functional closed subset of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We have to show that F is I-functional closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose that g : S −→ F is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So x ∈ Y as Y is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore x ∈ F since F is an I-functional closed subset of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus F is an I-functional closed subset of X, so F is a closed subset of X and hence a closed subset of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional spaces are preserved by topological sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Any quotient space of an I-functional space is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be an I-functional space and let f : X −→ Y be a quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let F ⊂ Y be I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If F is not closed, f −1(F) is not closed (as f is a quotient mapping) and so f −1(F) is not I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there exists a function g : S −→ f −1(F) which is I-convergent to x ̸∈ f −1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since F is I-functional closed and f is continuous, we obtain that f ◦g : S −→ F is I-convergent to f(x) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This contradicts that x ̸∈ f −1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Every I-functional space is a quotient of some metric space provided I = I0, the ideal defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be an I-functional space and let tn = 1 n + 1, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function f : S −→ R by f(s) = tn if s ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f is I-convergent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take Y = { 1 n + 1 : n ∈ N} ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The topology of Y is induced from the usual metric topology of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly O ⊂ Y is open if and only if either 0 ̸∈ O or if 0 ∈ O then f(s) ∈ O if s ∈ A for some A ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let S = {g : S −→ X : g is I-convergent to some g0 ∈ g(S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Writing {(g(s) : s ∈ S)} = Z, let d be a metric on Z × Y = {(Z, y) : y ∈ Y } defined by d(Z, a), (Z, b)) = |a − b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now consider the topological sum L = � Z∈S Z × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We observe that A ⊂ L is open if and only if {y ∈ Y : (Z, y) ∈ A} is open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consider the mapping Φ : L −→ X defined by Φ(Z, 0) = g0 and Φ(Z, f(s)) = g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly Φ is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now we show that Φ is a quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN Let U ⊂ X be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then for every g : S −→ X, I-convergent to a ∈ U, {s ∈ S : g(s) ∈ U} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If (Z, 0) ∈ Φ−1(U) then g0 ∈ U, also {s ∈ S : g(s) ∈ U} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Write E = {s ∈ S : g(s) ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' By the definition of Φ, Φ(Z, f(s)) = g(s) ∈ U for each s ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore, Φ−1(U) is open in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Again if U is not open in X, then there exists a function g : S −→ (X \\ U) which is I-convergent to g0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently {y ∈ Y : (Z, y) ∈ Φ−1(U)} = {0}, which is not open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence Φ−1(U) is not open in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional continuity In this section our main object of investigation is the notion of I-functional continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Recall that a mapping f from a space X to another space Y is called sequentially continuous [2] provided for any sequentially open set U in Y, f −1(U) is sequentially open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' It is proved in [2] that a mapping f : X −→ Y is sequentially continuous if and only if f preserves the convergence of sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=', for each sequence (xn : n ∈ N) in X converging to x, the sequence (f(xn) : n ∈ N) converges to f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In [36], authors introduced the notion of I-continuity in terms of I-open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Extending this notion in the language of functions, we introduce following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be an ideal of S and f : X −→ Y be a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then (i) f is called an I-functional convergence preserving mapping provided for a function g : S −→ X, I-convergent to x, f ◦ g is I-convergent to f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) f is called I-functional continuous provided for any I-functional open set U in Y , f −1(U) is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We call f simply functional continuous if we take functional open set instead of I-functional open set in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y ⊂ X and let U be I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then U ∩ Y is I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ Y be I-convergent to y ∈ U ∩Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then {s ∈ S : g(s) ∈ U} ∈ I+ (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4) and therefore {s ∈ S : g(s) ∈ U ∩ Y } ∈ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S and let U ⊂ Y ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose that U is I-functional open in Y and Y is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then U is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If U is not I-functional open in X then there exits a mapping f : S −→ X, I-converging to some a ∈ U and f −1(U) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As Y is I-functional open in X, and I is maximal, f −1(Y ) ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore f −1(Y \\ U) = f −1(Y ) \\ f −1(U) ∈ ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 11 F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let F = f −1(Y \\ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a mapping φ : S −→ Y by φ(s) = � f(s) if s ∈ F a if s ̸∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Evidently φ is I-convergent to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also φ−1(U) ∈ I, contradicts that U is I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be a space and let U be a cover of X by I-functional open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then a mapping f : X −→ Y is I-functional continuous if and only if for each U ∈ U the restriction f|U is I-functional continuous provided I is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be I-functional continuous and let U ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose that V ⊂ Y is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then (f|U)−1(V ) = f −1(V ) ∩ U is I-functional open in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conversely let the condition hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2 for any I-functional open set V ⊂ Y, f −1(V ) ∩ U is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As X = � U, f −1(V ) = � U∈U (f|U)−1(V ) and is I-functional open as each is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ In [36, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2], it was shown that every continuous mapping preserves I-convergence of sequences and if a mapping preserves I-convergence of se- quences then the mapping is I-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Here also, similar kind of results hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X, Y be two spaces and f : X −→ Y be a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) If f is continuous then f preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) If f preserves I-functional convergence then f is I-functional continu- ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The examples given below, show that the converses of preceding Proposition are not generally true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take X = � S(I) as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 and let Y = X be endowed with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly i : S −→ X, the inclusion function is not convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let S′ ⊂ S and i|S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If S′ ∈ I then i can not converge to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Otherwise take an infinite S′′ ⊂ S′ satisfying |S′ \\ S′′| = |S′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If S′′ ∈ I then (S \\ S′′) ∪ {∞} is a neighbourhood of ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So i|S′ again can not be convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Finally if S′′ ∈ F(I) then S′′ ∪ {∞} is a neighbourhood of ∞ but it does not contain all but finitely many terms of i|S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So i|S′ is not convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore there is no convergent function from S to X except for eventual constant mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So f preserves Ifin-functional convergence trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, f is also functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But evidently f is not con- tinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � S(I) and let Y = {1, 0} be endowed with discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also let I be a non-maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there is A ⊂ S for which both A ∈ I+ and S \\ A ∈ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a mapping g : X −→ Y by g(x) = 1 if x ∈ A and g(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As (S \\ A) ∪ {∞} is I-functional open, so g is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But g does not preserve I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be an I-functional continuous mapping and let g : S −→ X be I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If V ⊂ Y is an I-functional open set containing f(x) then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4, {s ∈ S : g(s) ∈ f −1(V )} ∈ I+ and thus {s ∈ S : (f ◦ g)(s) ∈ V } ∈ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This observation leads to the following result immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then a mapping f : X −→ Y is I-functional continuous if and only if it preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For the next result we recall the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I is called a P-ideal if for any (An)n∈ω from F(I) there is A ∈ F such that A \\ An is finite for each n [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a P-ideal and X be a first-countable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f : X −→ Y is I-functional continuous if and only if it preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' From [29], it follows that I-convergence implies I∗-convergence of func- tions, as I is a P-ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be I-functional continuous and let g : S −→ X be I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there is A ∈ I such that g ⇃S\\A−→ X is convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let U be an open neighbourhood of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f −1(U) is I-functional open in X, and so is a functional open set containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conse- quently g(s) ∈ f −1(U) for all s ∈ S \\(A∪F) (for a suitable finite subset F of S) and hence (f ◦ g)(s) ∈ U for all s ∈ S \\ (A ∪ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' By admissibility of I it follows that {s ∈ S : (f ◦ g)(s) ∈ U} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The converse result follows directly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Next we investigate the interrelationships between the notions of continuity and I-f-continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a mapping from an I-functional space X to another space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f is continuous if and only if f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since every open set is I-functional open and X is an I-functional space the converse follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 13 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a mapping from a functional space X to another space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (1) f is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (2) f preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (3) f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (4) f is functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (1) =⇒ (2) follows from the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (2) =⇒ (3) follows di- rectly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As each functional space is I-functional space and continuity implies functional continuity, preceding theorem establishes that (3) =⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Finally (4) =⇒ (1) since X is a functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be a sequential space and I be as defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then for a mapping f : X −→ Y, f is continuous if and only if f preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let V ⊂ Y be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f −1(V ) is not open then f −1(V ) is not I-functional open (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there is a function g : S −→ X which is I-convergent to x ∈ f −1(V ) such that {s ∈ S : g(s) ∈ f −1(V )} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So f ◦ g : S −→ Y is I-convergent to f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But {s ∈ S : g(s) ∈ f −1(V )} ∈ I =⇒ {s ∈ S : (f ◦ g)(s) ∈ V } ∈ I, which contradicts that V is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Converse is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f : X −→ Y preserves J -functional convergence for each J ∈ Θ(I) then f preserves I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f : X −→ Y is J -functional continuous for each J ∈ Θ(I) then f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The result follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' There exists a mapping which preserves Ifin-functional conver- gence but is not J -functional continuous for J ∈ Θ(Ifin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � S(J ) and let Y = S, endowed with discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a mapping f : X −→ Y by f(s) = s if s ∈ S and f(∞) = a for some particular a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' There is no function from S to X which is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence f preserves Ifin-functional convergence but is not J -functional continuous since S \\ {a} is J -functional closed and f −1(S \\ {a}) = S is not J -functional closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � i∈Λ Xi have the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then πi : X −→ Xi is I-functional continuous for each i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ 14 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X = � i∈Λ Xi have the product topology and let Y be a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then a mapping f : Y −→ X is I-functional continuous if and only if πi ◦ f is so for each i ∈ Λ provided I is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let πi ◦ f be I-functional continuous for each i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ Y be I-convergent to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then πi ◦ f ◦ g : S −→ Xi is I-convergent to (πi ◦ f)(y) for each i ∈ Λ (by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3, it follows that f ◦ g is I-convergent to f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore f is I-functional continuous (by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conversely let f be I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Uα ⊂ Xi be I-functional open in Xi for some i ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now (πi ◦ f)−1(Uα) = f −1(π−1 i (Uα)) where π−1 i (Uα) is I-functional open in X (by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently f −1(π−1 i (Uα)) is I-functional open in Y as f is I-functional continuous □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional quotient and I-functional covering mappings In the literature (see the papers [2,26–28]), the notions of quotient, sequen- tially quotient and sequence covering mappings play an important role in study- ing sequential spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' These notions have been extended using ideal convergence of sequences to I-quotient and I-covering mappings in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In this section we intend to further extend these concepts by defining them in terms of functions over an arbitrary set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X, Y be two spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Recall that an onto mapping f : X −→ Y is said to be a quotient mapping provided U is open in Y if and only if f −1(U) is open in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' f is said to be sequentially quotient [2] provided U is sequentially open in Y if and only if f −1(U) is sequentially open in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' f is said to be I-quotient [36] provided U is I-open in Y if f −1(U) is I-open in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' f is said to be sequence covering [2] if whenever (yn : n ∈ N) is a sequence in Y converging to y ∈ Y, there exists a sequence (xn : n ∈ N) in X satisfying xn ∈ f −1(yn) for all n ∈ N and x ∈ f −1(y) such that xn converges to x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' f is said to be I-covering [36] if whenever (yn : n ∈ N) is a sequence in Y, I-converging to y ∈ Y, there exists a sequence (xn : n ∈ N) in X satisfying xn ∈ f −1(yn) for all n ∈ N and x ∈ f −1(y) such that (xn : n ∈ N) is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Our next definitions are introduced following this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a mapping from a space X onto another space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) f is said to be I-functional quotient provided U is I-functional open in Y if and only if f −1(U) is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) f is said to be I-functional covering if for any g : S −→ Y , I-converging to y ∈ Y, there exists a function h : S −→ X satisfying (f ◦ h)(s) = g(s) for all s ∈ S and x ∈ f −1(y) such that h is I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 15 We call f simply functional quotient if we take functional open set instead of I-functional open set in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be a space and Y be a non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Further let f : X −→ Y be an onto mapping and I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' There exists a strongest topology on Y w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='t which f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let J = {V ⊂ Y : f −1(V ) is I-functional open in X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then J is a topology on Y w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='t which f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Next let J ′ be any other topology on Y w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='t which f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then for every J ′-functional open set V ⊂ Y, f −1(V ) is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So for each V ∈ J ′, V ∈ J and hence J ′ ⊂ J as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ In the above Theorem f is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A mapping f : X −→ Y is said to be I-functional open provided f(U) is I-functional open in Y whenever U is I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Every I-functional continuous, I-functional open onto map- ping is I-functional quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y and g : Y −→ Z be two mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then the following results hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) If f and g are I-functional quotient mappings then g ◦ f is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) If f and g ◦ f are I-functional quotient mappings then g is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) If g, f are I-functional continuous then g ◦ f is also I-functional con- tinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Again for any V ⊂ Z, (g ◦ f)−1(V ) = (f −1(g−1(V ))) and therefore g ◦ f is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) Let V ⊂ Z such that g−1(V ) is I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So (f −1(g−1(V ))) is I-functional open in X as f is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now (g ◦ f)−1(V ) = (f −1(g−1(V ))) and g ◦ f being I-functional quotient, together imply that V is I-functional open in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Next let V be I-functional open in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then as g ◦ f is I-functional continuous and f is I-functional quotient, we have g−1(V ) is I-functional open in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional quotient mappings are preserved by finite prod- ucts provided I is a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let fi : Xi −→ Yi be an I-functional quotient mapping for i = 1, 2, · · · , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We define a mapping f : N � i=1 Xi −→ N � i=1 Yi by f(x1, x2, · · · , xN) = (f1(x1), f2(x2), · · · , fN(xN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3, f is I-functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' It is also onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Next let U ⊂ N � i=1 Yi be such that f −1(U) is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there exists a function g : S −→ N � i=1 Yi, I-convergent to y ∈ U and {s ∈ S : g(s) ∈ U} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently πi◦g : S −→ Yi is I-convergent to πi(y) for i = 1, 2, · · · , N (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ F(I) for each i = 1, 2, · · · , N then {s ∈ S : g(s) ∈ U} = N � i=1 {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ F(I), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So there exists i such that {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ̸∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Maximality of I implies that {s ∈ S : (πi ◦ g)(s) ∈ πi(U)} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently πi(U) is not I-functional open in Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus f −1 i (πi(U)) is not I-functional open as fi is I-functional quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But (πi ◦ f −1)(U) = f −1 i (πi(U)), which contradicts the fact that f −1(U) is I-functional open (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ The interrelationships results among I-quotient, quotient and I-covering map- pings that have been studied in [36] can be further generalized as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a mapping from a space X onto a space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (i) If f is I-functional continuous and an I-functional covering mapping then f is an I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) If f is one-to-one and an I-functional quotient mapping then f is an I-functional covering provided I is a maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (1) Let f : X −→ Y be an I-functional continuous and I-functional covering mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Suppose U ⊂ Y is such that f −1(U) is I-functional open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If U is not I-functional open there exists a function g : S −→ Y, I-converging to y ∈ U for which {s ∈ S : g(s) ∈ U} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As f is I-functional covering there is a function h : S −→ X, I-convergent to x ∈ X such that f ◦ h = g and f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also {s ∈ S : g(s) ∈ U} = {s ∈ S : (f ◦ h)(s) ∈ U} = {s ∈ S : h(s) ∈ f −1(U)} which implies that f −1(U) is not I-functional open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (ii) Let f : X −→ Y be an one-to-one and I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ Y be I-convergent to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As f is one-to-one and onto, for each s ∈ S there exists an unique xs ∈ X such that f(xs) = g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a function h : S −→ X by h(s) = xs and let f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If h is not I-convergent to x, there exists an open set O containing x such that {s ∈ S : h(s) ∈ O} ̸∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since I is maximal, {s ∈ S : h(s) ∈ O} ∈ I, so {s ∈ S : (f ◦ h)(s) ∈ f(O)} ∈ I (as f is one-to-one) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' {s ∈ S : g(s) ∈ f(O)} ∈ I (because f ◦ h = g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now f(O) is ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 17 I-functional open in Y as f −1(f(O)) = O is I-functional open in X, and f is I-functional quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ The next result establishes when an I-functional quotient mapping becomes a quotient mapping and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a continuous mapping from an I-functional space X to another space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f is a quotient map if and only if f is I-functional quotient and Y is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' First let f : X −→ Y be a quotient mapping and let F ⊂ Y be not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f −1(F) is not closed in X as f is a quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So f −1(F) is not I-functional closed since X is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4 there exists a function g : S −→ f −1(F) which is I-convergent to x ∈ X \\ f −1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore, f ◦ g : S −→ F is I-convergent to f(x) ∈ Y \\ F and consequently F is not I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus Y is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let F ⊂ Y be such that f −1(F) is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now if F is not I-functional closed, F is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Thus f −1(F) is not closed (as f is a quotient mapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since X is an I-functional space, f −1(F) is not I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conversely let f be an I-functional quotient mapping and let Y be an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take F ⊂ Y such that f −1(F) is closed in X and so I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since f is an I-functional quotient mapping F is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Y is an I-functional space, thus F is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This concludes that f is a quotient map- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Generally an I-functional quotient mapping is not quotient and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The next result characterises I-functional spaces in terms of the interrelations of I-functional quotient and quotient mappings, which can be proved in the same way as that of [36, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let X be a space and I be a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then X is an I-functional space if and only if each I-functional quotient mapping onto X is quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' An onto mapping p : X −→ Y is I-functional quotient if and only if it has the property that for any space W and a mapping f : Y −→ W, I-functional continuity of f ◦ p implies that of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' First let p : X −→ Y be I-functional quotient and let f be a mapping from Y to some space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take f ◦ p, an I-functional continuous mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let F ⊂ W be I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then (f ◦p)−1(F) = p−1(f −1(F)) is I-functional closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since p : X −→ Y is I-functional quotient, f −1(F) is I-functional closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conversely let the condition hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consider F ⊂ Y such that p−1(F) is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let W = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define a mapping f : Y −→ W by f(y) = 1 18 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN if y ∈ F and f(y) = 0 if y ∈ Y \\ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So (f ◦ p)(x) = 1 if x ∈ p−1(F) and (f ◦p)(x) = 0 if x ̸∈ p−1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, the topology on W is induced by f ◦p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As (f ◦p)−1({1}) = p−1(F) is I-functional closed in X, {1} is I-functional closed in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore by I-functional continuity of f, F is I-functional closed in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consequently p : X −→ Y is I-functional quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ If I has a pseudounion, then the next theorem readily follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f is a mapping from a space X onto Y and if I has a pseu- dounion then following results hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (1) f is I-functional continuous if and only if f is functional continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (2) f is I-functional quotient if and only if f is functional quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f : X −→ Y be a mapping and let I have a pseudounion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f is I-functional quotient if and only if for any function g : S −→ Y, I-converging to p there exists a function h : S −→ X which is I-convergent to some x ∈ X so that (f ◦ h)(S) ⊂ g(S) and f(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a I-functional quotient mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f is functional quo- tient (by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ Y be I-convergent to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since I has pseudounion, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, we obtain a function g′ : S −→ Y which is convergent to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now f −1(g′(S) \\ {p}) is not functional closed, and so we get a function k : S −→ f −1(g′(S) \\ {p}), converging to a ̸∈ f −1(g′(S) \\ {p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now (f ◦ k) : S −→ g′(S) is convergent to f(a) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Conversely let the condition hold and let F ⊂ Y so that f −1(F) is an I-functional closed set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let g : S −→ F be I-convergent to y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then there exists an infinite set S′ ⊂ S and g|S′ is convergent to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Φ = g|S′ and h : S −→ S′ be an onto mapping satisfying (Φ ◦ h)(s) = g(s) for every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then (Φ◦ h) is convergent to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So, there is a function k : S −→ f −1((Φ◦ h)(S)) that is I-convergent to x ∈ f −1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As f −1(F) is I-functional closed, x ∈ f −1(F) so y ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional Fr´echet-Uryshon space Recall that a space X is Fr´echet-Uryshon [13] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=', statistically Fr´echet- Uryshon [10], I-Fr´echet-Uryshon [36]) if for each A ⊂ X with x ∈ cl(A) there exists a sequence in A which is convergent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' statistically convergent, I-convergent) to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We can extend these notions to I-functional Fr´echet-Uryshon space in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A space X is called an I-functional Fr´echet-Uryshon space if for each A ⊂ X and each x ∈ cl(A) there exists a function f : S −→ A, I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 19 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Every I-functional Fr´echet-Uryshon space is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We modify [34, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1] to show that the converse of the above Propo- sition does not hold for a maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let I be a maximal ideal of S and let X be a non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For every a ∈ S, let ga be a function from S to X and take Ga = ga(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For fixed a ∈ S assume that ga(x) ̸= ga(y) for x ̸= y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Take G = {xa : a ∈ S} with xa ̸= xb for a ̸= b ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consider a point ∞ outside �{Ga : a ∈ S} � G and take X = �{Ga : a ∈ S} � G �{∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' The topology on X is defined in the following way: (1) Each point ga(s) is isolated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (2) For each a ∈ S, an open neighbourhood of xa is taken as a set of the form {xa} ∪ Ma where Ma = {ga(s) : s ∈ F} for some F ∈ F(I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' (3) Each open neighbourhood of ∞ is a set of the form {∞}∪M ∪{Ma : a ∈ M} where M = {xa : a ∈ F} for some F ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' First to show that X is an I-functional space, take Y ⊂ X which is I-functional closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also let y ∈ cl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now one of the three cases may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Case 1: If y ∈ �{Ga : a ∈ S} then {y} is an open neighbourhood of y and y ∈ cl(Y ) implies that y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Case 2: If y ∈ G, then y = xa for some a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If possible let y ̸∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Now if {s ∈ S : ga(s) ∈ Y } ∈ I then U = (Ga\\Y )∪{xa} is an open neighbourhood of xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Also U ∩Y = ∅ contradicts that y ∈ cl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We define a function h : S −→ Y ∩Ga by h(s) = ga(s) if ga(s) ∈ Y and h(s) = ga(s0) for some fixed s0 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then h is I-convergent to xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Because Y is I-functional closed, xa = y ∈ Y, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Therefore y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Case 3: Consider the case when y = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If A = {s ∈ S : xs ∈ Y } ∈ I put V = X \\ �{Ga : a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then V is an open neighbourhood of ∞ and V ∩ Y = ∅, which contradicts ∞ ∈ cl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence as in Case 2, we obtain a function f : S −→ Y, I-convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As Y is I-functional closed, ∞ ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' To prove that X is not an I-functional Fr´echet-Uryshon space, take ∞ ∈ cl(X \\ ({xa : a ∈ S} ∪ {∞})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let h : S −→ X \\ ({xa : a ∈ S} ∪ {∞}) be I-convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Define Aa = h(S) ∩ Ga, a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If for each a ∈ S, {s ∈ S : ga(s) ∈ h(S)} ∈ I then V = {∞} ∪ {Ga \\ Aa : a ∈ S} is an open set containing ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' But V ∩ h(S) = ∅ and this contradicts that h is I-convergent to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So there is a ∈ S such that {s ∈ S : ga(s) ∈ h(S)} ∈ F(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As in case 2, a function k : S −→ h(S) ∩ Ga can be constructed which is I-convergent to xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Next if {s ∈ S : h(s) ∈ Ga} ∈ I then k : S −→ h(S) ∩ Ga can be made to I-converge to a point different from xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This contradicts Lemma ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='. Now if {s ∈ S : h(s) ∈ Ga} ∈ F(I) then k can be made to be I-convergent to ∞ ̸= xa, 20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' DAS, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' SAMANTA, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' LIN which again is a contradiction as X is a Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Considering functions over S instead of functions over N, we restate [36, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A space X is an I-functional Fr´echet-Uryshon space if and only if each subset of X is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Subspaces of an I-functional Fr´echet-Uryshon space are I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional Fr´echet-Uryshon spaces are preserved by topological sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let {Xa : a ∈ Λ} be a disjoint family of I-functional Fr´echet-Uryshon spaces and let X = � a∈Λ Xa be its topological sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' From Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2, X is an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' For every Y ⊂ X and for every a ∈ Λ, Y ∩Xa is an I-functional Fr´echet-Uryshon space in Xa, and consequently is an I-functional space in Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence the topological sum � a∈Λ Y ∩ Xa becomes an I-functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' As Y is an I-functional space, therefore by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, X is an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Although the line of proof of the next result is analogous with that of [36, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='9], it has its own significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Every space is a continuous and I-functional covering image of an I-functional Fr´echet-Uryshon space provided I is a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In general product of two I-functional Fr´echet-Uryshon spaces may not be an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This follows from a modification of an Ex- ample from [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Actually product of two I-functional Fr´echet-Uryshon spaces need not be an I-functional space either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A mapping f : X −→ Y is called pseudo-open [1] if for each y ∈ Y and an open subset U in X with f −1(y) ⊂ U, f(U) is a neighbourhood of y in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f be a pseudo-open mapping from an I-functional Fr´echet- Uryshon space X onto a space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f preserves I-functional convergence then Y is an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let f preserve I-functional convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let A ⊂ Y and choose y ∈ cl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' If f −1({y}) ∩ (cl(f −1(A))) = ∅ then f −1(y) ∈ X \\ (cl(f −1(A))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since f is pseudo-open, y ∈ int(f(X \\ cl(f −1(A))) = int(f(int(X \\ f −1(A)))) ⊂ int(f(X \\ f −1(A))) = int(Y \\ A) = Y \\ cl(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' So, y ∈ Y \\ cl(A), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' ON CERTAIN GENERALIZED NOTIONS USING I-CONVERGENCE IN TOPOLOGICAL SPACES 21 Therefore, there exists x ∈ f −1({y}) ∩ (cl(f −1(A))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Since X is an I-functional Fr´echet-Uryshon space, there is a function g : S −→ f −1(A), I-convergent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Hence f ◦ g : S −→ A is I-convergent to f(x) = y, which consequently implies that Y is an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' I-functional Fr´echet-Uryshon spaces are preserved by continu- ous pseudo-open mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A space Y is an I-functional Fr´echet-Uryshon space if every mapping onto Y that preserves I-functional convergence is pseudo-open provided I is a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y be a space and let g : S −→ Y be I-convergent to yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Consider Sg = g(S)∪{yg} and S is the family of all functions on S which are I-convergent in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A topology on Sg is defined as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4 and is denoted by SI g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Clearly g is I-convergent to yg in SI g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' In view of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1, one can conclude that SI g is an I-functional Fr´echet-Uryshon space since every subset of Sg is open or closed in SI g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Z = � g∈S SI g be the topological sum of {SI g }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='2, Z is an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We define a mapping f : Z −→ Y such that f|Sg : SI g −→ (Sg, τSg) is the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then f preserves I-functional convergence and therefore by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='4 Y is an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ But suitably modifying [36, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='7], we can obtain the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Let Y be an I-functional Fr´echet-Uryshon space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Then every I-functional covering mapping from a space onto Y is pseudo-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' We end the section with the following interesting observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' A space X is an I-functional Fr´echet-Uryshon space if and only if every continuous I-functional covering mapping onto X is pseudo-open provided I is a maximal ideal of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' This follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='6, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='3 and corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' □ References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Liu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Lin, On topological spaces defined by I-convergence, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Iranian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=', (Accepted), July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content=' Department of Mathematics, Jadavpur University, Kolkata-32, West Bengal, In- dia Email address: pratulananda@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='in, samantaupasana@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='com † Department of Mathematics, Ningde Normal University, Ningde, 352100, Fujian, People’s Republic of China Email address: shoulin60@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNAyT4oBgHgl3EQfy_n2/content/2301.00696v1.pdf'} diff --git a/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/2301.13386v1.pdf.txt b/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/2301.13386v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c551e4d7fbdce8763afbcf0ce8f454fb26536ebb --- /dev/null +++ b/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/2301.13386v1.pdf.txt @@ -0,0 +1,673 @@ +Emergence of extreme events in a quasi-periodic oscillator +Premraj Durairaj1, Sathiyadevi Kanagaraj1, Suresh Kumarasamy1, Karthikeyan Rajagopal1,2 +1Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai-600 069, Tamilnadu, India. +2Department of Electronics and Communications Engineering, University Centre for +Research and Development, Chandigarh University, Mohali, 140 413, Punjab, India. +(Dated: February 1, 2023; Received :) +Extreme events are unusual and rare large-amplitude fluctuations that occur can unexpectedly in +nonlinear dynamical systems. Events above the extreme event threshold of the probability distri- +bution of a nonlinear process characterize extreme events. Different mechanisms for the generation +of extreme events and their prediction measures have been reported in the literature. +Based on +the properties of extreme events, such as rare in the frequency of occurrence and extreme in am- +plitude, various studies have shown that extreme events are both linear and nonlinear in nature. +Interestingly, in this work, we report on a special class of extreme events which are nonchaotic and +nonperiodic. These nonchaotic extreme events appear in between the quasi-periodic and chaotic +dynamics of the system. We report the existence of such extreme events with various statistical +measures and characterization techniques. +PACS numbers: 05.45.-a +Extreme events are unanticipated, rare events that oc- +cur in many natural and engineering systems. Extreme +events (EE) can exist in various forms, including floods, +cyclones, droughts, pandemics, power outages, material +ruptures, explosions, chemical contamination, and stock +market crashes, among others [1]. Such events have a +severe impact on real-world situations. Thus, it is nec- +essary to understand the relevant mechanism and its +generic characteristics for the occurrence of EE in order +to prevent such EE. As a result, the researchers focused +on exploring the EE in diverse nonlinear oscillators [2–5], +maps [6], and neural networks [7]. Further, the extreme +events have also been identified in a super-fluid helium +[8], plasma[9], optical fibers [10], lasers [11], and capillary +wave [12] etc. +However, depending on the characteristics of a dynam- +ical system, the occurrence of EE has been discovered +under a variety of mechanisms, including internal crises, +on-off intermittency, blowout bifurcations, stick-slip bi- +furcations, and so on [6, 11, 13–15]. For instance, prior +studies reveal that EE can arise as a result of the abrupt +expansion and destruction of chaotic attractors produced +by internal or external crises [11, 14]. Further, interior +crises are found to be a critical mechanism for the oc- +currence of EE, when the trajectory of chaotic attractors +reaches the stable manifold of a saddle or unstable peri- +odic orbit, which increases the size of the chaotic attrac- +tors. Such a sudden expansion of the chaotic attractor +may result in EE. In addition, Pomeau-Manneville in- +termittency is identified as another mechanism for the +existence of EE. Such intermittency can occur when the +periodic oscillations are interspersed by chaotic bursts, +which further results in very large amplitude events. EEs +can also exist through the following other mechanisms. +The sliding bifurcation near the discontinuous boundary +can cause EE. The trajectory of the attractors might hop +between coexisting attractors due to noise in multi-stable +systems, which can cause unusually large events. This is +referred to as noise-induced intermittency. The trajec- +tory of the attractors in coupled systems departs from +the synchronization manifold to the transverse direction +of the manifold. During such a transition, a synchroniza- +tion error of dynamics can show zero or nonzero and is +referred to as on-off intermittency [16]. +Moreover, previous studies discovered that extreme or +rare events can occur as a result of chaotic or stochas- +tic processes [16]. +In particular, the appearance of +EE has been reported in micro-electromechanical can- +tilevers with discontinuous boundaries and diode lasers +with phase-conjugate feedback [17, 18]. By applying the +harmonic pump modulation to the fiber laser the emer- +gence of Rogue waves has been identified [2, 19–21]. The +EE in stochastic transport on networks has been demon- +strated using multiple random walks on complex net- +works [23, 24]. Now the interesting question is whether +extreme events can be induced by nonchaotic signals. +In literature, a study has shown nonchaotic and non- +periodic have been well studied in the name of strange +nonchaotic dynamics, which arises during the attractor +transition from quasi-periodicity to chaos [31]. One can +find the generation mechanisms of these strange non- +chaotic attractors in literature [25, 26, 31]. The results +in the present work show that similar to the strange +nonchaotic dynamics, the nonperiodic and nonchaotic +dynamics show large-amplitude extreme events. +The +present study opens a new area of study where the non- +chaotic nonlinear process can also lead to extreme events +and the same has not been found reported. +To show the nonchaotic extreme events, we consider +the Morse Oscillator (MO) which is used to describe the +motion of diatomic molecules. Importantly, the MO has +made substantial contributions in the fields of classical, +semi-classical, and quantum mechanics [27, 28]. +The +MO was used for photo-dissociation molecules without +any damping. +In the presence of driving and damp- +ing, the MO was exploited for multi-photon excitation +arXiv:2301.13386v1 [physics.data-an] 31 Jan 2023 + +2 +of molecules, pumping the local mode of polyatomic +molecules [29]. We consider the quasi-periodically forced +MO and its dynamical equation can be written as +˙x = y +˙y = fsin(ω1t) + gsin(ω2t) + e−2x − e−x − γy +(1) +where x, and y are the state variables of the system and +γ is a damping parameter. The amplitudes of the first +and second force are represented by f and g and the +corresponding frequencies are denoted by ω1 and ω2, re- +spectively. +FIG. 1: (a) Time evolution of xn for the nonchaotic dynamics +with forcing amplitudes as (a) f = g = 0.255, and (b) f = +g = 0.278. The xn is the nth local peaks of the variable x. +The horizontal black dot-dashed and red dashed lines are the +critical threshold lines defining the extreme events (refer to +the text for the meaning of N and A). (c) The probability +distribution function corresponds to the extreme events and +(d) return interval (R) (inter-event interval) with respect to +the probability of recurrence times (PR) of the EE for (b). +The filled circles and solid lines in (d) represent the numerical +data and the corresponding power-law fit. We fixed the other +parameter values as γ = 0.35, ω1 = 0.3, and ω2 = ( +√ +5−1 +2 +). +To manifest the existence of extreme events, we first +depicted the time evolution of the x-variable in Fig. 1(a) +and Fig. 1(b) by fixing the amplitude of the first and sec- +ond forcing as f = g = 0.255 and f = g = 0.278. We +observe from Fig. 1(a) that some of the oscillation(event) +has larger amplitudes, while the rest of them take lower +amplitudes. To check the larger amplitude oscillations +satisfy the extreme events criteria defined in the litera- +ture, we use the following relation: +xEE =< xn > +Nσxn, +(2) +where xEE is the critical amplitude threshold and N is a +multiplication factor. The mean and standard deviation +of the variable x is represented by < xn > and σxn, re- +spectively. Here, the xn (an event) are the local peaks +of the variable x. An event or a local peak can satisfy +extreme event criteria if it has a value higher than the +critical threshold defined by Eq. (2) with N ≥ 4. To con- +firm the presence of EE, we plotted the critical threshold +on the time series for N = 5 and N = 4 in Figs. 1(a) +and 1(b). We used two different N values depending on +the time series. Though the choice of N is arbitrary, we +set the minimum N value as 4 in the present study. We +also find the critical value of Nmax for a range of each f +value – the details will be discussed below. In both cases, +we can see that some of the large amplitude events cross +the threshold line, confirming the presence of EE. Since +the choice of N is arbitrary in the previous criterion, we +use another criterion defined by the abnormality index; +An = Hfn +H1/3 [17], where Hfn is the difference between the +maximum height of the event n and the mean height of +its population, Hfn = xn − ⟨xn⟩n and H1/3 is the av- +erage value among the highest one-third values of Hfn. +If an event xn has abnormality An greater than 2 then +the event is termed an extreme event. We find that both +cases in Figs. 1(a) and 1(b) satisfy the above criterion +with abnormality index A = 3 denoted by a dashed hori- +zontal line in the plots. It is evident that a few rare large +amplitude events cross the abnormality index line. We +computed the probability distribution function (PDF) in +Fig. 1(c) for the time series shown in Fig. 1(b). The EE +critical threshold at N = 4 is plotted as a vertical dashed +line on the PDF diagram. In the plot, the events with +a finite probability above the critical threshold line char- +acterize the extreme events. We can plot similar proba- +bility distribution for Fig. 1(a), however, for simplicity, +we have plotted the PDF corresponding to Fig. 1(b). +The above analysis shows that the observed behavior +satisfies the extreme events criterion in the amplitudes. +Another important characteristic of extreme events is +an inter-event interval. The inter-event interval defines +the frequency occurrence of the events and should not +have discrete values (discrete values mean the periodic +occurrence of events), rather it should have a distribu- +tion over a range. In order to examine the distribution +of events in the observed time series, we find inter-event +intervals (R) between successive extreme events. Subse- +quently, we find the probability of such inter-event in- +tervals (PR) as shown in Fig. 1(d). Inter-event interval +and its probability obey power-law relations as given by +log10(PR) = a log10(R)b, where a and b are constants +with values a = −0.006 and b = 2.96, respectively. The +obtained numerical values are depicted in a filled circle, + +(a) +7.0 +N=5 +A=3 +X3.5 +0.0 +(b) +2.30×106 +4.60×106 +7.0 +N=4 +X 3.5 +0.0 +0 +3.50×105 +7.00×105 +(c) +n +(d) +-2 +N= 4: +log R vs. log PR +0.1 +Power law +PDF +PR +0.01 +log +4 +0.001 +0.0001 +6 +0.0 +3.5 +7.0 +6.9 +7.9 +8.9 +9.7 +x. +log R3 +and a continuous line shows the corresponding power- +law fit. The route for the emergence of EE and its tran- +sitions is further estimated below using Lyapunov ex- +ponents (LE), amplitude maxima Xmax, critical factor +Nmax, and two-parameter analysis. +FIG. 2: +(a) The two-parameter bifurcation diagram in (f, g) +space. Using the range of Lyapunov exponents (λ) (denoted +by the color bar) the dynamical regions are marked. (b) The +maximum Lyapunov exponents as a function of forcing am- +plitude f(= g), (c) maximum amplitude of the events xmax +(red) and the corresponding Nmax (Eq. 3) of the event (blue) +by varying the magnitude of f(= g). The black line repre- +sents the extreme events critical threshold (xEE) drawn from +Eq. (2) for N=4. The other parameter values are fixed as the +same as in Fig. 1. +To illustrate the global dynamical transition of the at- +tractors and route of the EE, the two-parameter diagram +is drawn in (f, g) space using the maximum LE as shown +in Fig. 2(a). The range of LE (shown in the color bar) +denotes the emergence of quasi-periodic, nonchaotic, and +chaotic attractors in the respective parameters of f and +g. If the forcing amplitudes f and g are small, attractors +have a maximum negative LE, indicating the presence of +a quasi-periodic (QP) attractor region. To better com- +prehend QP attractors, we plotted their time-evolution +and phase portrait trajectories in Supplementary Mate- +rial Fig. S1 a(i, ii) for f = g = 0.23, which show their +bounded nature. Thus, the EE critical threshold for this +attractor is greater than the amplitude of QP attrac- +tors. +By increasing f and g values, the QP attractor +transits to a chaotic (CH) attractor via strange and non- +chaotic dynamics in which the LE takes the values from +negative (near zero) to positive. To distinguish between +the strange nonchaotic and chaotic attractors, the time- +evolution and phase portrait trajectories are shown in +Figs. S1 b(i,ii) and Figs. S1 c(i,ii) in the supplementary +materials by fixing f = g = 0.278 and f = g = 0.33, +respectively. +Also, the frequency spectra can be used +to distinguish quasiperiodic, SNA and chaos. We have +the frequency spectrum analysis in the Supplementary +material in Fig. S4 (a-c). When compared to the chaotic +attractor (which has a greater number of large ampli- +tude oscillations), we found the SNA shows fewer large +amplitude oscillations. The supplemental material’s Fig. +S1 can be consulted for more information. Furthermore, +to show the dynamical transitions clearly, we displayed +maximum Lyapunov exponents in Fig. 2(b) by keeping +the parameter (f = g) and varying it along the diagonal +dashed line shown in Fig. 2(a). In Fig. 2(b), the maxi- +mum LE is illustrated as a function of forcing amplitudes +f and g (f = g) in the range (0.23 < f(= g) < 0.32). We +observe that when the forcing amplitudes are minimum +in the mentioned range, LE takes negative values, in- +dicating quasi-periodic dynamics. While increasing the +parameter, the transition of LE from negative to posi- +tive values indicates the dynamical transition of quasi- +periodic behavior to chaotic behavior. Furthermore, we +found that the negative values of LE near-zero exhibit +strange nonchaotic behavior; extreme events are seen in +this region. The literature has shown that the EEs occur +under chaotic dynamics [16] through distinct routes and +stochastic processes like stochastic transport on networks +has been demonstrated using multiple random walks on +complex networks [23, 24]. Among the various routes, the +occurrence of EEs in nonchaotic dynamics is new and it +has not been reported to the best of our knowledge. +To validate the occurrence of EEs in the SNA region, +we find the maximum amplitude xmax, extreme event +threshold xEE, and maximum value of N (Nmax) of a +given time series. In Fig. 2(c), we have plotted the above +quantities by varying the magnitude of f = g. The plot +explains the regime of extreme events in the following +way. During the non-extreme regime, the critical thresh- +old xEE is larger than the xmax. +It means that the +threshold is larger than the large amplitude oscillations +and does not satisfy the extreme events criterion. While +in the EE regime, the xmax is larger than the EE crit- +ical threshold xEE (shaded EE region). +This explains +that extreme events have a larger amplitude than the ex- +treme event criterion. Note that the SNA regime in the + +(a) +0.32 +0.02 +0.0 +0.29 +0.02 +0.04 +0.26 +-0.06 +0.08 +0.23 +0.23 +0.26 +0.29 +0.32 +-0.10 +(b) 0.03 +Chaotic +0.00 +-0.03 +Quasi-Periodic +SNA +-0.06 +0.23 +0.26 +0.29 +0.32 +12 +(c) +Chaotic +10 +E +XEE +8 +Quasi-Periodic +Xmax +6 +Z +N=5.6114 +N=4 +4 +N. +2 +0.23 +0.26 +0.29 +0.324 +parameter range f ∈ 0.28 +to 0.2912 shows no extreme +events. As we discussed above, we fixed N = 4 as an +arbitrary constant from the literature [30]. However, the +maximum value of the N can be determined by rewriting +Eq. (2), as +Nmax = max(xEE) − ⟨xn⟩ +σxn +. +(3) +In the SNA region shown in Fig. 2(c), we found that the +SNA +QP1 +QP2 +QP3 +x0 +y0 +FIG. 3: Basin of attraction for f = g = 0.278. QP1, QP2, +and QP3 are the quasi-periodic attractor-1, quasi-periodic +attractor-2, and quasi-periodic attractor-3, respectively. SNA +represents the strange nonchaotic attractor. +We fixed the +other parameter values the same as in Fig. 1. +multiplication factor taking values between 4 ≤ Nmax ≤ +5.611 when the forcing amplitudes in the range from +0.256 to 0.28 denoted by shaded transparent pattern. +The plot of Nmax shows that depending on the parameter +choice, the arbitrary value can be chosen N∈ {4, 5.611}. +Thus above results satisfy all the criteria proposed for +the extreme events and justify the existence of EEs in +the SNA regime. +As we discussed earlier, the observed EEs are non- +chaotic and nonperiodic. At the same time, the param- +eters corresponding to the strange nonchaotic EEs show +multiple stable behaviors. The multi-stable behavior can +be seen from the basins of attraction drawn for a range +of initial conditions. Figure 3 is drawn by varying the +initial states x0 and y0 of the system for the parameters +given in Fig. 1 caption. We can see that basin of non- +chaotic and nonperiodic behavior or SNA is embedded +within the basin of quasi-periodic dynamics. Outside the +SNA basin, we have found three different basins which +contain quasi-periodic attractors. +All the three differ- +ent quasi-periodic attractor basins and the SNA basin, +denoted by QP1, QP2, QP3, and SNA respectively in +Fig. 3. In supplemental material Fig. (S2), each of the +quasi-periodic attractors is depicted. Figure 3 shows that +extreme events occur for specific values of initial condi- +tions. The size of these basins changes as we vary the +parameter within the EEs regime marked in Fig. 2. +EE +NEE +f +g +FIG. 4: Two parameter phase diagram in (f, g) space (plot- +ted using Eq. 2 for fixed initial condition (x0, y0) = (0.3, 0.2)), +to distinguish the existence of extreme events (EE) and non- +extreme events (NEE), respectively. We fixed the other pa- +rameter values as the same as in Fig. 1. +Similarly, to determine the regime of the extreme +event in the parametric space between f and g, a two- +parameter diagram is drawn as shown in Fig. 4. +The +white regime in the plot shows the extreme events for the +combinations of parameter (f, g) separated with the help +of Eq. +2 from the non-extreme events (NEE– denoted +by blue color). By comparing Fig. 2(a) with Fig. 4 we +can say that EEs occur in the SNA region (however some +of the SNA parameter regime may not contain EEs). +-20 +30 +-60 +10 +80 +lm[x(α,N)] +Re[x(α,N)] +(b) +0 +2.7 +5.5 +3 +5 +7 +log10|x(α,N)|2 +log10 N +(a)(a) +(b) +FIG. 5: +Singular continuous spectrum for fixing the forc- +ing amplitudes f, g = 0.278. +(a) The logarithmic plot of +|x(α, N)|2 against N. +The red and black lines denote the +numerical values and the corresponding power-law fit. +(b) +Fractal path in the complex plane of x. The other parameter +values are defined as γ = 0.35, ω1 = 0.3, ω2 = ( +√ +5−1 +2 +). +To show the generality of the existence of EEs in the +SNA regime, we present the regime of EEs for γ = 0.4 +in the supplementary material Figs. S3 (a),(b). This re- +sult validates the presence of strange nonchaotic extreme +events in the selected parameter regime. In the following +section, we characterize the observed behavior as strange +and nonchaotic in nature. For this purpose, we perform + +10 +5 +0 +-5 +-10 +-15 +-20 +-5 +0 +5 +10 +15 +200.32 +0.29 +0.26 +0.23 +0.23 +0.26 +0.29 +0.325 +singular continuous spectrum analysis and distribution +of finite-time Lyapunov exponents. +To validate the strange nonchaotic dynamics, we plot +singular continuous spectrum [31] in Fig. 5 using par- +tial Fourier sum of the signal x given by X(α, N) = +�N +m=1 xme2πimα, where α is proportional to the exter- +nal frequency (ω1) and N is the length of the time series. +The red and black lines show the singular continuous +spectrum and the corresponding power-law fit. When N +is considered as time, |X(α, N)|2 grows with N, that is +|X(α, N)|2 ∼ N β, where β is the slope. When the signal +possesses the properties of strange nonchaotic dynamics, +the corresponding slope values lie between 1 < β < 2. +For this case, the slope value β = 1.576 confirms the exis- +tence of strange nonchaotic dynamics shown in Fig. 5(a). +The corresponding path of Brownian motion with fractal +structure in complex [Re(x), Im(x)] plane also confirms +the strange nonchaotic dynamics in Fig. 5(b). + 0.0001 + 0.001 + 0.01 + 0.1 + 1 +-0.08 -0.06 -0.04 -0.02 + 0 + 0.02 0.04 +P +–λ +FIG. 6: Finite time Lyapunov exponent with respect to prob- +ability distribution function (PDF) for SNAs by fixing the +three distinct finite time periods T = 500 (red line), T = 1000 +(blue dashed line), and T = 1500 (black dotted line) with +f, g = 0.278. +The strange nonchaotic dynamics are also validated +using another statistical characterization known as the +distribution of finite-time Lyapunov exponents. The dis- +tribution takes both positive and negative values, but +the area under the curve is maximum in the negative +regime for strange nonchaotic dynamics. Figure 6 plot- +ted for three different finite time intervals T = 500, 1000, +and 1500, the distribution has a large negative region +compared to the positive region showing nonchaotic dy- +namics. From these analyses, the observed dynamics are +strange (nonperiodic) as well as nonchaotic, which also +shows the large amplitude and rare events. +The present letter shows a mechanism of the emergence +of extreme events in a quasi-periodically forced Morse os- +cillator. As a function of forcing amplitude, we found the +transition from quasi-periodic (QP) to chaotic (CH) at- +tractor via strange nonchaotic extreme events. During +such extreme event dynamics, we found a long excur- +sion of trajectories that are away from the bounded at- +tractor, while the chaotic attractors show many higher +amplitude peaks. To confirm the existence of EEs, we +estimated the critical threshold, and it is observed that +the higher amplitude peaks in the EE cross the critical +threshold while the peaks in the CH and QP attractor do +not. The dynamical transitions of the attractors and the +occurrence of nonchaotic EE dynamics are manifested +through maximum Lyapunov exponents. The observed +extreme events are further validated using the probabil- +ity distribution and return interval (inter-event interval) +with respect to the probability of recurrence times of the +EE. Extreme events are abnormal and unexpected events +that occur in many natural and man-made systems. Un- +derstanding the mechanism or route can help to antici- +pate the onset of EEs. Early works on extreme events +show the chaotic nature of the extreme events because +of the rare and extreme amplitude properties of extreme +events. The present study shows an unknown emergence +of extreme events that are nonchaotic and nonperiodic +extreme events. This finding shed light on the new direc- +tion where extreme events can happen as a nonchaotic +process. +We gratefully acknowledge this work is funded by +the Center for Nonlinear Systems, Chennai Institute +of +Technology +(CIT), +India, +vide +funding +number +CIT/CNS/2022/RP-016. +[1] S. +Albeverio, +V. +Jentsch +and +H. +Kantz, +Extreme +Events in Nature and Society, The Frontiers Collection +(Springer, Berlin, 2006). +[2] R. H¨ohmann, U. Kuhl, H. J. St¨ockmann, L. Kaplan and +E. J. Heller, Phys. Rev. Lett., 104, (2010) 093901. +[3] J. J. Metzger, R. Fleischmann and T. Geisel, Phys. Rev. +Lett., 112, (2014) 203903. +[4] A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty and +J. M. Dudley, Sci. Rep., 5, (2015) 12822. +[5] S. Birkholz, C. Br´ee, I. Veseli´c, A. Demircan and G. +Steinmeyer, Sci. Rep., 6, (2016) 35207. +[6] D. Premraj, K. Suresh, S. A. Pawar, L. 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A, 27, (1994) +5209. + diff --git a/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/load_file.txt b/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2535af3d4135f1cb985dae1dd6501f3968a59a1 --- /dev/null +++ b/pdFQT4oBgHgl3EQfsDaJ/content/tmp_files/load_file.txt @@ -0,0 +1,587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf,len=586 +page_content='Emergence of extreme events in a quasi-periodic oscillator Premraj Durairaj1, Sathiyadevi Kanagaraj1, Suresh Kumarasamy1, Karthikeyan Rajagopal1,2 1Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai-600 069, Tamilnadu, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2Department of Electronics and Communications Engineering, University Centre for Research and Development, Chandigarh University, Mohali, 140 413, Punjab, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (Dated: February 1, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Received :) Extreme events are unusual and rare large-amplitude fluctuations that occur can unexpectedly in nonlinear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Events above the extreme event threshold of the probability distri- bution of a nonlinear process characterize extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Different mechanisms for the generation of extreme events and their prediction measures have been reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Based on the properties of extreme events, such as rare in the frequency of occurrence and extreme in am- plitude, various studies have shown that extreme events are both linear and nonlinear in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Interestingly, in this work, we report on a special class of extreme events which are nonchaotic and nonperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' These nonchaotic extreme events appear in between the quasi-periodic and chaotic dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We report the existence of such extreme events with various statistical measures and characterization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' PACS numbers: 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='-a Extreme events are unanticipated, rare events that oc- cur in many natural and engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Extreme events (EE) can exist in various forms, including floods, cyclones, droughts, pandemics, power outages, material ruptures, explosions, chemical contamination, and stock market crashes, among others [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Such events have a severe impact on real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Thus, it is nec- essary to understand the relevant mechanism and its generic characteristics for the occurrence of EE in order to prevent such EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' As a result, the researchers focused on exploring the EE in diverse nonlinear oscillators [2–5], maps [6], and neural networks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Further, the extreme events have also been identified in a super-fluid helium [8], plasma[9], optical fibers [10], lasers [11], and capillary wave [12] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' However, depending on the characteristics of a dynam- ical system, the occurrence of EE has been discovered under a variety of mechanisms, including internal crises, on-off intermittency, blowout bifurcations, stick-slip bi- furcations, and so on [6, 11, 13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' For instance, prior studies reveal that EE can arise as a result of the abrupt expansion and destruction of chaotic attractors produced by internal or external crises [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Further, interior crises are found to be a critical mechanism for the oc- currence of EE, when the trajectory of chaotic attractors reaches the stable manifold of a saddle or unstable peri- odic orbit, which increases the size of the chaotic attrac- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Such a sudden expansion of the chaotic attractor may result in EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In addition, Pomeau-Manneville in- termittency is identified as another mechanism for the existence of EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Such intermittency can occur when the periodic oscillations are interspersed by chaotic bursts, which further results in very large amplitude events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' EEs can also exist through the following other mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The sliding bifurcation near the discontinuous boundary can cause EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The trajectory of the attractors might hop between coexisting attractors due to noise in multi-stable systems, which can cause unusually large events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' This is referred to as noise-induced intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The trajec- tory of the attractors in coupled systems departs from the synchronization manifold to the transverse direction of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' During such a transition, a synchroniza- tion error of dynamics can show zero or nonzero and is referred to as on-off intermittency [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Moreover, previous studies discovered that extreme or rare events can occur as a result of chaotic or stochas- tic processes [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In particular, the appearance of EE has been reported in micro-electromechanical can- tilevers with discontinuous boundaries and diode lasers with phase-conjugate feedback [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' By applying the harmonic pump modulation to the fiber laser the emer- gence of Rogue waves has been identified [2, 19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The EE in stochastic transport on networks has been demon- strated using multiple random walks on complex net- works [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Now the interesting question is whether extreme events can be induced by nonchaotic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In literature, a study has shown nonchaotic and non- periodic have been well studied in the name of strange nonchaotic dynamics, which arises during the attractor transition from quasi-periodicity to chaos [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' One can find the generation mechanisms of these strange non- chaotic attractors in literature [25, 26, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The results in the present work show that similar to the strange nonchaotic dynamics, the nonperiodic and nonchaotic dynamics show large-amplitude extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The present study opens a new area of study where the non- chaotic nonlinear process can also lead to extreme events and the same has not been found reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To show the nonchaotic extreme events, we consider the Morse Oscillator (MO) which is used to describe the motion of diatomic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Importantly, the MO has made substantial contributions in the fields of classical, semi-classical, and quantum mechanics [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The MO was used for photo-dissociation molecules without any damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In the presence of driving and damp- ing, the MO was exploited for multi-photon excitation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='13386v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='data-an] 31 Jan 2023 2 of molecules, pumping the local mode of polyatomic molecules [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We consider the quasi-periodically forced MO and its dynamical equation can be written as ˙x = y ˙y = fsin(ω1t) + gsin(ω2t) + e−2x − e−x − γy (1) where x, and y are the state variables of the system and γ is a damping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The amplitudes of the first and second force are represented by f and g and the corresponding frequencies are denoted by ω1 and ω2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1: (a) Time evolution of xn for the nonchaotic dynamics with forcing amplitudes as (a) f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='255, and (b) f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The xn is the nth local peaks of the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The horizontal black dot-dashed and red dashed lines are the critical threshold lines defining the extreme events (refer to the text for the meaning of N and A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (c) The probability distribution function corresponds to the extreme events and (d) return interval (R) (inter-event interval) with respect to the probability of recurrence times (PR) of the EE for (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The filled circles and solid lines in (d) represent the numerical data and the corresponding power-law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We fixed the other parameter values as γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='35, ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='3, and ω2 = ( √ 5−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To manifest the existence of extreme events, we first depicted the time evolution of the x-variable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(b) by fixing the amplitude of the first and sec- ond forcing as f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='255 and f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(a) that some of the oscillation(event) has larger amplitudes, while the rest of them take lower amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To check the larger amplitude oscillations satisfy the extreme events criteria defined in the litera- ture, we use the following relation: xEE =< xn > +Nσxn, (2) where xEE is the critical amplitude threshold and N is a multiplication factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The mean and standard deviation of the variable x is represented by < xn > and σxn, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Here, the xn (an event) are the local peaks of the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' An event or a local peak can satisfy extreme event criteria if it has a value higher than the critical threshold defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (2) with N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To con- firm the presence of EE, we plotted the critical threshold on the time series for N = 5 and N = 4 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(a) and 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We used two different N values depending on the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Though the choice of N is arbitrary, we set the minimum N value as 4 in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We also find the critical value of Nmax for a range of each f value – the details will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In both cases, we can see that some of the large amplitude events cross the threshold line, confirming the presence of EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Since the choice of N is arbitrary in the previous criterion, we use another criterion defined by the abnormality index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' An = Hfn H1/3 [17], where Hfn is the difference between the maximum height of the event n and the mean height of its population, Hfn = xn − ⟨xn⟩n and H1/3 is the av- erage value among the highest one-third values of Hfn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' If an event xn has abnormality An greater than 2 then the event is termed an extreme event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We find that both cases in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(a) and 1(b) satisfy the above criterion with abnormality index A = 3 denoted by a dashed hori- zontal line in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' It is evident that a few rare large amplitude events cross the abnormality index line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We computed the probability distribution function (PDF) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(c) for the time series shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The EE critical threshold at N = 4 is plotted as a vertical dashed line on the PDF diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In the plot, the events with a finite probability above the critical threshold line char- acterize the extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We can plot similar proba- bility distribution for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(a), however, for simplicity, we have plotted the PDF corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The above analysis shows that the observed behavior satisfies the extreme events criterion in the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Another important characteristic of extreme events is an inter-event interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The inter-event interval defines the frequency occurrence of the events and should not have discrete values (discrete values mean the periodic occurrence of events), rather it should have a distribu- tion over a range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In order to examine the distribution of events in the observed time series, we find inter-event intervals (R) between successive extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Subse- quently, we find the probability of such inter-event in- tervals (PR) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Inter-event interval and its probability obey power-law relations as given by log10(PR) = a log10(R)b, where a and b are constants with values a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='006 and b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='96, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The obtained numerical values are depicted in a filled circle, (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 N=5 A=3 X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='30×106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='60×106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 N=4 X 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='50×105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='00×105 (c) n (d) 2 N= 4: log R vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' log PR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='1 Power law PDF PR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='01 log 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0001 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='7 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' log R3 and a continuous line shows the corresponding power- law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The route for the emergence of EE and its tran- sitions is further estimated below using Lyapunov ex- ponents (LE), amplitude maxima Xmax, critical factor Nmax, and two-parameter analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2: (a) The two-parameter bifurcation diagram in (f, g) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Using the range of Lyapunov exponents (λ) (denoted by the color bar) the dynamical regions are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (b) The maximum Lyapunov exponents as a function of forcing am- plitude f(= g), (c) maximum amplitude of the events xmax (red) and the corresponding Nmax (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 3) of the event (blue) by varying the magnitude of f(= g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The black line repre- sents the extreme events critical threshold (xEE) drawn from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (2) for N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The other parameter values are fixed as the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To illustrate the global dynamical transition of the at- tractors and route of the EE, the two-parameter diagram is drawn in (f, g) space using the maximum LE as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The range of LE (shown in the color bar) denotes the emergence of quasi-periodic, nonchaotic, and chaotic attractors in the respective parameters of f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' If the forcing amplitudes f and g are small, attractors have a maximum negative LE, indicating the presence of a quasi-periodic (QP) attractor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To better com- prehend QP attractors, we plotted their time-evolution and phase portrait trajectories in Supplementary Mate- rial Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S1 a(i, ii) for f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23, which show their bounded nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Thus, the EE critical threshold for this attractor is greater than the amplitude of QP attrac- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' By increasing f and g values, the QP attractor transits to a chaotic (CH) attractor via strange and non- chaotic dynamics in which the LE takes the values from negative (near zero) to positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To distinguish between the strange nonchaotic and chaotic attractors, the time- evolution and phase portrait trajectories are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S1 b(i,ii) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S1 c(i,ii) in the supplementary materials by fixing f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278 and f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='33, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Also, the frequency spectra can be used to distinguish quasiperiodic, SNA and chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We have the frequency spectrum analysis in the Supplementary material in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S4 (a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' When compared to the chaotic attractor (which has a greater number of large ampli- tude oscillations), we found the SNA shows fewer large amplitude oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The supplemental material’s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S1 can be consulted for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Furthermore, to show the dynamical transitions clearly, we displayed maximum Lyapunov exponents in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(b) by keeping the parameter (f = g) and varying it along the diagonal dashed line shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(b), the maxi- mum LE is illustrated as a function of forcing amplitudes f and g (f = g) in the range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 < f(= g) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We observe that when the forcing amplitudes are minimum in the mentioned range, LE takes negative values, in- dicating quasi-periodic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' While increasing the parameter, the transition of LE from negative to posi- tive values indicates the dynamical transition of quasi- periodic behavior to chaotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Furthermore, we found that the negative values of LE near-zero exhibit strange nonchaotic behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' extreme events are seen in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The literature has shown that the EEs occur under chaotic dynamics [16] through distinct routes and stochastic processes like stochastic transport on networks has been demonstrated using multiple random walks on complex networks [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Among the various routes, the occurrence of EEs in nonchaotic dynamics is new and it has not been reported to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To validate the occurrence of EEs in the SNA region, we find the maximum amplitude xmax, extreme event threshold xEE, and maximum value of N (Nmax) of a given time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(c), we have plotted the above quantities by varying the magnitude of f = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The plot explains the regime of extreme events in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' During the non-extreme regime, the critical thresh- old xEE is larger than the xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' It means that the threshold is larger than the large amplitude oscillations and does not satisfy the extreme events criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' While in the EE regime, the xmax is larger than the EE crit- ical threshold xEE (shaded EE region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' This explains that extreme events have a larger amplitude than the ex- treme event criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Note that the SNA regime in the (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='03 Chaotic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='03 Quasi-Periodic SNA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='32 12 (c) Chaotic 10 E XEE 8 Quasi-Periodic Xmax 6 Z N=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='6114 N=4 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='324 parameter range f ∈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='28 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='2912 shows no extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' As we discussed above, we fixed N = 4 as an arbitrary constant from the literature [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' However, the maximum value of the N can be determined by rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (2), as Nmax = max(xEE) − ⟨xn⟩ σxn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (3) In the SNA region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(c), we found that the SNA QP1 QP2 QP3 x0 y0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 3: Basin of attraction for f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' QP1, QP2, and QP3 are the quasi-periodic attractor-1, quasi-periodic attractor-2, and quasi-periodic attractor-3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' SNA represents the strange nonchaotic attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We fixed the other parameter values the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' multiplication factor taking values between 4 ≤ Nmax ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='611 when the forcing amplitudes in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='256 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='28 denoted by shaded transparent pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The plot of Nmax shows that depending on the parameter choice, the arbitrary value can be chosen N∈ {4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='611}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Thus above results satisfy all the criteria proposed for the extreme events and justify the existence of EEs in the SNA regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' As we discussed earlier, the observed EEs are non- chaotic and nonperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' At the same time, the param- eters corresponding to the strange nonchaotic EEs show multiple stable behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The multi-stable behavior can be seen from the basins of attraction drawn for a range of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Figure 3 is drawn by varying the initial states x0 and y0 of the system for the parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1 caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We can see that basin of non- chaotic and nonperiodic behavior or SNA is embedded within the basin of quasi-periodic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Outside the SNA basin, we have found three different basins which contain quasi-periodic attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' All the three differ- ent quasi-periodic attractor basins and the SNA basin, denoted by QP1, QP2, QP3, and SNA respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In supplemental material Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (S2), each of the quasi-periodic attractors is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Figure 3 shows that extreme events occur for specific values of initial condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The size of these basins changes as we vary the parameter within the EEs regime marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' EE NEE f g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 4: Two parameter phase diagram in (f, g) space (plot- ted using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2 for fixed initial condition (x0, y0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='2)), to distinguish the existence of extreme events (EE) and non- extreme events (NEE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We fixed the other pa- rameter values as the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Similarly, to determine the regime of the extreme event in the parametric space between f and g, a two- parameter diagram is drawn as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The white regime in the plot shows the extreme events for the combinations of parameter (f, g) separated with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2 from the non-extreme events (NEE– denoted by blue color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 2(a) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 4 we can say that EEs occur in the SNA region (however some of the SNA parameter regime may not contain EEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 20 30 60 10 80 lm[x(α,N)] Re[x(α,N)] (b) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='5 3 5 7 log10|x(α,N)|2 log10 N (a)(a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 5: Singular continuous spectrum for fixing the forc- ing amplitudes f, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (a) The logarithmic plot of |x(α, N)|2 against N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The red and black lines denote the numerical values and the corresponding power-law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' (b) Fractal path in the complex plane of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The other parameter values are defined as γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='35, ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='3, ω2 = ( √ 5−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To show the generality of the existence of EEs in the SNA regime, we present the regime of EEs for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='4 in the supplementary material Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' S3 (a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' This re- sult validates the presence of strange nonchaotic extreme events in the selected parameter regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' In the following section, we characterize the observed behavior as strange and nonchaotic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' For this purpose, we perform 10 5 0 5 10 15 20 5 0 5 10 15 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='325 singular continuous spectrum analysis and distribution of finite-time Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To validate the strange nonchaotic dynamics, we plot singular continuous spectrum [31] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 5 using par- tial Fourier sum of the signal x given by X(α, N) = �N m=1 xme2πimα, where α is proportional to the exter- nal frequency (ω1) and N is the length of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The red and black lines show the singular continuous spectrum and the corresponding power-law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' When N is considered as time, |X(α, N)|2 grows with N, that is |X(α, N)|2 ∼ N β, where β is the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' When the signal possesses the properties of strange nonchaotic dynamics, the corresponding slope values lie between 1 < β < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' For this case, the slope value β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='576 confirms the exis- tence of strange nonchaotic dynamics shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The corresponding path of Brownian motion with fractal structure in complex [Re(x), Im(x)] plane also confirms the strange nonchaotic dynamics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='08 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='04 P –λ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' 6: Finite time Lyapunov exponent with respect to prob- ability distribution function (PDF) for SNAs by fixing the three distinct finite time periods T = 500 (red line), T = 1000 (blue dashed line), and T = 1500 (black dotted line) with f, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content='278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The strange nonchaotic dynamics are also validated using another statistical characterization known as the distribution of finite-time Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The dis- tribution takes both positive and negative values, but the area under the curve is maximum in the negative regime for strange nonchaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Figure 6 plot- ted for three different finite time intervals T = 500, 1000, and 1500, the distribution has a large negative region compared to the positive region showing nonchaotic dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' From these analyses, the observed dynamics are strange (nonperiodic) as well as nonchaotic, which also shows the large amplitude and rare events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The present letter shows a mechanism of the emergence of extreme events in a quasi-periodically forced Morse os- cillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' As a function of forcing amplitude, we found the transition from quasi-periodic (QP) to chaotic (CH) at- tractor via strange nonchaotic extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' During such extreme event dynamics, we found a long excur- sion of trajectories that are away from the bounded at- tractor, while the chaotic attractors show many higher amplitude peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' To confirm the existence of EEs, we estimated the critical threshold, and it is observed that the higher amplitude peaks in the EE cross the critical threshold while the peaks in the CH and QP attractor do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The dynamical transitions of the attractors and the occurrence of nonchaotic EE dynamics are manifested through maximum Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The observed extreme events are further validated using the probabil- ity distribution and return interval (inter-event interval) with respect to the probability of recurrence times of the EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Extreme events are abnormal and unexpected events that occur in many natural and man-made systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Un- derstanding the mechanism or route can help to antici- pate the onset of EEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Early works on extreme events show the chaotic nature of the extreme events because of the rare and extreme amplitude properties of extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' The present study shows an unknown emergence of extreme events that are nonchaotic and nonperiodic extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' This finding shed light on the new direc- tion where extreme events can happen as a nonchaotic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' We gratefully acknowledge this work is funded by the Center for Nonlinear Systems, Chennai Institute of Technology (CIT), India, vide funding number CIT/CNS/2022/RP-016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' Albeverio, V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} +page_content=' A, 27, (1994) 5209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFQT4oBgHgl3EQfsDaJ/content/2301.13386v1.pdf'} diff --git a/ptAzT4oBgHgl3EQf5v4s/vector_store/index.faiss b/ptAzT4oBgHgl3EQf5v4s/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ab0f03d8663f057ff57ca2fde93b99f67ecc7bc4 --- /dev/null +++ b/ptAzT4oBgHgl3EQf5v4s/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7fcff498daab768ddbba5eb13fafa390c2f03b123d8eb42b7c5db046c587c9dc +size 3145773 diff --git a/rdE1T4oBgHgl3EQf2wXf/content/tmp_files/2301.03483v1.pdf.txt b/rdE1T4oBgHgl3EQf2wXf/content/tmp_files/2301.03483v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc43b5ead12bf0fda5bcb2d383793b14764c5ba2 --- /dev/null +++ b/rdE1T4oBgHgl3EQf2wXf/content/tmp_files/2301.03483v1.pdf.txt @@ -0,0 +1,1081 @@ +arXiv:2301.03483v1 [hep-ph] 9 Jan 2023 +Is d∗(2380) a compact hexaquark state? +ManYing Pana,∗ Xinmei Zhub,† and Jialun Pinga‡ +aDepartment of Physics and Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, +Nanjing Normal University, Nanjing 210023, People’s Republic of China and +bDepartment of Physics, Yangzhou University, Yangzhou 225009, P.R. China +The most fascinating dibaryon in the non-strange quark sector is d∗(2380), which was reported +by WASA-at-COSY Collaboration and confirmed by A2@MAMI Collaboration. The reported mass +and width are M ≈ 2.37 GeV, Γ ≈ 70 MeV and the quantum numbers IJP = 03+. The structure of +d∗(2380) is still in controversy. In the present calculation, the powerful method in few-body system, +Gaussian expansion method is employed to explore the structure of d∗(2380) in the framework of +constituent quark models without assuming the presupposed structure. The results show that the +radius of d∗(2380) is around 0.7 fm, a very compact object. Because of the compact structure, the +color singlet-singlet component has a large overlap with the color octet-octet one, two colorless, +large overlapped ∆s dominate the state is possible. +I. +INTRODUCTION +In addition to the popular “XYZ” particles and the +hidden charm pentaquarks, the dibaryon states are also +important exotic hadron states. +Generally, any object +with a baryon number B = 2 can be called a dibaryon. +Since the baryon number of each quark is 1/3, the +dibaryon is composed of six valence quarks. +The pro- +posal of looking for dibaryons was in the same year as the +publication of quark model by Gell-Mann [1]. In 1964, +based on SU(6) symmetry of strong interaction, Dyson +and Xuong predicted the possible existence of dibaryon +states and obtained the mass of these particles by a mass +formula [2], the predicted mass of D03 is surprisingly close +to that of d∗(2380) later found [3–9]. +Deuteron is a state with B = 2, which was discovered +by Urey, Brickwedde and Murphy in 1932 [10]. It is a +loosely bound state of proton and neutron with quan- +tum numbers IJ = 01. +At the quark level, the con- +tent of deuteron is uuuddd, these six quarks could also +make up ∆∆, so whether the deuteron contains a non- +nucleon component is a meaningful subject [11, 12]. Due +to the large separation between proton and neutron in +deuteron [13], it can safely be regarded as a molecule. +Of course, the dibaryon state may also be a more ex- +otic compact six quark structure, that is, the state can- +not represented by two well separated color singlet quark +clusters. It is more interesting because it is a new form +of matter. +d∗ with quantum numbers IJ = 03 is ex- +pected to be a compact object in quark model calcu- +lations. +Dibaryon search has experienced a long and +eventful history, there are many twists and turns dur- +ing the searches of dibaryons, A comprehensive review +of dibaryons can be found in the references [14, 15], the +status of d∗ can be seen in [16]. +After the initial study of d∗ in 1964, the further study +∗E-mail: 211001005@stu.njnu.edu.cn +†E-mail: zxm yz@126.com +‡E-mail: jlping@njnu.edu.cn (corresponding author) +of dibaryon state related to d∗ was traced back to 1977. +Inspired by the anomalous results of proton polarization +in the γd → pn reaction [17], Kamae and Fujita inves- +tigated the possible existence of deep bound dibaryon +state, in which they calculated the ∆∆ atate with quan- +tum numbers IJ = 03 and IJ = 30 using the non rela- +tivistic one boson exchange model, and obtained a bind- +ing energy of about 100 MeV [18]. In fact, the experi- +mental results are coincide with the present results of the +state d∗ [3–9]. +In 1989, Goldman et al proposed “an inevitable non- +strange dibaryon” based on the basic idea of constituent +quark model [19], which was named d∗. The following +realistic calculations in quark delocalization and color +screening model confirmed the prediction of d∗ [20–22]. +In the framework of chiral quark model, The results +shown that there are attractions between two ∆’s, the +dynamical calculation with the help of the resonating +group method (RGM) obtained small binding energy, +22.2-64.8 MeV for d∗ and a compact structure, the root- +mean-square radius (RMS) is about 0.84-1.01 fm [23]. +To guide the experimental searching, a nucleon-nucleon +(NN) scattering phase shifts calculation including d∗ +was performed, the phase shifts of D-wave NN scat- +tering show a clear resonance structure, with the mass +2273-2404 MeV and width 33-149 MeV [24]. The break- +through comes in 2009, CELSIUS/WASA-at-COSY Col- +laboration reported their results on double pionic fusion +reaction pn → dπ0π0, a resonance with mass and width +2.36 GeV and 80 MeV is needed to describe the exper- +imental data [3]. the subsequent series of experiments +confirmed the resonance and fixed quantum numbers. +The updated results are the resonance mass is around +2.37 MeV, the decay width is about 70 MeV and quan- +tum numbers are IJP = 03+. It is a dibaryon d∗, a spin +excitation of deuteron. Especially, the re-analysis of NN +scattering amplitude in 3D3-3G3 partial waves by incor- +porating new data suggest a pole which corresponding to +d∗ [7]. The theoretical study a d∗ resonance in the cou- +pled 3D3-3G3 partial waves of nucleon-nucleon scattering +reproduced the experimental data [25]. The recent po- +larization experiment of A2 Collaboration at MAMI also + +2 +TABLE I: The transformation coefficients between physical +bases and symmetry bases. [ν] and [µ] denote the symmetry +of orbital and spin-flavor for six-quark systems. +[ν][µ] = [6][33] +[ν][µ] = [42][33] +∆∆ +− +� +1/5 +− +� +4/5 +CC +− +� +4/5 +� +1/5 +find signatures of the d∗(2380) hexaquark in d(γ,p⃗n) [26]. +After the experiment discovery, more researches are de- +voted to the structure and the narrow decay width of d∗. +To understand the narrow decay width of d∗, the assump- +tion that the dominant component of d∗ is hidden color +channel was proposed [27]. The assumption comes from +the transformation between the physical bases (denoted +by two q3 state) and the symmetry bases (denoted by the +orbital symmetry and isospin-spin symmetry) [28, 29]. +From the table, one can see that if the orbital symmetry +of d∗ is [6] in the symmetry bases, all six quarks occupy +the same orbital state, then in the physical bases, the +hidden color channel (CC) is the dominant component +(80%). Really in the resonating-group-method (RGM) +approach, by including the hidden-color channel (CC), +the calculation of d∗ in chiral quark model gave that d∗ +has a mass of about 2.38 − 2.42 GeV and a root-mean- +square radius (RMS) of about 0.76 − 0.88 fm, and the +fraction of CC component in the d∗ is found to be about +66%-68% [30]. +However, there is a mis-understanding +of the above transformation table. If the orbital sym- +metry of six-quark state d∗ is [6], then only one basis +in the symmetry bases scheme is available, the corre- +sponding available physical basis must be one, too, the +color-singlet channel ∆∆ is the same as the hidden-color +channel CC. In the RGM approach, the overlap between +∆∆ and CC is about 1 when the separation between two +clusters are small, for example ⟨∆∆|CC⟩ = 0.98 with +separation s = 0.5 fm. +Very recently, Huang showed +a revised quark model investigation of d∗(2380) [31], he +pointed out that there are some inadequacies in the pre- +vious quark model calculations, it would be imprecise +to set size parameter to be same for all the considered +baryons. In the updated the chiral quark model calcu- +lation, he found the effects of hidden-color channel are +much less important, which is different from their previ- +ous work [30]. +As for the structure of d∗, the most quark model cal- +culation show that it is compact object [23, 24, 32]. +However, in the three-body Faddeev equation approach +of πN∆, the extended object is invoked to explain +d∗ [33, 34]. +In lattice QCD approach, the similar re- +sults with that of quark model calculations are obtained, +the short-range strong attraction between two ∆’s leads +to the quasi-bound states with compact structure [35]. +In quark model calculations, RGM is often employed. +It is an approximation method for few-body system, in +which the system is separated into two sub-clusters and +the structures of the sub-clusters are frozen in the dy- +namical calculation. In this way, the few-body problem +was turned into two-body one. +It is expected to be a +good approximation in nucleon-nucleon scattering study. +It maybe not suitable for studying the structure and the +percentage of hidden-color channel in d∗. In the present +work, the powerful few-body method, gaussian expansion +method (GEM) [36] is invoked to determine the contri- +bution of hidden-color channels and root-mean-square ra- +dius of d∗. +This paper is structured as follows: Sec. +II briefly +introduced the quark models, the construction of hex- +aquark wave functions and GEM. The calculated results +and discussions are presented in Sec. III. The summary +of our investigation is given in the last section. +II. +MODELS AND WAVE FUNCTIONS +To check the model dependence of the calculation, two +quark models are used, one is the na´ıve quark model, +another is the chiral quark model. The calculations are +limited to the ground states, so only the central parts of +Hamiltonian are given below. +A. +Naive Quark Model +In the na´ıive quark model, the Hamiltonian includes +kinetic energy term, color confinement potential and one +gluon exchange potential, which can be written as: +H = +6 +� +i=1 +� +mi + p2 +i +2mi +� +− TCM + +6 +� +j>i=1 +Vij, +(1) +Vij = V C +ij + V G +ij , +V C +ij += −acλc +i · λc +j(r2 +ij + V0), +(2) +V G +ij += αs +4 λc +i · λc +j +� 1 +rij +− σi · σj +6mimj +e−rij/r0(µ) +rijr2 +0(µ) +� +, +(3) +r0(µ) = ˆr0/µ. +Where TCM is center of mass kinetic energy, other sym- +bols have their usual meanings. +B. +Chiral Quark Model +Chiral quark model was setup based on the dynamic +breaking of chiral symmetry [37]. +In addition to the +color confinement and one-gluon-exchange potentials, the +Goldstone boson and its chiral partner exchange poten- +tials are invoked. The Hamiltonian in chiral quark model +is written as [38]: + +3 +H = +6 +� +i=1 +� +mi + p2 +i +2mi +� +− TCM + +� +i 0 represents the privacy protection level, and δ stands for the probability to break ϵ-DP. +A smaller ϵ means that it is more difficult to distinguish between the two neighboring datasets D and D′, +thus resulting in stronger privacy protection. It should be noted that ϵ is also referred to as the privacy +loss, which quantifies the amount of privacy compromised through a single round of information (model +parameters) exchange with the PS. In practical FL systems, it is crucial to monitor the total privacy loss +over multiple communication rounds of model parameters exchange with the PS. Though its computation +is quite involved, its upper bound can be computed using the moment accountant method [36], which so +far yields the tightest bound on the total privacy loss. +Lemma 1 +[10, Theorem 3.22] Suppose the query function g accesses the dataset D via randomized +mechanism M. Then, the noise scale added to query function g to guarantee +� +ϵ, δ +� +-DP is given by +σ = 2s2 ln(1.25/δ) +ϵ2 +, +(2) +where s is the ℓ2-norm sensitivity of the function g defined by +s ≜ max +D,D′ +��g(D) − g +� +D′���. +(3) +According to the privacy amplification theorem [21], it has been known that, running on a randomly +generated subset of a dataset, the DP mechanism can yield stronger privacy protection than running on +the entire dataset. This fact implies that the noise variance required for achieving a predefined DP level +can be reduced when partial data are randomly selected at each iteration. The privacy analysis to be +addressed in Section IV-B relies on the following privacy amplification theorem. +Theorem 1 (Privacy Amplification Theorem +[21]) Suppose that a mechanism M is (ϵ, δ)-DP over a +given dataset D with size n. Consider the subsampling mechanism that outputs a sample from the uniform +distribution over all subsets Ds ⊆ D with size b. Then, when ϵ ≤ 1, executing M mechanism on the +subset Ds guarantees (ϵ′, δ′)-DP, where ϵ′ and δ′ are given by +ϵ′ = 2qϵ, δ′ = qδ, +(4) +where q = b/n is the data sampling ratio. +Proof: The proof mainly follows the work [21] by considering both data sampling with replacement +case and that without replacement case. Specially, when ϵ ≤ 1 and data are uniformly sampled with +replacement, data subsampling mechanism guarantees (ln(1 + q(exp(ϵ) − 1), qδ)-DP [21], we have +ln(1 + q(exp(ϵ) − 1)) +(a) +≤ q(exp(ϵ) − 1) +(b) +≤ 2qϵ, +(5) + +7 +where (a) and (b) hold because ln(1 + x) ≤ x and exp(x) − 1 ≤ 2x when 0 < x ≤ 1 [37]. +When data are uniformly sampled without replacement, the data subsampling mechanism yields (ϵ′′, qδ)- +DP [21], where +ϵ′′ = ln +� +1 + (1 − (1 − 1 +n))b(exp(ϵ) − 1) +� +(a) +≤ ln +� +1 + q(exp(ϵ) − 1) +� (b) +≤ 2qϵ, +(6) +where (b) follows because of (5), and (a) holds since +(1 − (1 − 1 +n))b ≤ b +n = q. +(7) +By combining (5) and (6), we complete the proof. +■ +According to Theorem 1, the privacy would be amplified when q ≤ 1/2. Note that, the privacy +amplification for local DP is pervasively adopted in existing FL literatures [9], [22] since only a small +portion of data being used in local SGD. +B. Centralized clustering model +Let X be a data matrix that contains n data samples and each sample has m features, i.e., X = +[x1, . . . , xn] ∈ Rm×n. The clustering task is to assign the n data samples of X into a predefined number +of k clusters such that the samples within a cluster are closer to each other than to those belonging to any +other cluster in terms of a certain distance metric. Among hundreds of clustering algorithms, the most +classic and popular one is the k-means which aims to obtain k non-overlapping clusters {Ci}k +i=1, i.e., +Ci ∩ Ci′ = ∅, ∀i ̸= i′ ∈ [k], �k +i=1 Ci = {xj}n +j=1, by minimizing the average Euclidean distance between +each cluster centroid and all the data samples within the cluster. +From the perspective of optimization, the k-means algorithm can be viewed as an ad hoc algorithm, +which handles the following matrix factorization by alternative minimization (AM) [38]: +min +W∈Rm×k,H ∥X − WH∥2 +F +(8a) +s.t. H(i, j) ∈ {0, 1}k×n, ∥H(:, j)∥0 = 1, ∀j, +(8b) +where W ∈ Rm×k is a matrix consisting of the k centroids, and H is an indicator matrix with only one +non-zero element (i.e., unity) in each column. Applying AM to problem (8) gives rise to the following +update rules of W and H at iteration t + 1: +Ht+1 = arg min +H +∥X − WtH∥2 +F , +s.t. H(i, j) ∈ {0, 1}k×n, ∥H(:, j)∥0 = 1, ∀j. +(9) + +8 +Wt+1 = arg +min +W∈Rm×k ∥X − WHt+1∥2 +F . +(10) +Closed-form solutions to (8) and (9) are respectively given by +Ht+1(l, j) = +� +� +� +� +� +1 +if l = arg min +u ∥X(:, j) − Wt(:, u)∥2, +0 +otherwise, +(11) +and +Wt+1(:, l) = +1 +|J t +l | +� +u∈J t +l +X(:, u), +(12) +where J t +l = {j|H(l; j) = 1}. Note that at the iteration t + 1, the l-th row of H is updated according +to the minimum distance from each data sample to the l-th centroid according to Wt, and then the l-th +centroid (i.e., the l-column of W) is updated as the average of the data belonging to cluster l according +to the l-th row of the updated indicator matrix. +Despite its empirical effectiveness, the k-means algorithm fails for the dataset with complex distribution +and data heterogeneity, and it is not suitable for distributed environments either, especially the FL setting. +The reasons are twofold. First, the non-convex k-means problem (8) is NP-hard due to involving binary +variables, implying that almost any algorithm (including k-means) is unable to work well. No wonder, +it’s performance is quite sensitive to the initial conditions, complex data distribution, and the obtained +solution easily trapped in bad local minima and so forth [39]. Moreover, the less data samples the +worse its performance, thus further downgrading its performance in FL scenarios, especially when the +data are sensitive and under privacy concern. Most existing FedC algorithms are based on k-means and +operate in computation-aggregation fashion, but their performance may get seriously downgraded under +FL scenarios, including massively distributed clients and severe client heterogeneity [40]. +Following the idea in [41], we replace the binary constraint (8b) with a norm-based equality constraint +and reformulate problem (8) as +min +W,H ∥X − WH∥2 +F + µh +2 ∥H∥2 +F + µw +2 ∥W∥2 +F +(13a) +s.t. H ≥ 0, ∥H(:, j)∥2 +1 = ∥H(:, j)∥2 +2 , ∀j ∈ [n]. +(13b) +where µh > 0 and µw > 0 are two positive parameters. Problem (13) is a non-convex and non-smooth +problem and it can be regarded as a relaxation of the k-means problem (8) as H has been relaxed as a +real k×n matrix, with at most one non-zero entry (not equal to one) in each column, though the equality +constraint (13b) is still non-convex. Moreover, the two regularization terms (i.e., the 2nd and the 3rd +terms in (13a)) are used to control the resulting scaling/counter-scaling ambiguity [38]. + +9 +Instead of solving problem (13), we consider the following problem by dropping the equality constraint +in (13b) and adding an associated penalty term in the objective function: +min +W,H ∥X − WH∥2 +F + µh +2 ∥H∥2 +F + µw +2 ∥W∥2 +F + ρ +2 +N +� +i=1 +�� +1⊤H(:, j) +�2 +− ∥H(:, j)∥2 +2 +� +(14a) +s.t. H ≥ 0, +(14b) +where ρ > 0 is a penalty parameter. The larger the value of ρ, the smaller the approximation error of +the equality constraint in (13b) and the more sparse the matrix H. It is remarkable that problem (14) +is much efficient to handle than problem (13) for two reasons. One is that (14b) is a simple convex +constraint; the other is that the assignment of each data sample to an unique cluster is not reliable for +problem (13) [42], [43]. Therefore, in contrast to the hard clustering performed by k-means, solving (14) +corresponds to seeking a soft clustering solution [44] instead. +C. Federated clustering model +To solve problem (14) under the FL network, we assume that the data matrix is partitioned and +distributed over N clients. i.e., X = [X1, X2, . . . , XN]. Specifically, each client i owns non-overlapping +data Xi ∈ Rm×ni, where ni is the number of data samples in client i and �N +i=1 ni = n. Under the FL +scenario, N could be large, and the data X1, X2, . . . , XN could be unbalanced and non-i.i.d. [45], [46]. +Figure 2 illustrates the federated clustering framework in which a central server coordinates the N clients +to accomplish the clustering task. As seen, the matrix H is partitioned in the same fashion as X, i.e., +H = [H1, H2, . . . , HN], as each column of H corresponds to a certain data sample in X. Then, one can +reformulate problem (14) as follows. +min +W, Hi, +i=1,...,N +F(W, H) ≜ 1 +N +N +� +i=1 +Fi(W, Hi) +(15a) +s.t. Hi ≥ 0, ∀i ∈ [N], +(15b) +where +Fi(W, Hi) ≜∥Xi − WHi∥2 +F + ρ +2(Tr(HiUH⊤ +i ) − ∥Hi∥2 +F ) + µh +2 ∥Hi∥2 +F + µw +2 ∥W∥2 +F +(16) +is the local objective function of each client i, and U ≜ 11⊤. +In contrast to the vanilla FL problem which contains only one shared optimization variable, problem +(15) involves two variables: one is W which is the cluster centroid matrix W shared among clients, and +the other one is Hi which is local cluster indicator matrix for Xi owned by client i. This apparently +brings challenges in the algorithm development, especially in the presence of non-i.i.d. data. In parallel, +as W is shared, there certainly exist possibilities of leaking clients’ privacy in the FL process. Recent + +10 +Client 1 += +𝑚 × 𝑛 +𝐗 +𝐖 +𝑚 × 𝑘 +𝐇 +𝑘 × 𝑛 +𝐖 += +𝐗𝟏 +𝐇𝟏 +𝑚 × 𝑛1 +𝑚 × 𝑘 +𝑘 × 𝑛1 +× +Client 2 +𝐖 += +𝐗𝟐 +𝐇𝟐 +𝑚 × 𝑛2 +𝑚 × 𝑘 +𝑘 × 𝑛2 +Client N +𝐖 += +𝐗𝑵 +𝐇𝑵 +𝑚 × 𝑛𝑁 +𝑚 × 𝑘 𝑘 × 𝑛𝑁 +× +× +× +Parameter server +•• +•• +•• +Fig. 2: The framework of federated clustering. +work [47] showed that the honest-but-curious server could infer clients’ private data from the uploaded +information in the federated matrix factorization framework. Consequently, it is inevitable to develop an +effective and privacy-preserving FL algorithm for problem (15). +III. PROPOSED ALGORITHM FOR PROBLEM (15) +In this section, we develop a FedC algorithm to solve (15), which judiciously updates W and Hi, i ∈ +[N], and adopts an amplified DP for rigorous privacy protection. +A. Update of W and Hi in FL +The key of algorithmic development to problem (15) is to specify how to perform the local update of +Hi and global update of W in the presence of FL challenges. Inspired by [24], we follow the same spirit +of local SGD and PCP, where a subset of clients are selected to locally update Hi and the associated +local copies Wi’s of W, and then upload these iterates to the server for global aggregation in each +round. In particular, for round t = 1, 2, . . ., +(a) Client sampling: We let the PS uniformly sample a small and fixed-size set St of K clients, i.e., +St ⊆ [N], |St| = K, and then broadcast the global Wt−1 to all clients. + +11 +(b) Local update: All clients are asked to obtain an approximate solution (Wt +i, Ht +i) to the following +local subproblem of (15). +(Wt +i, Ht +i) = arg +min +W,Hi≥0 Fi(W, Hi). +(17) +After that, each client i ∈ St uploads Wt +i to the PS. +(c) Global aggregation: After receiving Wt +i from all clients i ∈ St, the PS aggregates them to produce +the new global Wt, i.e., +Wt = 1 +K +� +i∈St +Wt +i. +(18) +In order to specify the local iterates (Wt +i, Ht +i), we propose to handle (17) by combining one-step AM +[48] and local SGD. That is, Ht +i is produced by applying multiple steps of gradient descent (GD) to +(17) with Wi fixed, and then Wt +i is updated similarly by fixing Hi. To be more specific, we first let all +clients perform Q1 ≥ 1 consecutive steps of projected GD with respect to Hi, i.e., for r = 1, . . . , Q1, +Ht,r +i = +� +Ht,r−1 +i +− 1 +γt +i +∇HiFi(Wt−1, Ht,r−1 +i +) +�+ +, +(19) +where γt +i > 0 is the learning rate. Then, they are asked to perform Q2 ≥ 1 consecutive steps of SGD +(no projection) with respect to W, i.e., for r = Q1 + 1, . . . , Qt, +Wt,r +i += Wt,r−1 +i +− 1 +ηt ∇W Fi(Wt,r−1 +i +, Ht,Q1 +i +; Bt,r +i ), +(20) +where Qt = Q1 + Qt +2 and ηt > 0 is a step size, and ∇W Fi(Wt,r−1 +i +, Ht +i; Bt,r +i ) is the stochastic gradient +computed using mini-batch dataset Bt,r +i +with size b. Lastly, (Wt +i, Ht +i) is obtained by setting Ht +i = Ht,Q1 +i +and Wt +i = Wt,Qt +i +. +B. Privacy concern and enhancement +Data security and privacy are primary concerns in FL systems. To enhance data privacy, we apply the +strategy of DP to the proposed algorithm. In particular, in each round t, we add an artificially Gaussian +noise matrix ξt +i ∈ Rm×k to Wt +i, where all the mk entries of ξt +i are i.i.d. Gaussian random variables with +zero mean and variance σ2 +i,t, thus yielding a new matrix � +Wt +i as follows: +� +Wt +i = Wt +i + ξt +i, +(21) +and then upload � +Wt +i to the PS. Then, (18) becomes +Wt+1 = 1 +K +� +i∈St +� +Wt +i. +(22) + +12 +The details of the proposed algorithm are summarized in Algorithm 1. Note that, the diminishing Qt +2 = +⌊ +�Q +t ⌋ + 1 (line 12) denotes the number of iterations in updating Wt,r +i +(lines 13-15) by (20), where the +mini-bacth dataset Bt,r +i +of size b used is addressed in the following remark: +Remark 1 For lines 13-15 of Algorithm 1, Qt +2b data samples are obtained from the dataset Di at each +round (i.e., the data sampling ratio qi,t = Qt +2b/ni), and then divided into Qt +2 mini-batch datasets Bt,r +i +of +size b for each inner iteration r. +It is acknowledged that the DP noise matrix ξt +i will bring about adverse effects on algorithm conver- +gence and learning performance. However, the performance degradation of Algorithm 1 will get worse +from round to round due to W perturbed by the DP noise and the coupling of W and H, on one hand. +The accumulated DP noise effects will also get more serious with the number of rounds on the other +hand. Therefore, Algorithm 1 is performance-sensitive to the DP noise in a complicated manner, such that +obtaining a satisfactory privacy-utility tradeoff through theoretical analysis becomes more intractable. +Nevertheless, the privacy amplification presented in Theorem 1, can be utilized to pursue the perfor- +mance analysis of Algorithm 1, in order to find the clue about the variance reduction of the DP-noise +for guaranteeing (ϵ, δ)-DP privacy protection level at each round. The details are presented in the next +section. +IV. THEORETICAL ANALYSIS +A. Assumptions +We need the following assumptions to analyze the privacy guarantee and convergence performance of +the proposed algorithm. +Assumption 1 Each local cost function Fi is continuously differentiable in both W and Hi. That is, +∇HiFi(Wt, ·) is Lipschitz continuous with constant Lt +Hi, and ∇W Fi(·, Ht +i) is Lipschitz continuous with +constant Lt +Wi, i.e., for any x, y, we have +∥∇HiFi(Wt, x) − ∇HiFi(Wt, y)∥ ≤ Lt +Hi∥x − y∥, +(23) +∥∇W Fi(x, Ht +i) − ∇W Fi(y, Ht +i)∥ ≤ Lt +Wi∥x − y∥. +(24) +According to Assumption 1 and [31], ∇W F(·, Ht) is Lipschitz continuous with a constant Lt +W = +� +(1/N) �N +i=1(Lt +Wi)2, together with upper and lower bounds for Lt +Hi and Lt +Wi, i.e., +LW ≥ Lt +Wi ≥ LW > 0, LH ≥ Lt +Hi ≥ LH > 0, ∀i, t. +(25) + +13 +Algorithm 1 DP-FedC algorithm +1: Input: initial values of W0 +1 = · · · = W0 +N = W0, initial values of {H0 +i }N +i=1, S0 = {1, . . . , N}, R +and �Q. +2: for round t = 1 to R do +3: +Server side: +4: +Compute Wt by (22). +5: +Uniformly sample a set of clients St ⊆ [N], and broadcast Wt to all clients. +6: +Client side: +7: +for client i ∈ [N] in parallel do +8: +Set Ht,0 +i += Ht−1 +i +and Wt,0 +i += Wt. +9: +for r = 1 to Q1 do +10: +Update Ht,r +i +by (19), and set Wt,r +i += Wt,r−1 +i +. +11: +end for +12: +Compute Qt +2 = ⌊ +�Q +t ⌋ + 1. +13: +for r = Q1 + 1 to Qt = Q1 + Qt +2 do +14: +Update Wt,r +i +by (20), and set Ht,r +i += Ht,r−1 +i +. +15: +end for +16: +end for +17: +Set Wt +i = Wt,Qt +i +and Ht +i = Ht,Qt +i +. +18: +for client i ∈ St in parallel do +19: +Compute � +Wt +i by (21). +20: +Upload � +Wt +i to the PS for next round of aggragation. +21: +end for +22: end for +Assumption 2 The gradient of each local cost function Fi, ∀i ∈ [N] is bounded, and Fi is lower bounded. +For any i ∈ [N], we have +∥∇W Fi(W, Hi; Bi)∥2 +F ≤ G2, ∀W, Hi ≥ 0, +(26a) +Fi(W, Hi) ≥ F > −∞, ∀W, Hi ≥ 0, +(26b) +where G is a constant, and Bi denotes the mini-batch dataset. + +14 +Assumption 3 Denote Bt +i as mini-batch dataset with size b randomly sampled from dataset Di. Then, +for any i ∈ [N], the following equations hold, +E[∇W Fi(Wt +i, Ht +i; Bt +i)] = ∇W Fi(Wt +i, Ht +i); +(27) +E[∥∇W Fi(Wt +i, Ht +i) − ∇W Fi(Wt +i, Ht +i; Bt +i)∥2 +F ] ≤ φ2 +b , +(28) +where φ is a constant. +Assumption 4 (ζ-non-i.i.d. data) All the local objective functions are ζ-non-i.i.d., namely, the following +condition holds: +∥∇W Fi(W, Hi) − ∇W F(W, H)∥2 +F ≤ ζ2, ∀W, H ≥ 0, +(29) +where ζ ≥ 0 is a constant. +Note that, (29) in Assumption 4 is a metric to measure the degree of non-i.i.d. data. The above assumptions +are widely adopted in the FL literature, especially for non-convex FL problems. +B. Privacy analysis +1) Privacy guarantee: The ℓ2-norm sensitivity [10] of Wt +i is stated in following Lemma. +Lemma 2 For any t ∈ [R] and i ∈ [N], the ℓ2-norm sensitivity of uploaded local model Wt +i is +st +i = 2GQt +2 +ηt +. +(30) +Proof: See the Appendix A. +According to Lemma 1 and Lemma 2, we have the following theorem, which can serve as a useful +reference for determining the variance of DP noise necessary to fulfill the associated DP-based FL. +Theorem 2 For any client i ∈ [N], suppose that ϵ ≤ 1, δ ≤ 1, and the data sampling ratio qi,t = Qt +2b/ni +(cf. Remark 1). Each entry of ξt +i is sampled from the Gaussian distribution with zero mean and variance +σ2 +i,t, where +σ2 +i,t = +32G2(Qt +2)2q2 +i,t ln(1.25qi,t/δ) +(ηt)2ϵ2 +. +(31) +Then each communication round of the proposed algorithm guarantees (ϵ, δ)-DP. +Proof: In each communication round of the proposed algorithm, each client i performs Qt +2 steps of SGD +w.r.t. W by (20), where the mini-batch dataset with size b used is randomly sampled without replacement +from local dataset Di. According to Lemma 1 and Theorem 1, the Gaussian noise with variance +σ2 +i,t = +2s2 +i,t ln(1.25/δ) +ϵ2 +(32) + +15 +can achieve at least (2qi,tϵ, qi,tδ)-DP for client i, where qi,t = Qt +2b/ni is data sampling ratio for client i. +Then, by plugging s2 +i,t given by (30) into (32), we obtain +σ2 +i,t = 4G2(Qt +2)2 ln(1.25/δ) +(ηt)2ϵ2 +, ∀i ∈ [N]. +(33) +By (33), one can achieve an (ϵ, δ)-DP for Wt +i, by replacing ϵ and δ in (33) with ϵ/2qi,t and δ/qi,t, +respectively, leads to (31). +■ +2) Total privacy loss: As done in [3], we also use the moments accountant method to estimate the +total privacy loss when the algorithm runs R communication rounds. +Theorem 3 Suppose that client i is uniformly sampled by the PS with a probability pi and the data +sampling ratio qi,t = Qt +2b/ni (cf. Remark 1), where Qt +2 = ⌊ +�Q +t ⌋ + 1. A proper noise with variance σ2 +i,t +given in Theorem 2 is added to guarantee (ϵ, δ)-DP at each round. Then, the total privacy loss ¯ϵi for +client i after R communication rounds is given by +¯ϵi = c0q2 +i,tϵ +� +piR +1 − qi,t +, ∀i ∈ [N], +(34) +where c0 is a constant. +Proof: The proof basically follows that of Theorem 1 reported in [3]. However, we further consider +privacy amplification. Thus, the desired result (34) can be obtained by replacing the ϵ with 2qi,tϵ in the +corresponding ¯ϵi in Theorem 1 of [3]. +■ +Theorem 3 shows that the bound of total privacy loss for the proposed DP-FedC is tighter than that +of the cutting-edge reported in [3] when p and q are appropriately chosen. +Remark 2 When clients are uniformly sampled by the PS with a probability pi, The (34) in Theorem 3 +demonstrates that Algorithm 1 guarantees (O(qϵ√pR), δ)-DP when running R rounds, where p and q +are given by +q = max +i,t +q2 +i,t +�1 − qi,t +, ∀i ∈ [N], t ∈ [R], +(35) +p = max +i +pi, ∀i ∈ [N], +(36) +where qi,t = Qt +2b/ni. +C. Convergence analysis +To find some convergence conditions, we define the following sequence +W +t,r = +� +� +� +� +� +� +� +1 +K +� +i∈St Wt,r +i , when r ∈ [Qt − 1], +1 +K +� +i∈St +� +Wt,Qt +i ++ ξt +i +� +, when r = Qt. +(37) + +16 +W +t,r is the instantaneous weighted average of local models. Motivated by [31], we use the following +terms as the optimality gap between a stationary solution of problem (15) +GH(W +t,r, Ht,r) ≜ +N +� +i=1 +(γt +i)2��Ht,r +i +− +� +Ht,r +i +− 1 +γt +i +∇HiFi(W +t,r, Ht,r +i ) +�+��2 +F , ∀r ∈ [Q1], +(38) +GW (W +t,r, Ht,r) ≜ ∥∇W F(W +t,r, Ht,r) +� +∥2 +F , ∀r ∈ [Qt] \ [Q1]. +(39) +If GH(W +t,r, Ht,r) = 0 and GW (W +t,r, Ht,r) = 0, then (W +t,r, Ht,r) is a stationary solution of problem +(15). The main theoretical result for the DP-FedC is given as follows. +Theorem 4 Let T = RQ1 + �R +t=1 Qt +2 be the total number of gradient evaluations per client and R +be the total number of communication rounds. Moreover, let Qt +2 = ⌊ +�Q +t ⌋ + 1, γt +i = α1Lt +H/2 and ηt = +α2Lt +W , where α1 > 1 and α2 ≥ Qt +2 +� +3(1 + L +2 +W /L2 +W ). Then, under Assumptions 1-4, the sequence +{(W +t,r, Ht,r)} yielded by Algorithm 1 satisfies +1 +T +� +R +� +t=1 +Q1 +� +r=1 +E[GH(W +t,r−1, Ht,r−1)] + +R +� +t=1 +Qt +� +r=Q1+1 +E[GW (W +t,r−1, Ht,r−1)] +� +≤2(α2 +1L +2 +H + 1) +T +� +α2LW +� +F(W +1,0, H1,0) − F +� ++ 16mkG2 ln(1.25/δ) �R +t=1(Qt +2)3 +α2ϵ2 ++ LW φ2 �R +t=1 Qt +2 +2KbLW α2 ++ ζ2 �R +t=1 Qt +2 +K ++ 4Nζ2 �R +t=1 Ct +1 +K2α2 +2 +� +, +(40) +where +Ct +1 = Qt +2(Qt +2 − 1)(2Qt +2 − 1). +(41) +Proof: See Appendix B. +■ +Theorem 4 provides an upper bound of the average total local SGDs over R communication rounds; +the smaller its value, the higher convergence rate and the smaller of the objective value in (15) achieved +by Algorithm 1. Based on Theorem 4, we have the following remarks. +Remark 3 (Convergence rate analysis) Since Qt +2 = ⌊ +�Q +t ⌋ + 1, then we have �R +t=1 Ct +1, �R +t=1 Qt +2 and +�R +t=1(Qt +2)3 all in O(R). According to (40), if we set Q1 = +√ +R, then the proposed algorithm converges +at a rate of O(1/ +√ +R). +Remark 4 (Impact of DP) The larger value of ϵ (or ¯ϵ), the smaller the upper bound in (40), implying +that the better learning performance (convergence rate and the objective value) and the weaker required +privacy protection level, namely a privacy-utility tradeoff. + +17 +Remark 5 (Impact of non-i.i.d. data and PCP) The smaller the value of ζ or the larger the value of +K, the smaller the upper bound in (40), implying that the smaller degree of non-i.i.d. data or the more +clients in PCP, the better learning performance (convergence rate and the objective value). +V. EXPERIMENT RESULTS +In this section, in terms of the achieved value of the objective function in (15) and clustering accuracy, +some experimental results are presented to evaluate the performance of the proposed DP-FedC algorithm +(Algorithm 1) including comparison with some state-of-the-art FedC algorithms. The experiment is +performed using two real datasets and each obtained result is the average over 5 independent runs with +the same randomly generated initial feasible points for all the algorithms under test. +A. Experiment setup +Datasets: The two real data sets used in the experiment are TCGA [34] and MNIST datasets. Specif- +ically, TCGA dataset was obtained from the Cancer Genome Atlas database which contains the gene +expression data of 5,314 cancer samples belonging to 20 cancer types. Each data sample in TCGA dataset +is a 5000 × 1 real vector containing the top-ranked 5000 features selected through Pearson’ Pearson’s +Chi-Squares Test [31]. The MNIST database contains 60,000 training images of 10 handwritten digits +and 10,000 test ones. We randomly select 10,000 images from the 60,000 training images as the dataset +in our experiment, where each data sample is a 784 × 1 real vector containing 784 features. +In the experiment, we distribute the samples of each dataset to N = 100 clients in the following two +ways: +(i) IID case: We follow the data partition method in [28] to obtain balanced and i.i.d. distributed data +for the two datasets. To be specific, the i.i.d. distributed data are generated by randomly assigning +the data samples to all clients. +(ii) non-IID case: For the TCGA dataset, we apply the k-means algorithm to cluster the dataset into +100 clusters, and the data samples belonging to the same cluster is assigned to one client. For the +MNIST dataset, we follow the partition method in [24] to obtain distributed data such that each +client’s dataset only contains two digits, thus yielding a highly unbalanced and non-i.i.d. dataset. +Parameter setting: In problem (15), if not mentioned specifically, we set the parameters µw = 0, +ρ = 10−7× ∥X∥2 +F +N +and µh = 10−10× ∥X∥2 +F +N +. As for the parameters in Algorithm 1, the step size γt +i = 1 +2Lt +Hi +where Lt +Hi is estimated to be λmax((Wt,0 +i )⊤Wt,0 +i ). Analogously, the step size ηt = 5Lt +W where Lt +W +is estimated to be λmax(Ht,Q1(Ht,Q1)⊤). Given the total privacy loss ¯ϵ, the privacy protection level ϵ + +18 +at each communication round is obtained by Theorem 3 for R = 100 and δ = 10−4. The mini-batch +dataset size b is set to 50. Other parameters are empirically chosen to our best. All the algorithms under +test runs until R = 100 is reached. Then the clustering accuracy is calculated as the ratio of the number +of correct classifications (no. of columns of all the estimated Hi, i ∈ [N], i.e., their maximum column +entries falling in the correct cluster) to the total number of data (i.e., n). +B. Impact of DP +Figure 3 depicts the objective value (value of F(W, H) in (15)) and the clustering accuracy versus +communication round with different values of ¯ϵ for both IID case and non-IID case, where K = 30, +Q1 = 10, and Qt +2 = ⌊ 10 +t ⌋ + 1. Some observations from Figs. 3(a)-(d), are as follows: +(i) The larger the value of ¯ϵ where the results without DP conceptually corresponds to ¯ϵ → ∞, the +smaller the objective value and the higher the clustering accuracy and convergence rate for both IID +case case and non-IID case; +(ii) The objective value is smaller and the clustering accuracy is higher for the IID case than for non- +IID case, and the performance gap between the two cases seems more appreciable in clustering +accuracy. +The above three observations also apply to Figs. 3(e)-(h). Moreover, the impact of non-i.i.d. data is more +serious for the TCGA dataset. These results are consistent with Remark 4 and Remark 5, so a proper +choice of ¯ϵ value is needed to achieve a good privacy-utility tradeoff. +C. Impact of the number of participated clients (K) +Figure 4 depicts the convergence performance of DP-FedC versus communication rounds under dif- +ferent values of K with ¯ϵ = 20, Q1 = 10, and Qt +2 = ⌊ 10 +t ⌋ + 1. It can be seen from Fig. 4(a), 4(b), 4(e) +and 4(f), that the objective value is smaller together with faster convergence rate either for larger K or +for the IID case. This is also true for the clustering accuracy, though the convergence rate on TCGA +for the IID case is only slightly better than for the non-IID case. These results are also consistent with +Remark 5. +D. Comparison with existing distributed clustering methods +We here compare the proposed DP-FedC algorithm with four benchmark algorithms in terms of +clustering performance. These algorithms include federated k-means (FKM) [14], federated fuzzy k- +means (FZKM) [12], distributed k-means++ (DK++) [25], distributed k-median (DKM) [28]. The first +two are state-of-the-art federated clustering algorithms while the latter two are traditional distributed + +19 +0 +20 +40 +60 +80 +100 +Communication rounds +0.3 +0.5 +0.8 +1 +Objective value +(a) MNIST, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.3 +0.5 +0.8 +1 +Objective value +(b) MNIST, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0 +0.1 +0.2 +0.3 +0.4 +Clustering accuracy +(c) MNIST, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0 +0.1 +0.2 +0.3 +0.4 +Clustering accuracy +(d) MNIST, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.1 +0.3 +0.5 +1 +Objective value +(e) TCGA, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.6 +0.8 +1 +1.5 +2 +Objective value +(f) TCGA, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.1 +0.2 +0.4 +0.6 +0.75 +Clustering accuracy +(g) TCGA, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.05 +0.1 +0.15 +0.2 +Clustering accuracy +(h) TCGA, non-IID +Fig. 3: Objective value and clustering accuracy versus communication rounds under IID case and non-IID +case for three ¯ϵ values. +0 +20 +40 +60 +80 +100 +Communication rounds +0.38 +0.4 +0.42 +0.44 +0.45 +Objective value +(a) MNIST, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.38 +0.4 +0.42 +0.44 +0.45 +Objective value +(b) MNIST, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.3 +0.33 +0.36 +0.39 +0.41 +Clustering accuracy +(c) MNIST, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.3 +0.33 +0.36 +0.39 +0.41 +Clustering accuracy +(d) MNIST, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.1 +0.2 +0.3 +0.4 +0.5 +Objective value +(e) TCGA, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.1 +0.3 +0.5 +0.7 +1 +Objective value +(f) TCGA, non-IID +0 +20 +40 +60 +80 +100 +Communication rounds +0.6 +0.65 +0.7 +0.75 +Clustering accuracy +(g) TCGA, IID +0 +20 +40 +60 +80 +100 +Communication rounds +0 +0.2 +0.4 +0.6 +Clustering accuracy +(h) TCGA, non-IID +Fig. 4: Objective value and clustering accuracy versus communication rounds under IID case and non-IID +case for K ∈ {10, 30, 100}. +clustering methods. As mentioned previously, they were basically developed by extending the k-means +algorithm and its variants. We add the artificial noise to DP noise that guarantees the (ϵ, δ)-DP at each +communication round in the implementation of the above four existing algorithms in our experiment. + +20 +TABLE I: Performance comparisons of all the algorithms under +test in terms of clustering accuracy (%). +Method +Dataset +TCGA +(without DP) +MNIST +(without DP) +TCGA +(¯ϵ = 20) +MNIST +(¯ϵ = 20) +DK++ [25] +65.4 +42.6 +50.1 +26.8 +DKM [28] +38.7 +43.3 +31.0 +26.1 +FKM [14] +70.2 +43.2 +58.4 +31.8 +FZKM [12] +72.8 +47.1 +66.9 +36.4 +DP-FedC +76.7 +50.5 +72.2 +43.1 +Then we apply the proposed algorithm to process the given dataset with parameters K = 30, Q1 = 10, +Qt +2 = 5, ¯ϵ = 20. However, the parameters used for the other four algorithms are taken from the associated +references together with K = 30, ¯ϵ = 20. +The obtained experimental results (for the clustering accuracy) are shown in Table I. It can be seen +from this table that the clustering accuracy performances of all the algorithms under test results for the +case of without DP noise are better than with DP noise used; The performance gap in between the +two cases for our DP-FedC algorithm is much smaller than for the other algorithms, implying that the +proposed algorithm is more robust again DP noise thanks to the privacy amplification strategy applied. +VI. CONCLUSION +We have presented a novel FedC algorithm, called DP-FedC, which operates following computation- +aggregation protocol, and considering PCP, multiple local SGD steps, mini-batch dataset, and data +heterogeneity. In particular, based on the privacy amplification theorem, a necessary variance of the +zero-mean DP noise added to all the local models at each communication round is obtained to guarantee +the required privacy protection level. The efficacy (convergence rate and clustering accuracy) of the +proposed DP-FedC is well supported by both theoretical convergence analysis and experimental results, +including its much superior performance over some state-of-the-art FedC algorithms. +APPENDIX A +PROOF OF LEMMA 2 +Assume Di and D′ +i are the neighboring datasets that differ in only one data sample. Without loss of +generality, let ui be the unique different element between Di and D′ +i, i.e., D′ +i ∪ {ui} = Di ∪ {ui}. For +clarity of the following proof, let us make the following notational correspondences: Wt,r +i +↔ Wt,r +Di, + +21 +Ht,r +i +↔ Ht,r +Di, and Bt,r +i +↔ Bt,r +Di. Then, for any r ∈ [Qt] \ [Q1], the ℓ2-sensitivity [10] of Wt +i is calculated +by +st +i = max +Di,D′ +i +��Wt +Di − Wt +D′ +i +�� += max +Di,D′ +i +��� +Qt +� +r=Q1+1 +Wt,r−1 +Di +− ∇W Fi(Wt,r−1 +Di +, Ht,r−1 +Di +; Bt,r +Di) +ηt +− +Qt +� +r=Q1+1 +� +Wt,r−1 +D′ +i +− +∇W Fi(Wt,r−1 +D′ +i +, Ht,r−1 +D′ +i +; Bt,r +D′ +i) +ηt +���� += max +Di,D′ +i +��� +� +Wt,Q1 +Di +− ∇W Fi(Wt,Q1 +Di , Ht,Q1 +Di ; Bt,Q1+1 +Di +) +ηt +− · · · − ∇W Fi(Wt,Qt−1 +Di +, Ht,Qt−1 +Di +; Bt,Qt +Di ) +ηt +� +− +� +Wt,Q1 +D′ +i +− +∇W Fi(Wt,Q1 +D′ +i , Ht,Q1 +D′ +i ; Bt,Q1+1 +D′ +i +) +ηt +− · · · − +∇W Fi(Wt,Qt−1 +D′ +i +, Ht,Qt−1 +D′ +i +; Bt,Qt +D′ +i ) +ηt +���� +(a) +≤ 2GQt +2 +ηt +, +(A.1) +where (a) holds because of Assumption 2, and Wt,Q1 +Di += Wt,Q1 +D′ +i +always holds. +■ +APPENDIX B +PROOF OF THEOREM 4 +According to (37) and (20), we have +W +t,r =W +t,r−1 − +1 +Kηt +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i ). +(B.1) +(Objective Descent w.r.t. H) According to [49, Lemma 3.2] and setting γt +i = α1Lt +H/2 ≤ α1LH/2 where +α1 > 1, we have +Fi(W +t,r, Ht,r +i ) − Fi(W +t,r−1, Ht,r−1 +i +) ≤ −α1 − 1 +2 +LH∥Ht,r−1 +i +− Ht,r +i ∥2 +F , ∀r ∈ [Q1]. +(B.2) +Taking expectation over two sides of (B.2) and then summing up from r = 1 to Q1 yields +E +� +Fi(W +t,Q1, Ht,Q1 +i +) +� +− E +� +Fi(W +t,0, Ht,0 +i ) +� +≤ −α1 − 1 +2 +LH +Q1 +� +r=1 +E +� +∥Ht,r−1 +i +− Ht,r +i ∥2 +F +� +, ∀r ∈ [Q1]. +(B.3) +By taking the summation over two sides of (B.3) from i = 1 to N, the objective function F descends +with local updates of H is given by +E[F(W +t,Q1, Ht,Q1)] − E[F(W +t,0, Ht,0)] ≤ −α1 − 1 +2 +LH +Q1 +� +r=1 +N +� +i=1 +E +� +∥Ht,r−1 +i +− Ht,r +i ∥2 +F +� +, ∀r ∈ [Q1]. +(B.4) +(Objective Descent w.r.t. W) Since Ht,r +i += Ht,r−1 +i +(cf. line 14 in Algorithm 1) and ∇W F(·, Ht,Q) +is Lipschitz continuous under Assumption 1. Then, by the descent lemma [49, Lemma 3.1], when r ∈ +[Qt − 1] \ [Q1], we have + +22 +E +� +F(W +t,r, Ht,r) +� +≤E +� +F(W +t,r−1, Ht,r−1) +� ++ Lt +W +2 E +� +∥W +t,r − W +t,r−1∥2 +F +� ++ E +� +⟨∇W F(W +t,r−1, Ht,r−1), W +t,r − W +t,r−1⟩ +� +. +(B.5) +When r = Qt, by Algorithm 1, (B.5) becomes, +E +� +F(W +t,Qt +, Ht,Qt) +� +≤E +� +F(W +t,Qt−1, Ht,Qt−1) +� ++ Lt +W +2 E +� +∥W +t,Qt +− W +t,Qt−1 + ξt∥2 +F +� +� +�� +� +≜(S.1) ++ E +� +⟨∇W F(W +t,Qt−1, Ht,Qt−1), W +t,Qt +− W +t,Qt−1⟩ +� +, +(B.6) +where ξt = 1 +K +�K +i=i ξt +i. The (S.1) can be further bounded by, +(S.1) =Lt +W +2 E +� +∥W +t,Qt +− W +t,Qt−1 + ξt∥2 +F +� +=Lt +W +2 E +� +∥W +t,Qt−1 − W +t,Qt +∥2 +F ] + Lt +W +2 E +� +∥ξt∥2 +F +� +(a) +≤ Lt +W +2 E +� +∥W +t,Qt−1 − W +t,Qt +∥2 +F +� ++ 16mkG2 ln(1.25/δ)(Qt +2)2 +α2ηtϵ2 +, +(B.7) +where (a) holds from ηt = α2Lt +W and +E +� +∥ξt∥2 +F +� (a) += mk +K +K +� +i=1 +32G2(Qt +2)2q2 +i,t ln(1.25qi,t/δ) +(ηt)2ϵ2 +(b) +≤ 32mkG2(Qt +2)2 ln(1.25/δ) +(ηt)2ϵ2 +. +(B.8) +In (B.8), (a) follows from (31). (b) holds because of qi,t ≤ 1. By (B.5), (B.6) and (B.7), for r ∈ [Qt]\[Q1], +we have, +E +� +F(W +t,r, Ht,r) +� +≤E +� +F(W +t,r−1, Ht,r−1) +� ++ Lt +W +2 +E +� +∥W +t,r − W +t,r−1∥2 +F +� +� +�� +� +≜(S.2) ++ E +� +⟨∇W F(W +t,r−1, Ht,r−1), W +t,r − W +t,r−1⟩ +� +� +�� +� +≜(S.3) ++ 16mkG2(Qt +2)2 ln(1.25/δ) +α2ηtϵ2 +. +(B.9) +The terms (S.2) and (S.3) can be bounded by the following Lemma 3 (proved in Appendix C-A) and +Lemma 4 (proved in Appendix C-B), respectively. +Lemma 3 For any t and r ∈ [Qt − 1] \ [Q1], we have +E +� +∥W +t,r − W +t,r−1∥2 +F +� +≤ +φ2 +Kb(ηt)2 + +1 +(ηt)2 E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� +. +(B.10) +Lemma 4 For any t and r ∈ [Qt − 1] \ [Q1], we have +E +� +⟨∇W F(W +t,r−1, Ht,r−1), W +t,r − W +t,r−1⟩ +� + +23 += − 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F + +�� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2 +F +� ++ ζ2 +Kηt + +1 +Kηt +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +. +(B.11) +Thus, substituting (B.10) and (B.11) into (B.9) gives rise to +E +� +F(W +t,r, Ht,r) +� +− E +� +F(W +t,r−1, Ht,r−1) +� +≤ − 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� ++ +Lt +W φ2 +2Kb(ηt)2 + ζ2 +Kηt ++ ( Lt +W +2(ηt)2 − 1 +2ηt )E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� ++ +1 +Kηt +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� ++ 16mkG2(Qt +2)2 ln(1.25/δ) +α2ηtϵ2 +(a) +≤ − 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� ++ +LW φ2 +2Kb(ηt)2 + ζ2 +Kηt ++ +1 +Kηt +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� ++ 16mkG2(Qt +2)2 ln(1.25/δ) +α2ηtϵ2 +, +(B.12) +where (a) follows due to ηt = α2Lt +W ≥ Lt +W and Lt +Wi ≤ LW . Then, rearranging the two sides of (B.12) +yields +E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +≤2ηt� +E +� +F(W +t,r−1, Ht,r−1) +� +− E +� +F(W +t,r, Ht,r) +�� ++ 2 +K +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� ++ LW φ2 +Kbηt ++ 32mkG2(Qt +2)2 ln(1.25/δ) +α2ϵ2 ++ 2ζ2 +K . +(B.13) +Summing (B.13) up from r = Q1 + 1 to Qt yields +Qt +� +r=Q1+1 +E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +≤2ηt� +E +� +F(W +t,Q1, Ht,Q1) +� +− E +� +F(W +t,Qt +, Ht,Qt) +�� ++ 2 +K +Qt +� +r=Q1+1 +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +� +�� +� +≜(S.4) ++ 32mkG2(Qt +2)3 ln(1.25/δ) +α2ϵ2 ++ 2ζ2Qt +2 +K ++ Qt +2LW φ2 +Kbηt +. +(B.14) +The term (S.4) can be bounded with the following lemma, which is proved in Appendix C-C. +Lemma 5 Let α2 ≥ Qt +2 +� +3(1 + L +2 +W /L2 +W ). For any t and r ∈ [Qt] \ [Q1], it holds that + +24 +Qt +� +r=Q1+1 +N +� +i=1 +(Lt +Wi)2E[∥W +t,r−1 − Wt,r−1 +i +∥2 +F ] ≤ 4Nζ2Ct +1 +Kα2 +2 +, +(B.15) +where Ct +1 ≜ Qt +2(Qt +2 − 1)(2Qt +2 − 1). +By applying Lemma 5 and plugging (B.15) into (B.14), we have +Qt +� +r=Q1+1 +E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +≤2ηt� +E +� +F(W +t,Q1, Ht,Q1)] − E[F(W +t,Qt +, Ht,Qt) +�� ++ 32mkG2(Qt +2)3 ln(1.25/δ) +α2ϵ2 ++ 2ζ2Qt +2 +K ++ Qt +2LW φ2 +Kbηt ++ 8Nζ2Ct +1 +K2α2 +2 +. +(B.16) +Combining (B.4) and (B.16) yields +Q1 +� +r=1 +N +� +i=1 +E +� +∥Ht,r−1 +i +− Ht,r +i ∥2 +F +� ++ +Qt +� +r=Q1+1 +E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +(a) +≤2ηt� +E +� +F(W +t,0, Ht,0)] − E[F(W +t,Qt +, Ht,Qt) +�� ++ 32mkG2(Qt +2)3 ln(1.25/δ) +α2ϵ2 ++ 2ζ2Qt +2 +K ++ Qt +2LW φ2 +Kbηt ++ 8Nζ2Ct +1 +K2α2 +2 +, +(B.17) +where (a) holds because of ηt ≥ 1/((α1 − 1)LH). +(Derivation of the Main Result) We next derive the convergence in terms of the optimal gap functions +in (38) and (39). From (B.17) and γt +i = α1Lt +H/2 and ηt = α2Lt +W , we have +Q1 +� +r=1 +E[GH(W +t,r−1, Ht,r−1)] = +Q1 +� +r=1 +N +� +i=1 +(γt +i)2E[∥Ht,r−1 +i +− Ht,r +i ∥2 +F ] +(a) +≤2α2 +1L +2 +H +� +α2LW E[F(W +t,0, Ht,0)] − E[F(W +t,Qt +, Ht,Qt)] + 16mkG2(Qt +2)3 ln(1.25/δ) +α2ϵ2 ++ ζ2Qt +2 +K ++ Qt +2LW φ2 +2KbLW α2 ++ 4Nζ2Ct +1 +K2α2 +2 +� +, +(B.18) +where (a) follows because γt +i ≤ α1LH/2 and α2LW ≤ ηt ≤ α2LW . Then, summing (B.18) up from +t = 1 to R yields +R +� +t=1 +Q1 +� +r=1 +E +� +GH(W +t,r−1, Ht,r−1) +� +≤2α2 +1L +2 +H +� +α2LW +� +F(W +1,0, H1,0) − F +� ++ 16mkG2 ln(1.25/δ) �R +t=1(Qt +2)3 +α2ϵ2 ++ ζ2 �R +t=1 Qt +2 +K ++ LW φ2 �R +t=1 Qt +2 +2KbLW α2 ++ 4Nζ2 �R +t=1 Ct +1 +K2α2 +2 +� +. +(B.19) +Similarly, from (B.17), we have + +25 +R +� +t=1 +Qt +� +r=Q1+1 +E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� += +R +� +t=1 +Qt +� +r=Q1+1 +E +� +GW (W +t,r−1, Ht,r−1)] +� +≤2α2LW +� +F(W +1,0, H1,0) − F +� ++ 32mkG2 ln(1.25/δ) �R +t=1(Qt +2)3 +α2ϵ2 ++ 2ζ2 �R +t=1 Qt +2 +K ++ LW φ2 �R +t=1 Qt +2 +KbLW α2 ++ 8Nζ2 �R +t=1 Ct +1 +K2α2 +2 +. +(B.20) +By combining (B.19) and (B.20), and then dividing two sides of summation result by T = RQ1+�R +t=1 Qt +2 +yields +1 +T +� +R +� +t=1 +Q1 +� +r=1 +E +� +GH(W +t,r−1, Ht,r−1) +� ++ +R +� +t=1 +Qt +� +r=Q1+1 +E +� +GW (W +t,r−1, Ht,r−1) +�� +≤2(α2 +1L +2 +H + 1) +T +� +α2LW +� +F(W +1,0, H1,0) − F +� ++ 16mkG2 ln(1.25/δ) �R +t=1(Qt +2)3 +α2ϵ2 ++ ζ2 �R +t=1 Qt +2 +K ++ LW φ2 �R +t=1 Qt +2 +2KbLW α2 ++ 4Nζ2 �R +t=1 Ct +1 +K2α2 +2 +� +. +(B.21) +This completes the proof. +■ +APPENDIX C +PROOFS OF KEY LEMMAS FOR THEOREM 4 +A. Proof of Lemma 3 +According to (B.1), we have +E +� +∥W +t,r − W +t,r−1∥2 +F +� += +1 +(ηt)2 E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i ) +��2� +(a) += +1 +(ηt)2 E +��� 1 +K +� +i∈St +� +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i ) − ∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +���2� ++ +1 +(ηt)2 E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� +(b) += +1 +(ηt)2K2 E +� � +i∈St +��∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i ) − ∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� ++ +1 +(ηt)2 E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� +(c) +≤ +1 +(ηt)2 E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2� ++ +φ2 +Kb(ηt)2 , +(C.1) +where (a) follows because E[∥Z∥2] = E[∥Z−E[Z]∥2]+∥E[Z]∥2; (b) follows because ∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i )− +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) is independent across the clients; (c) holds due to Assumption 3. +■ + +26 +B. Proof of Lemma 4 +E +� +⟨∇W F(W +t,r−1, Ht,r−1), W +t,r − W +t,r−1⟩ +� +(a) += − 1 +ηt E +�� +∇W F(W +t,r−1, Ht,r−1), 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +; Bt,r +i ) +�� +(b) += − 1 +ηt E +�� +∇W F(W +t,r−1, Ht,r−1), 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +�� +(c) += − 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +− 1 +2ηt E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2 +F +� ++ 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) − 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2 +F +� +, +(C.2) +where (a) holds due to (20); (b) follows from Assumption 3; (c) follows from the basic identity +⟨Z1, Z2⟩ = 1 +2(∥Z1∥2 + ∥Z2∥2 − ∥Z1 − Z2∥2). The last term in (C.2) can be further bounded by +E +���∇W F(W +t,r−1, Ht,r−1) − 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2 +F +� +=E +��� 1 +K +� +i∈St +� +∇W F(W +t,r−1, Ht,r−1) − ∇W Fi(W +t,r−1, Ht,r−1) ++ ∇W Fi(W +t,r−1, Ht,r−1) − ∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +���2 +F +� +≤ 2 +K2 E +��� � +i∈St +� +∇W F(W +t,r−1, Ht,r−1)∇W Fi(W +t,r−1, Ht,r−1) +���2 +F +� ++ 2 +K2 E +��� � +i∈St +� +∇W Fi(W +t,r−1, Ht,r−1) − ∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +���2 +F +� +(a) +≤ 2ζ2 +K + 2 +K E +� � +i∈St +(Lt +Wi)2∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +≤2ζ2 +K + 2 +K +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +, +(C.3) +where the first term in the RHS of (a) comes from Assumption 4, and the second term in the RHS of +(C.8) follows because of Assumption 1. Then, Plugging (C.3) into (C.2) yields +E +� +⟨∇W F(W +t,r−1, Ht,r−1), W +t,r − W +t,r−1⟩ +� +≤ − 1 +2ηt E +���∇W F(W +t,r−1, Ht,r−1) +��2 +F +� +− 1 +2ηt E +��� 1 +K +� +i∈St +∇W Fi(Wt,r−1 +i +, Ht,r−1 +i +) +��2 +F +� ++ ζ2 +Kηt + +1 +Kηt +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +. +(C.4) +Thus, we complete the proof. +■ +C. Proof of Lemma 5 +According to the definition of W +t,r−1, for ∀r ∈ [Qt] \ [Q1], we have +W +t,r−1 = 1 +K +� +i∈St +Wt,r−1 +i + +27 +(a) += 1 +K +� +i∈St +� +Wt − 1 +ηt +r−1 +� +j=Q1 +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) +� += Wt − +1 +ηtK +r−1 +� +j=Q1 +� +i∈St +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ), +(C.5) +where (a) is obtained by applying (20), that is +Wt,r−1 +i += Wt − 1 +ηt +r−1 +� +j=Q1 +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ). +(C.6) +As a result, by (C.5) and (C.6), we have +E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +=E +����Wt − +1 +ηtK +r−1 +� +j=Q1 +� +i∈St +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) − +� +Wt − 1 +ηt +r−1 +� +j=Q1 +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) +���� +2 +F +� += +1 +(ηt)2 E +���� 1 +K +r−1 +� +j=Q1 +� +i∈St +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) − +r−1 +� +j=Q1 +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) +��� +2 +F +� +≤(r − Q1) +(ηt)2 +r−1 +� +j=Q1 +E +���� 1 +K +� +i∈St +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i ) − ∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +; Bt,j +k ) +��� +2 +F +� +(b) += (r − Q1) +(ηt)2 +r−1 +� +j=Q1 +��� 1 +K +� +i∈St +� +∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +) +���� +2 +F +(c) +≤ (r − Q1) +(ηt)2K +r−1 +� +j=Q1 +N +� +i=1 +���∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +) +��� +2 +F , +(C.7) +where (b) holds since ∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +; Bt,j +i )−∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +; Bt,j +k ) is independent across +the clients; (c) follows by using the inequality ∥ �N +i=1 zi∥2 ≤ N �N +i=1 ∥zi∥2 for any vectors zi and any +positive integer N. Then, the term ∥∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +)∥2 +F in (C.7) can +be bounded by +��∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +) +��2 +F +≤ +���∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fi(W +t,j−1, Ht,j−1 +i +) + ∇W Fi(W +t,j−1, Ht,j−1 +i +) − ∇W F(W +t,j−1, Ht,j−1) ++ ∇W F(W +t,j−1, Ht,j−1) − +� +∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +) − ∇W Fk(W +t,j−1, Ht,j−1 +k +) + ∇W Fk(W +t,j−1, Ht,j−1 +k +) +− ∇W F(W +t,j−1, Ht,j−1) + ∇W F(W +t,j−1, Ht,j−1) +���� +2 +F +≤4∥∇W Fi(Wt,j−1 +i +, Ht,j−1 +i +) − ∇W Fi(W +t,j−1, Ht,j−1 +i +)∥2 +F ++ 4∥∇W Fi(W +t,j−1, Ht,j−1 +i +) − ∇W F(W +t,j−1, Ht,j−1)∥2 +F ++ 4∥∇W Fk(Wt,j−1 +k +, Ht,j−1 +k +) − ∇W Fk(W +t,j−1, Ht,j−1 +k +)∥2 +F ++ 4∥∇W Fk(W +t,j−1, Ht,j−1 +k +) − ∇W F(W +t,j−1, Ht,j−1)∥2 +F +(d) +≤4(Lt +Wi)2∥W +t,j−1 − Wt,j−1 +i +∥2 +F + 4(Lt +Wk)2∥W +t,j−1 − Wt,j−1 +k +∥2 +F + 8ζ2, +(C.8) + +28 +where (d) follows from Assumption 4. Then, substituting (C.8) into (C.7) gives rise to +Qt +� +r=Q1+1 +N +� +i=1 +(Lt +Wi)2E +� +∥W +t,r−1 − Wt,r−1 +i +∥2 +F +� +≤ +Qt +� +r=Q1+1 +N +� +i=1 +(Lt +Wi)2�(r − Q1 − 1) +K(ηt)2 +r−2 +� +j=Q1 +N +� +i=1 +� +4(Lt +Wi)2∥W +t,j−1 − Wt,j−1 +i +∥2 +F ++ 8ζ2 + 4(Lt +Wk)2∥W +t,j−1 − Wt,j−1 +k +∥2 +F +�� +(e) += N +K +Qt +� +r=Q1+1 +4(r − Q1 − 1) +(ηt/Lt +W )2 +r−2 +� +j=Q1 +N +� +i=1 +(Lt +Wi)2∥W +t,j−1 − Wt,j−1 +i +∥2 +F + N +K +Qt +� +r=Q1+1 +8(r − Q1 − 1)2 +(ηt/Lt +W )2 +ζ2 ++ N +K +Qt +� +r=Q1+1 +4(r − Q1 − 1) +(ηt/Lt +W )2 +r−2 +� +j=Q1 +N +� +i=1 +(Lt +Wi)2 · +�Lt +Wi +Lt +W +�2∥W +t,j−1 − Wt,j−1 +i +∥2 +F +(f) +≤ 2NQt +2(Qt +2 − 1) +Kα2 +2 +(1 + L +2 +W +L2 +W +) +Qt +� +r=Q1+1 +N +� +i=1 +(Lt +Wi)2��W +t,j−1 − Wt,j−1 +i +��2 +F + 4NQt +2(Qt +2 − 1)(2Qt +2 − 1)ζ2 +3Kα2 +2 +, +(C.9) +where (e) follows since (Lt +W )2 = (1/N) �N +i=1(Lt +Wi)2; (f) follows due to +(Lt +Wi)2 +(Lt +W )2 ≤ L +2 +W +L2 +W and ηt = α2Lt +W , +and +Qt +� +r=Q1+1 +(r − 1 − Q1) +r−2 +� +j=Q1 +aj ≤ +Qt +� +r=Q1+1 +Qt +2(Qt +2 − 1) +2 +ar−1, ∀aj > 0, +(C.10) +and +Qt +� +r=Q1+1 +(r − 1 − Q1)2 = Qt +2(Qt +2 − 1)(2Qt +2 − 1) +6 +. +(C.11) +Since α2 ≥ Qt +2 +� +3(1 + L +2 +W /L2 +W ), implies α2 +2 ≥ 2Qt +2(Qt +2 − 1)(1 + L +2 +W /L2 +W ). 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Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth +problems,” Mathematical Programming, vol. 146, pp. 459–494, 2014. + diff --git a/wNAzT4oBgHgl3EQfCPpJ/content/tmp_files/load_file.txt b/wNAzT4oBgHgl3EQfCPpJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f31b21784756fc9b13d2898b6c62af45747ce131 --- /dev/null +++ b/wNAzT4oBgHgl3EQfCPpJ/content/tmp_files/load_file.txt @@ -0,0 +1,1207 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf,len=1206 +page_content='1 Differentially Private Federated Clustering over Non-IID Data Yiwei Li, Student Member, IEEE, Shuai Wang, Member, IEEE, Chong-Yung Chi, Life Fellow, IEEE, Tony Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Quek, Fellow, IEEE Abstract Federated clustering (FedC) is an adaptation of centralized clustering in federated settings, which aims to cluster data based on a global similarity measure while keeping all data locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Two of the main challenges of FedC are the non-identically and independently distributed (non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=') nature of data across different sources, as well as the need for privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In this paper, we propose a differentially private federated clustering (DP-FedC) algorithm to deal with these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Unlike most existing algorithms without considering privacy, the proposed DP-FedC algorithm is designed to handle non-convex and non-smooth problems by using differential privacy techniques to guarantee privacy, together with privacy amplification assisted tradeoff between learning performance and privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then some theoretical analyses of the performance and privacy of the proposed DP-FedC are presented, showing the impact of privacy protection, data heterogeneity, and partial client participation on learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Finally, some experimental results are presented to demonstrate the efficacy (including analytical results) of the proposed DP-FedC algorithm together with its superior performance over state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Keywords−Federated clustering, differential privacy, privacy amplification, non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' INTRODUCTION Federated learning (FL), as a novel distributed paradigm, enables massively distributed clients to jointly learn a machine learning (ML) model under the orchestration of a parameter server (PS) while refraining the clients’ sensitive data from being exposed [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' FL has received tremendous attention in the past few years as it seriously takes numerous practical challenges into account, including limited communication resources and data heterogeneity and client privacy protection in the learning process [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Under these challenges, most FL algorithms follow a computation-aggregation protocol by which the local update of model parameters and server aggregation are repeated in a round-by-round fashion until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' FedAvg [4] is a typical one, which improves communication efficiency by adopting This work is supported by the Ministry of Science and Technology, Taiwan, under Grants MOST 111-2221-E-007-035-MY2 and MOST 110-2221-E-007-031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It is also supported in part by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Future Communications Research & Development Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Li and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Chi are with Institute of Communications Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan (e-mail: lywei0306@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='com, cychi@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Wang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Quek are with Information Systems Technology and Design, Singapore University of Technology and Design, 487372, Singapore (e-mail: shuaiwang@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='cn, tonyquek@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='00955v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='DC] 3 Jan 2023 2 partial client participation (PCP) and multiple steps of local stochastic gradient descent (local SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Nevertheless, the challenge of data heterogeneity where the clients have non-identically and independently distributed (non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=') data is acknowledged to be the main bottleneck to FL deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Numerous efforts have been devoted to analyzing the adverse effects of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data on algorithm convergence and developing effective approaches to mitigate [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In parallel, FL still suffers from privacy leakage as the clients’ sensitive information could be inferred by adversaries through the exchanged model parameters between the clients and the PS [6]–[8], as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The differential privacy (DP) technique has recently gained increasing popularity in enhancing privacy of FL thanks to its algorithmic simplicity, support by rigorous mathematical theory, and negligible system overheads [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Despite the recent rapid progress of FL, substantial attention has been given to supervised learning and the problem of unsupervised learning, especially data clustering, is under-investigated in FL [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Clustering in the FL setting, called federated clustering (FedC), aims to obtain a partition of data points distributed over massive clients based on a global similarity measure while keeping them on respective clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' As clustering is one of most suitable missions for ML and has a great deal of applications, the FedC and its implementation is believed to be in impending need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' On the other hand, recent years have witnessed an incessant springing up of FedC applications, which again motivates research efforts in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For example, in e-commerce applications, FedC is widely used to group the online customers of multiple institutions with sensitive features, such as personal details, purchase orders, and bank transaction records, to identify their specific interests for precise service recommendation [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' FedC has also found numerous and successful applications in clustering FL clients to improve supervised FL [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Examples include seeking the most representative clients by FedC as active ones in each round to boost the algorithm convergence [15], and applying cluster-wise model personalization after FedC to account for data heterogeneity [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In this paper, an effective FedC algorithm is proposed, that considers both non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data and DP-based privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In FedC scenarios where data heterogeneity is prevalent, the global cluster information may be seriously deficient for each client as all the data in hands may belong to just a few clusters, and the correct cluster structure might become apparent when the local datasets are combined [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, effectively transferring the centralized clustering algorithms to FedC, such as k-means, is almost formidable due to the privacy concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Directly applying them to FedC by following the computation-aggregation protocol would result in serious significant performance degradation [11], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In addition, different from supervised FL, the process of FedC involves the iterative constrained optimization of both cluster centroids and cluster assignments of all data samples, which again brings more difficulties to algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' As for privacy protection, such coupling optimization necessitates a more careful and fine-grained 3 Parameter server Global model IoT device Local model Local model Local model Local model Adversary IoT device IoT device IoT device Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 1: The framework of FL system in the presence of adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' design and analysis of the DP-based FedC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In particular, it is well-known that DP protects privacy at the cost of learning performance loss [19], and balancing the tradeoff between protection level and convergence performance, so-called the privacy-utility tradeoff, is essential in practical FL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' To improve the privacy-utility tradeoff, privacy amplification [20], [21] has been pervasively adopted in many DP-based FL (DP-FL) applications [14], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The privacy amplification can reduce the variance of noise added to locally uploaded models without sacrificing the privacy protection level, thereby mitigating the adverse effects of DP [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In addition to the challenges posed by non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data and privacy protection, the practical application of FedC algorithms in FL systems requires careful consideration of communication cost and straggler effect [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' These factors and concerns not only affect the algorithm design, but also make the associated theoretical algorithm performance analysis much more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, the involvement of cluster centroids and data’s cluster-membership assignment in FedC further complicates the design of DP, and it is not clear how to achieve a good privacy-utility tradeoff in FedC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Related works To the best of our knowledge, only limited existing works target data clustering under FL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' While many successful methods have been reported for traditional distributed clustering, they are simply parallel implementations of the centralized clustering algorithms [25]–[27] or implementations through clustering representative data samples collected from distributed clients [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Apparently, the critical challenges of FL, such as massive clients, limited communication resources and data heterogeneity, were 4 never considered, and the demand for privacy protection was also overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The recent works in [12], [16], [30]–[32] have considered the FL scenarios and presented FedC algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, most of them were developed by combining the simple centralized k-means algorithm (and its variants) with FedAvg [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Specifically, in each communication round, the clients employ k- means algorithms to obtain the local cluster centroids, which are then uploaded to the PS to produce the global clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For example, the work [16] extended k-means to the FL system but operating only one communication round, together with rigorous performance analysis under the data heterogeneity consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Different from [16], the works [12], [30] adopted the fuzzy k-means to perform local clustering, while the global centroids are obtained from the received local centroids by k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The work [32] proposed a federated spectral clustering approach to train a generative model for each cluster, such that each data sample can be classified into only one cluster using the generated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Nevertheless, the above-mentioned FedC algorithms may suffer from unsatisfactory clustering perfor- mance as the k-means algorithm only works well in clustering datasets that are evenly spread around the centroids but fails in clustering datasets of complex and heterogeneous cluster structure [17], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, the effect of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data, which is ubiquitous in FL systems, has not been thoroughly studied, thus without presenting any improving actions either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In addition, few of these algorithms considered critical privacy protection issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The work [14] first adopted a secret sharing approach to protect privacy in the federated k-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, such a strategy requires complicated encryption protocols and substantial extra communication and computation cost [33], thus not applicable to large-scale FL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' There is a lack of research efforts on the study of how to secure the FedC algorithms through the DP mechanism together with the ensuing privacy-utility tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Contributions Motivated by the aforementioned issues of existing FedC methods, we propose a differentially private FedC algorithm, called DP-FedC, with the data heterogeneity and privacy protection taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: 1) Based on matrix factorization, a novel FedC problem is formulated for practical FL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then by handling this problem, a DP-FedC algorithm under the computation-aggregation protocol is developed, that alternatively update local cluster centroids and indicator matrices (indicating each sample and the cluster it belongs) through allowing multiple local SGD steps and partial client participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Furthermore, a novel privacy amplification strategy is also proposed to reduce DP noise for better learning performance without sacrificing the privacy protection level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 5 2) Two theoretical analyses for the proposed DP-FedC algorithm are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' One is a privacy analysis, showing that a tightest upper bound of the total privacy loss so far, to the best of our knowledge, is in (O(qϵ√pR), δ)-DP after total R communication rounds, where 0 < p, q ≤ 1 are defined in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The other is a convergence analysis, showing the convergence rate O(1/ √ R) even for the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 3) Extensive experimental results are provided to demonstrate the effectiveness of the proposed DP- FedC algorithm on real world datasets, including TCGA cancer gene data [34], and the MNIST hand-writing digits data [35], and its much superior performance over state-of-the-art distributed clustering and FedC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Synopsis: Section II presents some preliminaries of DP and the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Section III presents the proposed DP-FedC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Section IV presents privacy analysis and convergence analyses of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Experiment results are presented in Section V, and lastly the conclusion is drawn in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Notation: E[·] represents the expectation over all randomness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Pr[·] represents the probability function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Rm×n denotes the set of m by n real-valued matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The (i, j)-th entry of matrix A ∈ Rm×n is denoted by A(i, j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A(i, :) and A(:, j) denote the i-th row and the j-th column of A, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A ≥ 0 means A(i, j) ≥ 0, ∀i, j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' [A]+ denotes the matrix by replacing all the negative elements in A with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' λmax(A) stands for the maximum eigenvalue of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ∥·∥F , ∥·∥ and ∥·∥0 are the matrix Frobenius norm, Euclidean norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', ℓ2-norm) and zero norm of vectors, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ⟨x, y⟩ = x⊤y represents the inner product operator, where the superscript ‘⊤’ denotes the vector transpose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For any integer N, [N] denotes the integer set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 1 denotes the all-one vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' {Ci}k i=1 denotes the set {C1, C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , Ck};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ⌊·⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' PRELIMINARIES AND PROBLEM FORMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Differential privacy In this work, we assume that any third party is untrustworthy, including the honest-but-curious server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The core privacy protection mechanism of the proposed DP-FedC is the well-known DP based random mechanism defined as follows: Definition 1 (ϵ, δ)-DP [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Consider two neighboring datasets D and D′, which differ in only one data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A randomized mechanism M is (ϵ, δ)-DP if for any two D, D′ and measurable subset O ⊆ Range(M), we have Pr[M(D) ∈ O] ≤ exp(ϵ) · Pr � M(D′) ∈ O � + δ, (1) 6 where ϵ > 0 represents the privacy protection level, and δ stands for the probability to break ϵ-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A smaller ϵ means that it is more difficult to distinguish between the two neighboring datasets D and D′, thus resulting in stronger privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It should be noted that ϵ is also referred to as the privacy loss, which quantifies the amount of privacy compromised through a single round of information (model parameters) exchange with the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In practical FL systems, it is crucial to monitor the total privacy loss over multiple communication rounds of model parameters exchange with the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Though its computation is quite involved, its upper bound can be computed using the moment accountant method [36], which so far yields the tightest bound on the total privacy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Lemma 1 [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='22] Suppose the query function g accesses the dataset D via randomized mechanism M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, the noise scale added to query function g to guarantee � ϵ, δ � DP is given by σ = 2s2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) ϵ2 , (2) where s is the ℓ2-norm sensitivity of the function g defined by s ≜ max D,D′ ��g(D) − g � D′���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (3) According to the privacy amplification theorem [21], it has been known that, running on a randomly generated subset of a dataset, the DP mechanism can yield stronger privacy protection than running on the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' This fact implies that the noise variance required for achieving a predefined DP level can be reduced when partial data are randomly selected at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The privacy analysis to be addressed in Section IV-B relies on the following privacy amplification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Theorem 1 (Privacy Amplification Theorem [21]) Suppose that a mechanism M is (ϵ, δ)-DP over a given dataset D with size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Consider the subsampling mechanism that outputs a sample from the uniform distribution over all subsets Ds ⊆ D with size b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, when ϵ ≤ 1, executing M mechanism on the subset Ds guarantees (ϵ′, δ′)-DP, where ϵ′ and δ′ are given by ϵ′ = 2qϵ, δ′ = qδ, (4) where q = b/n is the data sampling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof: The proof mainly follows the work [21] by considering both data sampling with replacement case and that without replacement case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Specially, when ϵ ≤ 1 and data are uniformly sampled with replacement, data subsampling mechanism guarantees (ln(1 + q(exp(ϵ) − 1), qδ)-DP [21], we have ln(1 + q(exp(ϵ) − 1)) (a) ≤ q(exp(ϵ) − 1) (b) ≤ 2qϵ, (5) 7 where (a) and (b) hold because ln(1 + x) ≤ x and exp(x) − 1 ≤ 2x when 0 < x ≤ 1 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' When data are uniformly sampled without replacement, the data subsampling mechanism yields (ϵ′′, qδ)- DP [21], where ϵ′′ = ln � 1 + (1 − (1 − 1 n))b(exp(ϵ) − 1) � (a) ≤ ln � 1 + q(exp(ϵ) − 1) � (b) ≤ 2qϵ, (6) where (b) follows because of (5), and (a) holds since (1 − (1 − 1 n))b ≤ b n = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (7) By combining (5) and (6), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ According to Theorem 1, the privacy would be amplified when q ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Note that, the privacy amplification for local DP is pervasively adopted in existing FL literatures [9], [22] since only a small portion of data being used in local SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Centralized clustering model Let X be a data matrix that contains n data samples and each sample has m features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', X = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , xn] ∈ Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The clustering task is to assign the n data samples of X into a predefined number of k clusters such that the samples within a cluster are closer to each other than to those belonging to any other cluster in terms of a certain distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Among hundreds of clustering algorithms, the most classic and popular one is the k-means which aims to obtain k non-overlapping clusters {Ci}k i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', Ci ∩ Ci′ = ∅, ∀i ̸= i′ ∈ [k], �k i=1 Ci = {xj}n j=1, by minimizing the average Euclidean distance between each cluster centroid and all the data samples within the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' From the perspective of optimization, the k-means algorithm can be viewed as an ad hoc algorithm, which handles the following matrix factorization by alternative minimization (AM) [38]: min W∈Rm×k,H ∥X − WH∥2 F (8a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' H(i, j) ∈ {0, 1}k×n, ∥H(:, j)∥0 = 1, ∀j, (8b) where W ∈ Rm×k is a matrix consisting of the k centroids, and H is an indicator matrix with only one non-zero element (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', unity) in each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Applying AM to problem (8) gives rise to the following update rules of W and H at iteration t + 1: Ht+1 = arg min H ∥X − WtH∥2 F , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' H(i, j) ∈ {0, 1}k×n, ∥H(:, j)∥0 = 1, ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (9) 8 Wt+1 = arg min W∈Rm×k ∥X − WHt+1∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (10) Closed-form solutions to (8) and (9) are respectively given by Ht+1(l, j) = � � � � � 1 if l = arg min u ∥X(:, j) − Wt(:, u)∥2, 0 otherwise, (11) and Wt+1(:, l) = 1 |J t l | � u∈J t l X(:, u), (12) where J t l = {j|H(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' j) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Note that at the iteration t + 1, the l-th row of H is updated according to the minimum distance from each data sample to the l-th centroid according to Wt, and then the l-th centroid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', the l-column of W) is updated as the average of the data belonging to cluster l according to the l-th row of the updated indicator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Despite its empirical effectiveness, the k-means algorithm fails for the dataset with complex distribution and data heterogeneity, and it is not suitable for distributed environments either, especially the FL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The reasons are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' First, the non-convex k-means problem (8) is NP-hard due to involving binary variables, implying that almost any algorithm (including k-means) is unable to work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' No wonder, it’s performance is quite sensitive to the initial conditions, complex data distribution, and the obtained solution easily trapped in bad local minima and so forth [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, the less data samples the worse its performance, thus further downgrading its performance in FL scenarios, especially when the data are sensitive and under privacy concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Most existing FedC algorithms are based on k-means and operate in computation-aggregation fashion, but their performance may get seriously downgraded under FL scenarios, including massively distributed clients and severe client heterogeneity [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Following the idea in [41], we replace the binary constraint (8b) with a norm-based equality constraint and reformulate problem (8) as min W,H ∥X − WH∥2 F + µh 2 ∥H∥2 F + µw 2 ∥W∥2 F (13a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' H ≥ 0, ∥H(:, j)∥2 1 = ∥H(:, j)∥2 2 , ∀j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (13b) where µh > 0 and µw > 0 are two positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Problem (13) is a non-convex and non-smooth problem and it can be regarded as a relaxation of the k-means problem (8) as H has been relaxed as a real k×n matrix, with at most one non-zero entry (not equal to one) in each column, though the equality constraint (13b) is still non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, the two regularization terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', the 2nd and the 3rd terms in (13a)) are used to control the resulting scaling/counter-scaling ambiguity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 9 Instead of solving problem (13), we consider the following problem by dropping the equality constraint in (13b) and adding an associated penalty term in the objective function: min W,H ∥X − WH∥2 F + µh 2 ∥H∥2 F + µw 2 ∥W∥2 F + ρ 2 N � i=1 �� 1⊤H(:, j) �2 − ∥H(:, j)∥2 2 � (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' H ≥ 0, (14b) where ρ > 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The larger the value of ρ, the smaller the approximation error of the equality constraint in (13b) and the more sparse the matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It is remarkable that problem (14) is much efficient to handle than problem (13) for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' One is that (14b) is a simple convex constraint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' the other is that the assignment of each data sample to an unique cluster is not reliable for problem (13) [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Therefore, in contrast to the hard clustering performed by k-means, solving (14) corresponds to seeking a soft clustering solution [44] instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Federated clustering model To solve problem (14) under the FL network, we assume that the data matrix is partitioned and distributed over N clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', X = [X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , XN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Specifically, each client i owns non-overlapping data Xi ∈ Rm×ni, where ni is the number of data samples in client i and �N i=1 ni = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Under the FL scenario, N could be large, and the data X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , XN could be unbalanced and non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Figure 2 illustrates the federated clustering framework in which a central server coordinates the N clients to accomplish the clustering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' As seen, the matrix H is partitioned in the same fashion as X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', H = [H1, H2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , HN], as each column of H corresponds to a certain data sample in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, one can reformulate problem (14) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' min W, Hi, i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=',N F(W, H) ≜ 1 N N � i=1 Fi(W, Hi) (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Hi ≥ 0, ∀i ∈ [N], (15b) where Fi(W, Hi) ≜∥Xi − WHi∥2 F + ρ 2(Tr(HiUH⊤ i ) − ∥Hi∥2 F ) + µh 2 ∥Hi∥2 F + µw 2 ∥W∥2 F (16) is the local objective function of each client i, and U ≜ 11⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In contrast to the vanilla FL problem which contains only one shared optimization variable, problem (15) involves two variables: one is W which is the cluster centroid matrix W shared among clients, and the other one is Hi which is local cluster indicator matrix for Xi owned by client i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' This apparently brings challenges in the algorithm development, especially in the presence of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In parallel, as W is shared, there certainly exist possibilities of leaking clients’ privacy in the FL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Recent 10 Client 1 = 𝑚 × 𝑛 𝐗 𝐖 𝑚 × 𝑘 𝐇 𝑘 × 𝑛 𝐖 = 𝐗𝟏 𝐇𝟏 𝑚 × 𝑛1 𝑚 × 𝑘 𝑘 × 𝑛1 × Client 2 𝐖 = 𝐗𝟐 𝐇𝟐 𝑚 × 𝑛2 𝑚 × 𝑘 𝑘 × 𝑛2 Client N 𝐖 = 𝐗𝑵 𝐇𝑵 𝑚 × 𝑛𝑁 𝑚 × 𝑘 𝑘 × 𝑛𝑁 × × × Parameter server •• •• •• Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 2: The framework of federated clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' work [47] showed that the honest-but-curious server could infer clients’ private data from the uploaded information in the federated matrix factorization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Consequently, it is inevitable to develop an effective and privacy-preserving FL algorithm for problem (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' PROPOSED ALGORITHM FOR PROBLEM (15) In this section, we develop a FedC algorithm to solve (15), which judiciously updates W and Hi, i ∈ [N], and adopts an amplified DP for rigorous privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Update of W and Hi in FL The key of algorithmic development to problem (15) is to specify how to perform the local update of Hi and global update of W in the presence of FL challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Inspired by [24], we follow the same spirit of local SGD and PCP, where a subset of clients are selected to locally update Hi and the associated local copies Wi’s of W, and then upload these iterates to the server for global aggregation in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In particular, for round t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', (a) Client sampling: We let the PS uniformly sample a small and fixed-size set St of K clients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', St ⊆ [N], |St| = K, and then broadcast the global Wt−1 to all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 11 (b) Local update: All clients are asked to obtain an approximate solution (Wt i, Ht i) to the following local subproblem of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (Wt i, Ht i) = arg min W,Hi≥0 Fi(W, Hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (17) After that, each client i ∈ St uploads Wt i to the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (c) Global aggregation: After receiving Wt i from all clients i ∈ St, the PS aggregates them to produce the new global Wt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', Wt = 1 K � i∈St Wt i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (18) In order to specify the local iterates (Wt i, Ht i), we propose to handle (17) by combining one-step AM [48] and local SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' That is, Ht i is produced by applying multiple steps of gradient descent (GD) to (17) with Wi fixed, and then Wt i is updated similarly by fixing Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' To be more specific, we first let all clients perform Q1 ≥ 1 consecutive steps of projected GD with respect to Hi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , Q1, Ht,r i = � Ht,r−1 i − 1 γt i ∇HiFi(Wt−1, Ht,r−1 i ) �+ , (19) where γt i > 0 is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, they are asked to perform Q2 ≥ 1 consecutive steps of SGD (no projection) with respect to W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', for r = Q1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , Qt, Wt,r i = Wt,r−1 i − 1 ηt ∇W Fi(Wt,r−1 i , Ht,Q1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ), (20) where Qt = Q1 + Qt 2 and ηt > 0 is a step size, and ∇W Fi(Wt,r−1 i , Ht i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ) is the stochastic gradient computed using mini-batch dataset Bt,r i with size b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Lastly, (Wt i, Ht i) is obtained by setting Ht i = Ht,Q1 i and Wt i = Wt,Qt i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Privacy concern and enhancement Data security and privacy are primary concerns in FL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' To enhance data privacy, we apply the strategy of DP to the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In particular, in each round t, we add an artificially Gaussian noise matrix ξt i ∈ Rm×k to Wt i, where all the mk entries of ξt i are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Gaussian random variables with zero mean and variance σ2 i,t, thus yielding a new matrix � Wt i as follows: � Wt i = Wt i + ξt i, (21) and then upload � Wt i to the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, (18) becomes Wt+1 = 1 K � i∈St � Wt i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (22) 12 The details of the proposed algorithm are summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Note that, the diminishing Qt 2 = ⌊ �Q t ⌋ + 1 (line 12) denotes the number of iterations in updating Wt,r i (lines 13-15) by (20), where the mini-bacth dataset Bt,r i of size b used is addressed in the following remark: Remark 1 For lines 13-15 of Algorithm 1, Qt 2b data samples are obtained from the dataset Di at each round (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', the data sampling ratio qi,t = Qt 2b/ni), and then divided into Qt 2 mini-batch datasets Bt,r i of size b for each inner iteration r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It is acknowledged that the DP noise matrix ξt i will bring about adverse effects on algorithm conver- gence and learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, the performance degradation of Algorithm 1 will get worse from round to round due to W perturbed by the DP noise and the coupling of W and H, on one hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The accumulated DP noise effects will also get more serious with the number of rounds on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Therefore, Algorithm 1 is performance-sensitive to the DP noise in a complicated manner, such that obtaining a satisfactory privacy-utility tradeoff through theoretical analysis becomes more intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Nevertheless, the privacy amplification presented in Theorem 1, can be utilized to pursue the perfor- mance analysis of Algorithm 1, in order to find the clue about the variance reduction of the DP-noise for guaranteeing (ϵ, δ)-DP privacy protection level at each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The details are presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' THEORETICAL ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Assumptions We need the following assumptions to analyze the privacy guarantee and convergence performance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Assumption 1 Each local cost function Fi is continuously differentiable in both W and Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' That is, ∇HiFi(Wt, ·) is Lipschitz continuous with constant Lt Hi, and ∇W Fi(·, Ht i) is Lipschitz continuous with constant Lt Wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', for any x, y, we have ∥∇HiFi(Wt, x) − ∇HiFi(Wt, y)∥ ≤ Lt Hi∥x − y∥, (23) ∥∇W Fi(x, Ht i) − ∇W Fi(y, Ht i)∥ ≤ Lt Wi∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (24) According to Assumption 1 and [31], ∇W F(·, Ht) is Lipschitz continuous with a constant Lt W = � (1/N) �N i=1(Lt Wi)2, together with upper and lower bounds for Lt Hi and Lt Wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', LW ≥ Lt Wi ≥ LW > 0, LH ≥ Lt Hi ≥ LH > 0, ∀i, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (25) 13 Algorithm 1 DP-FedC algorithm 1: Input: initial values of W0 1 = · · · = W0 N = W0, initial values of {H0 i }N i=1, S0 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' , N}, R and �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 2: for round t = 1 to R do 3: Server side: 4: Compute Wt by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 5: Uniformly sample a set of clients St ⊆ [N], and broadcast Wt to all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 6: Client side: 7: for client i ∈ [N] in parallel do 8: Set Ht,0 i = Ht−1 i and Wt,0 i = Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 9: for r = 1 to Q1 do 10: Update Ht,r i by (19), and set Wt,r i = Wt,r−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 11: end for 12: Compute Qt 2 = ⌊ �Q t ⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 13: for r = Q1 + 1 to Qt = Q1 + Qt 2 do 14: Update Wt,r i by (20), and set Ht,r i = Ht,r−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 15: end for 16: end for 17: Set Wt i = Wt,Qt i and Ht i = Ht,Qt i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 18: for client i ∈ St in parallel do 19: Compute � Wt i by (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 20: Upload � Wt i to the PS for next round of aggragation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 21: end for 22: end for Assumption 2 The gradient of each local cost function Fi, ∀i ∈ [N] is bounded, and Fi is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For any i ∈ [N], we have ∥∇W Fi(W, Hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bi)∥2 F ≤ G2, ∀W, Hi ≥ 0, (26a) Fi(W, Hi) ≥ F > −∞, ∀W, Hi ≥ 0, (26b) where G is a constant, and Bi denotes the mini-batch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 14 Assumption 3 Denote Bt i as mini-batch dataset with size b randomly sampled from dataset Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, for any i ∈ [N], the following equations hold, E[∇W Fi(Wt i, Ht i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt i)] = ∇W Fi(Wt i, Ht i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (27) E[∥∇W Fi(Wt i, Ht i) − ∇W Fi(Wt i, Ht i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt i)∥2 F ] ≤ φ2 b , (28) where φ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Assumption 4 (ζ-non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data) All the local objective functions are ζ-non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', namely, the following condition holds: ∥∇W Fi(W, Hi) − ∇W F(W, H)∥2 F ≤ ζ2, ∀W, H ≥ 0, (29) where ζ ≥ 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Note that, (29) in Assumption 4 is a metric to measure the degree of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The above assumptions are widely adopted in the FL literature, especially for non-convex FL problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Privacy analysis 1) Privacy guarantee: The ℓ2-norm sensitivity [10] of Wt i is stated in following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Lemma 2 For any t ∈ [R] and i ∈ [N], the ℓ2-norm sensitivity of uploaded local model Wt i is st i = 2GQt 2 ηt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (30) Proof: See the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' According to Lemma 1 and Lemma 2, we have the following theorem, which can serve as a useful reference for determining the variance of DP noise necessary to fulfill the associated DP-based FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Theorem 2 For any client i ∈ [N], suppose that ϵ ≤ 1, δ ≤ 1, and the data sampling ratio qi,t = Qt 2b/ni (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Each entry of ξt i is sampled from the Gaussian distribution with zero mean and variance σ2 i,t, where σ2 i,t = 32G2(Qt 2)2q2 i,t ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25qi,t/δ) (ηt)2ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (31) Then each communication round of the proposed algorithm guarantees (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof: In each communication round of the proposed algorithm, each client i performs Qt 2 steps of SGD w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' W by (20), where the mini-batch dataset with size b used is randomly sampled without replacement from local dataset Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' According to Lemma 1 and Theorem 1, the Gaussian noise with variance σ2 i,t = 2s2 i,t ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) ϵ2 (32) 15 can achieve at least (2qi,tϵ, qi,tδ)-DP for client i, where qi,t = Qt 2b/ni is data sampling ratio for client i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, by plugging s2 i,t given by (30) into (32), we obtain σ2 i,t = 4G2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) (ηt)2ϵ2 , ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (33) By (33), one can achieve an (ϵ, δ)-DP for Wt i, by replacing ϵ and δ in (33) with ϵ/2qi,t and δ/qi,t, respectively, leads to (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ 2) Total privacy loss: As done in [3], we also use the moments accountant method to estimate the total privacy loss when the algorithm runs R communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Theorem 3 Suppose that client i is uniformly sampled by the PS with a probability pi and the data sampling ratio qi,t = Qt 2b/ni (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Remark 1), where Qt 2 = ⌊ �Q t ⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A proper noise with variance σ2 i,t given in Theorem 2 is added to guarantee (ϵ, δ)-DP at each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, the total privacy loss ¯ϵi for client i after R communication rounds is given by ¯ϵi = c0q2 i,tϵ � piR 1 − qi,t , ∀i ∈ [N], (34) where c0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof: The proof basically follows that of Theorem 1 reported in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, we further consider privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Thus, the desired result (34) can be obtained by replacing the ϵ with 2qi,tϵ in the corresponding ¯ϵi in Theorem 1 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ Theorem 3 shows that the bound of total privacy loss for the proposed DP-FedC is tighter than that of the cutting-edge reported in [3] when p and q are appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Remark 2 When clients are uniformly sampled by the PS with a probability pi, The (34) in Theorem 3 demonstrates that Algorithm 1 guarantees (O(qϵ√pR), δ)-DP when running R rounds, where p and q are given by q = max i,t q2 i,t �1 − qi,t , ∀i ∈ [N], t ∈ [R], (35) p = max i pi, ∀i ∈ [N], (36) where qi,t = Qt 2b/ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Convergence analysis To find some convergence conditions, we define the following sequence W t,r = � � � � � � � 1 K � i∈St Wt,r i , when r ∈ [Qt − 1], 1 K � i∈St � Wt,Qt i + ξt i � , when r = Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (37) 16 W t,r is the instantaneous weighted average of local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Motivated by [31], we use the following terms as the optimality gap between a stationary solution of problem (15) GH(W t,r, Ht,r) ≜ N � i=1 (γt i)2��Ht,r i − � Ht,r i − 1 γt i ∇HiFi(W t,r, Ht,r i ) �+��2 F , ∀r ∈ [Q1], (38) GW (W t,r, Ht,r) ≜ ∥∇W F(W t,r, Ht,r) � ∥2 F , ∀r ∈ [Qt] \\ [Q1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (39) If GH(W t,r, Ht,r) = 0 and GW (W t,r, Ht,r) = 0, then (W t,r, Ht,r) is a stationary solution of problem (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The main theoretical result for the DP-FedC is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Theorem 4 Let T = RQ1 + �R t=1 Qt 2 be the total number of gradient evaluations per client and R be the total number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, let Qt 2 = ⌊ �Q t ⌋ + 1, γt i = α1Lt H/2 and ηt = α2Lt W , where α1 > 1 and α2 ≥ Qt 2 � 3(1 + L 2 W /L2 W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, under Assumptions 1-4, the sequence {(W t,r, Ht,r)} yielded by Algorithm 1 satisfies 1 T � R � t=1 Q1 � r=1 E[GH(W t,r−1, Ht,r−1)] + R � t=1 Qt � r=Q1+1 E[GW (W t,r−1, Ht,r−1)] � ≤2(α2 1L 2 H + 1) T � α2LW � F(W 1,0, H1,0) − F � + 16mkG2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) �R t=1(Qt 2)3 α2ϵ2 + LW φ2 �R t=1 Qt 2 2KbLW α2 + ζ2 �R t=1 Qt 2 K + 4Nζ2 �R t=1 Ct 1 K2α2 2 � , (40) where Ct 1 = Qt 2(Qt 2 − 1)(2Qt 2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (41) Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ Theorem 4 provides an upper bound of the average total local SGDs over R communication rounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' the smaller its value, the higher convergence rate and the smaller of the objective value in (15) achieved by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Based on Theorem 4, we have the following remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Remark 3 (Convergence rate analysis) Since Qt 2 = ⌊ �Q t ⌋ + 1, then we have �R t=1 Ct 1, �R t=1 Qt 2 and �R t=1(Qt 2)3 all in O(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' According to (40), if we set Q1 = √ R, then the proposed algorithm converges at a rate of O(1/ √ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Remark 4 (Impact of DP) The larger value of ϵ (or ¯ϵ), the smaller the upper bound in (40), implying that the better learning performance (convergence rate and the objective value) and the weaker required privacy protection level, namely a privacy-utility tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 17 Remark 5 (Impact of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data and PCP) The smaller the value of ζ or the larger the value of K, the smaller the upper bound in (40), implying that the smaller degree of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data or the more clients in PCP, the better learning performance (convergence rate and the objective value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' EXPERIMENT RESULTS In this section, in terms of the achieved value of the objective function in (15) and clustering accuracy, some experimental results are presented to evaluate the performance of the proposed DP-FedC algorithm (Algorithm 1) including comparison with some state-of-the-art FedC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The experiment is performed using two real datasets and each obtained result is the average over 5 independent runs with the same randomly generated initial feasible points for all the algorithms under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Experiment setup Datasets: The two real data sets used in the experiment are TCGA [34] and MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Specif- ically, TCGA dataset was obtained from the Cancer Genome Atlas database which contains the gene expression data of 5,314 cancer samples belonging to 20 cancer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Each data sample in TCGA dataset is a 5000 × 1 real vector containing the top-ranked 5000 features selected through Pearson’ Pearson’s Chi-Squares Test [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The MNIST database contains 60,000 training images of 10 handwritten digits and 10,000 test ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' We randomly select 10,000 images from the 60,000 training images as the dataset in our experiment, where each data sample is a 784 × 1 real vector containing 784 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In the experiment, we distribute the samples of each dataset to N = 100 clients in the following two ways: (i) IID case: We follow the data partition method in [28] to obtain balanced and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' distributed data for the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' To be specific, the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' distributed data are generated by randomly assigning the data samples to all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (ii) non-IID case: For the TCGA dataset, we apply the k-means algorithm to cluster the dataset into 100 clusters, and the data samples belonging to the same cluster is assigned to one client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For the MNIST dataset, we follow the partition method in [24] to obtain distributed data such that each client’s dataset only contains two digits, thus yielding a highly unbalanced and non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Parameter setting: In problem (15), if not mentioned specifically, we set the parameters µw = 0, ρ = 10−7× ∥X∥2 F N and µh = 10−10× ∥X∥2 F N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' As for the parameters in Algorithm 1, the step size γt i = 1 2Lt Hi where Lt Hi is estimated to be λmax((Wt,0 i )⊤Wt,0 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Analogously, the step size ηt = 5Lt W where Lt W is estimated to be λmax(Ht,Q1(Ht,Q1)⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Given the total privacy loss ¯ϵ, the privacy protection level ϵ 18 at each communication round is obtained by Theorem 3 for R = 100 and δ = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The mini-batch dataset size b is set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Other parameters are empirically chosen to our best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' All the algorithms under test runs until R = 100 is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then the clustering accuracy is calculated as the ratio of the number of correct classifications (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' of columns of all the estimated Hi, i ∈ [N], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', their maximum column entries falling in the correct cluster) to the total number of data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Impact of DP Figure 3 depicts the objective value (value of F(W, H) in (15)) and the clustering accuracy versus communication round with different values of ¯ϵ for both IID case and non-IID case, where K = 30, Q1 = 10, and Qt 2 = ⌊ 10 t ⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Some observations from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 3(a)-(d), are as follows: (i) The larger the value of ¯ϵ where the results without DP conceptually corresponds to ¯ϵ → ∞, the smaller the objective value and the higher the clustering accuracy and convergence rate for both IID case case and non-IID case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (ii) The objective value is smaller and the clustering accuracy is higher for the IID case than for non- IID case, and the performance gap between the two cases seems more appreciable in clustering accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The above three observations also apply to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 3(e)-(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Moreover, the impact of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' data is more serious for the TCGA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' These results are consistent with Remark 4 and Remark 5, so a proper choice of ¯ϵ value is needed to achieve a good privacy-utility tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Impact of the number of participated clients (K) Figure 4 depicts the convergence performance of DP-FedC versus communication rounds under dif- ferent values of K with ¯ϵ = 20, Q1 = 10, and Qt 2 = ⌊ 10 t ⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 4(a), 4(b), 4(e) and 4(f), that the objective value is smaller together with faster convergence rate either for larger K or for the IID case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' This is also true for the clustering accuracy, though the convergence rate on TCGA for the IID case is only slightly better than for the non-IID case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' These results are also consistent with Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Comparison with existing distributed clustering methods We here compare the proposed DP-FedC algorithm with four benchmark algorithms in terms of clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' These algorithms include federated k-means (FKM) [14], federated fuzzy k- means (FZKM) [12], distributed k-means++ (DK++) [25], distributed k-median (DKM) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The first two are state-of-the-art federated clustering algorithms while the latter two are traditional distributed 19 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 1 Objective value (a) MNIST, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 1 Objective value (b) MNIST, non-IID 0 20 40 60 80 100 Communication rounds 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 Clustering accuracy (c) MNIST, IID 0 20 40 60 80 100 Communication rounds 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 Clustering accuracy (d) MNIST, non-IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 1 Objective value (e) TCGA, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 2 Objective value (f) TCGA, non-IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='75 Clustering accuracy (g) TCGA, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 Clustering accuracy (h) TCGA, non-IID Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 3: Objective value and clustering accuracy versus communication rounds under IID case and non-IID case for three ¯ϵ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='45 Objective value (a) MNIST, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='45 Objective value (b) MNIST, non-IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='41 Clustering accuracy (c) MNIST, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='41 Clustering accuracy (d) MNIST, non-IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 Objective value (e) TCGA, IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7 1 Objective value (f) TCGA, non-IID 0 20 40 60 80 100 Communication rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='75 Clustering accuracy (g) TCGA, IID 0 20 40 60 80 100 Communication rounds 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6 Clustering accuracy (h) TCGA, non-IID Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 4: Objective value and clustering accuracy versus communication rounds under IID case and non-IID case for K ∈ {10, 30, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' As mentioned previously, they were basically developed by extending the k-means algorithm and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' We add the artificial noise to DP noise that guarantees the (ϵ, δ)-DP at each communication round in the implementation of the above four existing algorithms in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' 20 TABLE I: Performance comparisons of all the algorithms under test in terms of clustering accuracy (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Method Dataset TCGA (without DP) MNIST (without DP) TCGA (¯ϵ = 20) MNIST (¯ϵ = 20) DK++ [25] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 DKM [28] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 FKM [14] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 FZKM [12] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4 DP-FedC 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1 Then we apply the proposed algorithm to process the given dataset with parameters K = 30, Q1 = 10, Qt 2 = 5, ¯ϵ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' However, the parameters used for the other four algorithms are taken from the associated references together with K = 30, ¯ϵ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The obtained experimental results (for the clustering accuracy) are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' It can be seen from this table that the clustering accuracy performances of all the algorithms under test results for the case of without DP noise are better than with DP noise used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The performance gap in between the two cases for our DP-FedC algorithm is much smaller than for the other algorithms, implying that the proposed algorithm is more robust again DP noise thanks to the privacy amplification strategy applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' CONCLUSION We have presented a novel FedC algorithm, called DP-FedC, which operates following computation- aggregation protocol, and considering PCP, multiple local SGD steps, mini-batch dataset, and data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' In particular, based on the privacy amplification theorem, a necessary variance of the zero-mean DP noise added to all the local models at each communication round is obtained to guarantee the required privacy protection level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The efficacy (convergence rate and clustering accuracy) of the proposed DP-FedC is well supported by both theoretical convergence analysis and experimental results, including its much superior performance over some state-of-the-art FedC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 2 Assume Di and D′ i are the neighboring datasets that differ in only one data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Without loss of generality, let ui be the unique different element between Di and D′ i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=', D′ i ∪ {ui} = Di ∪ {ui}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For clarity of the following proof, let us make the following notational correspondences: Wt,r i ↔ Wt,r Di, 21 Ht,r i ↔ Ht,r Di, and Bt,r i ↔ Bt,r Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, for any r ∈ [Qt] \\ [Q1], the ℓ2-sensitivity [10] of Wt i is calculated by st i = max Di,D′ i ��Wt Di − Wt D′ i �� = max Di,D′ i ��� Qt � r=Q1+1 Wt,r−1 Di − ∇W Fi(Wt,r−1 Di , Ht,r−1 Di ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r Di) ηt − Qt � r=Q1+1 � Wt,r−1 D′ i − ∇W Fi(Wt,r−1 D′ i , Ht,r−1 D′ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r D′ i) ηt ���� = max Di,D′ i ��� � Wt,Q1 Di − ∇W Fi(Wt,Q1 Di , Ht,Q1 Di ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,Q1+1 Di ) ηt − · · · − ∇W Fi(Wt,Qt−1 Di , Ht,Qt−1 Di ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,Qt Di ) ηt � − � Wt,Q1 D′ i − ∇W Fi(Wt,Q1 D′ i , Ht,Q1 D′ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,Q1+1 D′ i ) ηt − · · · − ∇W Fi(Wt,Qt−1 D′ i , Ht,Qt−1 D′ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,Qt D′ i ) ηt ���� (a) ≤ 2GQt 2 ηt , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) where (a) holds because of Assumption 2, and Wt,Q1 Di = Wt,Q1 D′ i always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ APPENDIX B PROOF OF THEOREM 4 According to (37) and (20), we have W t,r =W t,r−1 − 1 Kηt � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) (Objective Descent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' H) According to [49, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2] and setting γt i = α1Lt H/2 ≤ α1LH/2 where α1 > 1, we have Fi(W t,r, Ht,r i ) − Fi(W t,r−1, Ht,r−1 i ) ≤ −α1 − 1 2 LH∥Ht,r−1 i − Ht,r i ∥2 F , ∀r ∈ [Q1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) Taking expectation over two sides of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) and then summing up from r = 1 to Q1 yields E � Fi(W t,Q1, Ht,Q1 i ) � − E � Fi(W t,0, Ht,0 i ) � ≤ −α1 − 1 2 LH Q1 � r=1 E � ∥Ht,r−1 i − Ht,r i ∥2 F � , ∀r ∈ [Q1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) By taking the summation over two sides of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) from i = 1 to N, the objective function F descends with local updates of H is given by E[F(W t,Q1, Ht,Q1)] − E[F(W t,0, Ht,0)] ≤ −α1 − 1 2 LH Q1 � r=1 N � i=1 E � ∥Ht,r−1 i − Ht,r i ∥2 F � , ∀r ∈ [Q1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4) (Objective Descent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' W) Since Ht,r i = Ht,r−1 i (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' line 14 in Algorithm 1) and ∇W F(·, Ht,Q) is Lipschitz continuous under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, by the descent lemma [49, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1], when r ∈ [Qt − 1] \\ [Q1], we have 22 E � F(W t,r, Ht,r) � ≤E � F(W t,r−1, Ht,r−1) � + Lt W 2 E � ∥W t,r − W t,r−1∥2 F � + E � ⟨∇W F(W t,r−1, Ht,r−1), W t,r − W t,r−1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5) When r = Qt, by Algorithm 1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5) becomes, E � F(W t,Qt , Ht,Qt) � ≤E � F(W t,Qt−1, Ht,Qt−1) � + Lt W 2 E � ∥W t,Qt − W t,Qt−1 + ξt∥2 F � � �� � ≜(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) + E � ⟨∇W F(W t,Qt−1, Ht,Qt−1), W t,Qt − W t,Qt−1⟩ � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6) where ξt = 1 K �K i=i ξt i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) can be further bounded by, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) =Lt W 2 E � ∥W t,Qt − W t,Qt−1 + ξt∥2 F � =Lt W 2 E � ∥W t,Qt−1 − W t,Qt ∥2 F ] + Lt W 2 E � ∥ξt∥2 F � (a) ≤ Lt W 2 E � ∥W t,Qt−1 − W t,Qt ∥2 F � + 16mkG2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ)(Qt 2)2 α2ηtϵ2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7) where (a) holds from ηt = α2Lt W and E � ∥ξt∥2 F � (a) = mk K K � i=1 32G2(Qt 2)2q2 i,t ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25qi,t/δ) (ηt)2ϵ2 (b) ≤ 32mkG2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) (ηt)2ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8) In (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8), (a) follows from (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (b) holds because of qi,t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7), for r ∈ [Qt]\\[Q1], we have, E � F(W t,r, Ht,r) � ≤E � F(W t,r−1, Ht,r−1) � + Lt W 2 E � ∥W t,r − W t,r−1∥2 F � � �� � ≜(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) + E � ⟨∇W F(W t,r−1, Ht,r−1), W t,r − W t,r−1⟩ � � �� � ≜(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) + 16mkG2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ηtϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='9) The terms (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) can be bounded by the following Lemma 3 (proved in Appendix C-A) and Lemma 4 (proved in Appendix C-B), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Lemma 3 For any t and r ∈ [Qt − 1] \\ [Q1], we have E � ∥W t,r − W t,r−1∥2 F � ≤ φ2 Kb(ηt)2 + 1 (ηt)2 E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='10) Lemma 4 For any t and r ∈ [Qt − 1] \\ [Q1], we have E � ⟨∇W F(W t,r−1, Ht,r−1), W t,r − W t,r−1⟩ � 23 = − 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) ��2 F + �� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2 F � + ζ2 Kηt + 1 Kηt N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='11) Thus, substituting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='10) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='11) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='9) gives rise to E � F(W t,r, Ht,r) � − E � F(W t,r−1, Ht,r−1) � ≤ − 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) ��2 F � + Lt W φ2 2Kb(ηt)2 + ζ2 Kηt + ( Lt W 2(ηt)2 − 1 2ηt )E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� + 1 Kηt N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � + 16mkG2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ηtϵ2 (a) ≤ − 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) ��2 F � + LW φ2 2Kb(ηt)2 + ζ2 Kηt + 1 Kηt N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � + 16mkG2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ηtϵ2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='12) where (a) follows due to ηt = α2Lt W ≥ Lt W and Lt Wi ≤ LW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, rearranging the two sides of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='12) yields E ���∇W F(W t,r−1, Ht,r−1) ��2 F � ≤2ηt� E � F(W t,r−1, Ht,r−1) � − E � F(W t,r, Ht,r) �� + 2 K N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � + LW φ2 Kbηt + 32mkG2(Qt 2)2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ϵ2 + 2ζ2 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='13) Summing (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='13) up from r = Q1 + 1 to Qt yields Qt � r=Q1+1 E ���∇W F(W t,r−1, Ht,r−1) ��2 F � ≤2ηt� E � F(W t,Q1, Ht,Q1) � − E � F(W t,Qt , Ht,Qt) �� + 2 K Qt � r=Q1+1 N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � � �� � ≜(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4) + 32mkG2(Qt 2)3 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ϵ2 + 2ζ2Qt 2 K + Qt 2LW φ2 Kbηt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='14) The term (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4) can be bounded with the following lemma, which is proved in Appendix C-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Lemma 5 Let α2 ≥ Qt 2 � 3(1 + L 2 W /L2 W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' For any t and r ∈ [Qt] \\ [Q1], it holds that 24 Qt � r=Q1+1 N � i=1 (Lt Wi)2E[∥W t,r−1 − Wt,r−1 i ∥2 F ] ≤ 4Nζ2Ct 1 Kα2 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='15) where Ct 1 ≜ Qt 2(Qt 2 − 1)(2Qt 2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' By applying Lemma 5 and plugging (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='15) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='14), we have Qt � r=Q1+1 E ���∇W F(W t,r−1, Ht,r−1) ��2 F � ≤2ηt� E � F(W t,Q1, Ht,Q1)] − E[F(W t,Qt , Ht,Qt) �� + 32mkG2(Qt 2)3 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ϵ2 + 2ζ2Qt 2 K + Qt 2LW φ2 Kbηt + 8Nζ2Ct 1 K2α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='16) Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='16) yields Q1 � r=1 N � i=1 E � ∥Ht,r−1 i − Ht,r i ∥2 F � + Qt � r=Q1+1 E ���∇W F(W t,r−1, Ht,r−1) ��2 F � (a) ≤2ηt� E � F(W t,0, Ht,0)] − E[F(W t,Qt , Ht,Qt) �� + 32mkG2(Qt 2)3 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ϵ2 + 2ζ2Qt 2 K + Qt 2LW φ2 Kbηt + 8Nζ2Ct 1 K2α2 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='17) where (a) holds because of ηt ≥ 1/((α1 − 1)LH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (Derivation of the Main Result) We next derive the convergence in terms of the optimal gap functions in (38) and (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' From (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='17) and γt i = α1Lt H/2 and ηt = α2Lt W , we have Q1 � r=1 E[GH(W t,r−1, Ht,r−1)] = Q1 � r=1 N � i=1 (γt i)2E[∥Ht,r−1 i − Ht,r i ∥2 F ] (a) ≤2α2 1L 2 H � α2LW E[F(W t,0, Ht,0)] − E[F(W t,Qt , Ht,Qt)] + 16mkG2(Qt 2)3 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) α2ϵ2 + ζ2Qt 2 K + Qt 2LW φ2 2KbLW α2 + 4Nζ2Ct 1 K2α2 2 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='18) where (a) follows because γt i ≤ α1LH/2 and α2LW ≤ ηt ≤ α2LW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, summing (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='18) up from t = 1 to R yields R � t=1 Q1 � r=1 E � GH(W t,r−1, Ht,r−1) � ≤2α2 1L 2 H � α2LW � F(W 1,0, H1,0) − F � + 16mkG2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) �R t=1(Qt 2)3 α2ϵ2 + ζ2 �R t=1 Qt 2 K + LW φ2 �R t=1 Qt 2 2KbLW α2 + 4Nζ2 �R t=1 Ct 1 K2α2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='19) Similarly, from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='17), we have 25 R � t=1 Qt � r=Q1+1 E ���∇W F(W t,r−1, Ht,r−1) ��2 F � = R � t=1 Qt � r=Q1+1 E � GW (W t,r−1, Ht,r−1)] � ≤2α2LW � F(W 1,0, H1,0) − F � + 32mkG2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) �R t=1(Qt 2)3 α2ϵ2 + 2ζ2 �R t=1 Qt 2 K + LW φ2 �R t=1 Qt 2 KbLW α2 + 8Nζ2 �R t=1 Ct 1 K2α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='20) By combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='19) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='20), and then dividing two sides of summation result by T = RQ1+�R t=1 Qt 2 yields 1 T � R � t=1 Q1 � r=1 E � GH(W t,r−1, Ht,r−1) � + R � t=1 Qt � r=Q1+1 E � GW (W t,r−1, Ht,r−1) �� ≤2(α2 1L 2 H + 1) T � α2LW � F(W 1,0, H1,0) − F � + 16mkG2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='25/δ) �R t=1(Qt 2)3 α2ϵ2 + ζ2 �R t=1 Qt 2 K + LW φ2 �R t=1 Qt 2 2KbLW α2 + 4Nζ2 �R t=1 Ct 1 K2α2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='21) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ APPENDIX C PROOFS OF KEY LEMMAS FOR THEOREM 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof of Lemma 3 According to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1), we have E � ∥W t,r − W t,r−1∥2 F � = 1 (ηt)2 E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ) ��2� (a) = 1 (ηt)2 E ��� 1 K � i∈St � ∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ) − ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ���2� + 1 (ηt)2 E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� (b) = 1 (ηt)2K2 E � � i∈St ��∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ) − ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� + 1 (ηt)2 E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� (c) ≤ 1 (ηt)2 E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2� + φ2 Kb(ηt)2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='1) where (a) follows because E[∥Z∥2] = E[∥Z−E[Z]∥2]+∥E[Z]∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (b) follows because ∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i )− ∇W Fi(Wt,r−1 i , Ht,r−1 i ) is independent across the clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (c) holds due to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ 26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof of Lemma 4 E � ⟨∇W F(W t,r−1, Ht,r−1), W t,r − W t,r−1⟩ � (a) = − 1 ηt E �� ∇W F(W t,r−1, Ht,r−1), 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,r i ) �� (b) = − 1 ηt E �� ∇W F(W t,r−1, Ht,r−1), 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) �� (c) = − 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) ��2 F � − 1 2ηt E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2 F � + 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) − 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2 F � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) where (a) holds due to (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (b) follows from Assumption 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (c) follows from the basic identity ⟨Z1, Z2⟩ = 1 2(∥Z1∥2 + ∥Z2∥2 − ∥Z1 − Z2∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' The last term in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) can be further bounded by E ���∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) − 1 K � i∈St ∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ) ��2 F � =E ��� 1 K � i∈St � ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) − ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) + ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) − ∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ) ���2 F � ≤ 2 K2 E ��� � i∈St � ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1)∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) ���2 F � + 2 K2 E ��� � i∈St � ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1) − ∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ) ���2 F � (a) ≤ 2ζ2 K + 2 K E � � i∈St (Lt Wi)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ∥2 F � ≤2ζ2 K + 2 K N � i=1 (Lt Wi)2E � ∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ∥2 F � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) where the first term in the RHS of (a) comes from Assumption 4, and the second term in the RHS of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8) follows because of Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, Plugging (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='3) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='2) yields E � ⟨∇W F(W t,r−1, Ht,r−1), W t,r − W t,r−1⟩ � ≤ − 1 2ηt E ���∇W F(W t,r−1, Ht,r−1) ��2 F � − 1 2ηt E ��� 1 K � i∈St ∇W Fi(Wt,r−1 i , Ht,r−1 i ) ��2 F � + ζ2 Kηt + 1 Kηt N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='4) Thus, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' ■ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Proof of Lemma 5 According to the definition of W t,r−1, for ∀r ∈ [Qt] \\ [Q1], we have W t,r−1 = 1 K � i∈St Wt,r−1 i 27 (a) = 1 K � i∈St � Wt − 1 ηt r−1 � j=Q1 ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) � = Wt − 1 ηtK r−1 � j=Q1 � i∈St ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5) where (a) is obtained by applying (20), that is Wt,r−1 i = Wt − 1 ηt r−1 � j=Q1 ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6) As a result, by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='5) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='6), we have E � ∥W t,r−1 − Wt,r−1 i ∥2 F � =E ����Wt − 1 ηtK r−1 � j=Q1 � i∈St ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) − � Wt − 1 ηt r−1 � j=Q1 ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) ���� 2 F � = 1 (ηt)2 E ���� 1 K r−1 � j=Q1 � i∈St ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) − r−1 � j=Q1 ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) ��� 2 F � ≤(r − Q1) (ηt)2 r−1 � j=Q1 E ���� 1 K � i∈St ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i ) − ∇W Fk(Wt,j−1 k , Ht,j−1 k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j k ) ��� 2 F � (b) = (r − Q1) (ηt)2 r−1 � j=Q1 ��� 1 K � i∈St � ∇W Fi(Wt,j−1 i , Ht,j−1 i ) − ∇W Fk(Wt,j−1 k , Ht,j−1 k ) ���� 2 F (c) ≤ (r − Q1) (ηt)2K r−1 � j=Q1 N � i=1 ���∇W Fi(Wt,j−1 i , Ht,j−1 i ) − ∇W Fk(Wt,j−1 k , Ht,j−1 k ) ��� 2 F , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7) where (b) holds since ∇W Fi(Wt,j−1 i , Ht,j−1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j i )−∇W Fk(Wt,j−1 k , Ht,j−1 k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Bt,j k ) is independent across the clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (c) follows by using the inequality ∥ �N i=1 zi∥2 ≤ N �N i=1 ∥zi∥2 for any vectors zi and any positive integer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, the term ∥∇W Fi(Wt,j−1 i , Ht,j−1 i ) − ∇W Fk(Wt,j−1 k , Ht,j−1 k )∥2 F in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7) can be bounded by ��∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) − ∇W Fk(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) ��2 F ≤ ���∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) − ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) + ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) − ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1) + ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1) − � ∇W Fk(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) − ∇W Fk(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) + ∇W Fk(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) − ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1) + ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1) ���� 2 F ≤4∥∇W Fi(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) − ∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i )∥2 F + 4∥∇W Fi(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ) − ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1)∥2 F + 4∥∇W Fk(Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) − ∇W Fk(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k )∥2 F + 4∥∇W Fk(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ) − ∇W F(W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1)∥2 F (d) ≤4(Lt Wi)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ∥2 F + 4(Lt Wk)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ∥2 F + 8ζ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8) 28 where (d) follows from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' Then, substituting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='8) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='7) gives rise to Qt � r=Q1+1 N � i=1 (Lt Wi)2E � ∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='r−1 i ∥2 F � ≤ Qt � r=Q1+1 N � i=1 (Lt Wi)2�(r − Q1 − 1) K(ηt)2 r−2 � j=Q1 N � i=1 � 4(Lt Wi)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ∥2 F + 8ζ2 + 4(Lt Wk)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 k ∥2 F �� (e) = N K Qt � r=Q1+1 4(r − Q1 − 1) (ηt/Lt W )2 r−2 � j=Q1 N � i=1 (Lt Wi)2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ∥2 F + N K Qt � r=Q1+1 8(r − Q1 − 1)2 (ηt/Lt W )2 ζ2 + N K Qt � r=Q1+1 4(r − Q1 − 1) (ηt/Lt W )2 r−2 � j=Q1 N � i=1 (Lt Wi)2 · �Lt Wi Lt W �2∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ∥2 F (f) ≤ 2NQt 2(Qt 2 − 1) Kα2 2 (1 + L 2 W L2 W ) Qt � r=Q1+1 N � i=1 (Lt Wi)2��W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 − Wt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='j−1 i ��2 F + 4NQt 2(Qt 2 − 1)(2Qt 2 − 1)ζ2 3Kα2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='9) where (e) follows since (Lt W )2 = (1/N) �N i=1(Lt Wi)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (f) follows due to (Lt Wi)2 (Lt W )2 ≤ L 2 W L2 W and ηt = α2Lt W , and Qt � r=Q1+1 (r − 1 − Q1) r−2 � j=Q1 aj ≤ Qt � r=Q1+1 Qt 2(Qt 2 − 1) 2 ar−1, ∀aj > 0, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='10) and Qt � r=Q1+1 (r − 1 − Q1)2 = Qt 2(Qt 2 − 1)(2Qt 2 − 1) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='11) Since α2 ≥ Qt 2 � 3(1 + L 2 W /L2 W ), implies α2 2 ≥ 2Qt 2(Qt 2 − 1)(1 + L 2 W /L2 W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content=' After rearranging (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} +page_content='9), we obtain Qt � r=Q1+1 N � i=1 (Lt Wi)2E � ∥W t,r−1 − Wt,r−1 i ∥2 F � ≤ 4NQt 2(Qt 2 − 1)(2Qt 2 − 1)ζ2 3K � α2 2 − 2Qt 2(Qt 2 − 1)(1 + L 2 W /L2 W ) (g) ≤ 4NQt 2(Qt 2 − 1)(2Qt 2 − 1)ζ2 Kα2 2 =4NCt 1ζ2 Kα2 2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNAzT4oBgHgl3EQfCPpJ/content/2301.00955v1.pdf'} 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Ho +,4, 5 W. N. Brandt,6, 7, 8 Catherine J. Grier +,9 +Patrick B. Hall +,10 Y. Homayouni +,6 Anton M. Koekemoer +,11 Donald P. Schneider +,6, 7 and +Jonathan R. Trump +12 +1Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +2Department of Astronomy, University of Michigan, Ann Arbor, MI, 48109, USA +3National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +4Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People’s Republic of China +5Department of Astronomy, School of Physics, Peking University, Beijing 100871, People’s Republic of China +6Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA +7Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA +8Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA +9Department of Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA +10Department of Physics & Astronomy, York University, 4700 Keele St., Toronto, ON M3J 1P3, Canada +11Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA +12Department of Physics, University of Connecticut, 2152 Hillside Rd Unit 3046, Storrs, CT 06269, USA +ABSTRACT +We measure the correlation between black-hole mass MBH and host stellar mass M∗ for a sample +of 38 broad-line quasars at 0.2 ≲ z ≲ 0.8 (median redshift zmed = 0.5). The black-hole masses are +derived from a dedicated reverberation mapping program for distant quasars, and the stellar masses +are estimated from two-band optical+IR HST imaging. Most of these quasars are well centered within +≲ 1 kpc from the host galaxy centroid, with only a few cases in merging/disturbed systems showing +larger spatial offsets. Our sample spans two orders of magnitude in stellar mass (∼ 109 −1011 M⊙) and +black-hole mass (∼ 107 − 109 M⊙), and reveals a significant correlation between the two quantities. +We find a best-fit intrinsic (i.e., selection effects corrected) MBH −M∗,host relation of log(MBH/M⊙) = +7.01+0.23 +−0.33 +1.74+0.64 +−0.64 log(M∗,host/1010M⊙), with an intrinsic scatter of 0.47+0.24 +−0.17 dex. Decomposing our +quasar hosts into bulges and disks, there is a similar MBH−M∗,bulge relation with slightly larger scatter, +likely caused by systematic uncertainties in the bulge-disk decomposition. The MBH −M∗,host relation +at zmed = 0.5 is similar to that in local quiescent galaxies, with negligible evolution over the redshift +range probed by our sample. With direct black-hole masses from reverberation mapping and a large +dynamical range of the sample, selection biases do not appear to affect our conclusions significantly. +Our results, along with other samples in the literature, suggest that the locally-measured black-hole +mass−host stellar mass relation is already in place at z ∼ 1. +Keywords: black hole physics – galaxies: active – quasars: general – surveys +1. INTRODUCTION +The observed scaling relations between supermassive +black hole (BH) masses and the properties of their host +galaxies (e.g., stellar mass and stellar velocity disper- +sion) in the local Universe are the foundation of mod- +ern BH−galaxy co-evolution models (Magorrian et al. +1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; +H¨aring & Rix 2004; G¨ultekin et al. 2009; McConnell & +Ma 2013; Kormendy & Ho 2013, and references therein). +The tight correlations suggest that active galactic nuclei +(AGNs) may play important roles in regulating star for- +mation in the host galaxies via self-regulated BH growth +and feedback processes (Silk & Rees 1998; Di Matteo +et al. 2005; Heckman & Best 2014). Studying BH scal- +ing relations beyond the local Universe is a key to un- +derstanding BH and galaxy (co-)evolution over cosmic +history. +Over the past two decades, various investigations have +built an inventory of BH and host measurements to +arXiv:2301.04177v1 [astro-ph.GA] 10 Jan 2023 + +ID2 +Li et al. +study the redshift evolution of BH−host relations up to +z ∼ 3, including the BH mass−stellar velocity disper- +sion (MBH − σ∗) relation (Treu et al. 2004; Woo et al. +2006, 2010; Shen et al. 2015a; Park et al. 2015; Sex- +ton et al. 2019), the BH mass−bulge/host luminosity +(MBH − L∗,bulge/host) relation (Peng et al. 2006a,b; De- +carli et al. 2010), the BH mass−bulge/host stellar mass +(MBH − M∗,bulge/host) relation (Jahnke et al. 2009; Ben- +nert et al. 2011; Dong & Wu 2016; Suh et al. 2020; Ding +et al. 2021), as well as expanding the local baselines +to include AGNs of different host properties and lower +BH masses (e.g., Jiang et al. 2011a,b; Greene et al. 2008; +Reines & Volonteri 2015; Bentz & Manne-Nicholas 2018; +Greene et al. 2020; Bennert et al. 2021; Zhao et al. 2021). +Some groups found deviations from local scaling rela- +tions as a function of z (Peng et al. 2006a,b; Merloni +et al. 2010; Woo et al. 2010; Park et al. 2015; Sexton +et al. 2019) while others found similar BH−host rela- +tions as in the local Universe (e.g., Jahnke et al. 2009; +Suh et al. 2020; Li et al. 2021a; Ding et al. 2022; Silver- +man et al. 2022), which is also supported by the tight +correlation between the BH accretion rate and star for- +mation rate in bulge-dominated galaxies at z = 0.5 − 3 +(e.g., Yang et al. 2019). +The measurements of BH mass−host relations can be +challenging beyond z ∼ 0.1 for several reasons. First, +direct BH mass measurements based on resolved stel- +lar/gas dynamics are difficult to obtain beyond the lo- +cal Universe where the BH sphere of influence can- +not be readily resolved. Reverberation mapping (RM; +Blandford & McKee 1982; Peterson 2014) is the primary +method of measuring BH masses for broad-line (BL) +AGN beyond the local Universe, but RM is resource- +intensive and only available for a small number of ob- +jects beyond z ∼ 0.1 (e.g., Bentz et al. 2013). A sec- +ondary BH mass recipe, the single-epoch (SE) virial +estimator, is based on the broad-line region (BLR) +radius−luminosity relation (the R−L relation) and can +be easily adapted for large samples of BLAGNs at higher +redshift. However, Shen & Kelly (2010) demonstrated +that there is a statistical bias in SE BH masses for +flux-limited samples from the uncertainties in these BH +masses. In addition, the applicability of SE masses to +the high-redshift and high-luminosity regime is not well +understood, primarily because the local RM AGNs used +to derive the R − L relation is not representative of the +general quasar population (e.g., Shen et al. 2015b; Fon- +seca Alvarez et al. 2020), and the extrapolated R − L +relations for broad Mg ii and C iv used for high-redshift +BLAGNs are not as well-studied as the local R − L re- +lation based on broad Hβ (Bentz et al. 2013). +Host-galaxy properties are also difficult to measure as +the unobscured AGN (where virial BH masses are feasi- +ble) usually far outshines the host galaxy. For imaging +studies, high-resolution images, such as those from the +Hubble Space Telescope (HST), are often necessary to +robustly decompose the quasar and host light. +How- +ever, rigorous image analysis reveals that host galaxies +of local AGNs (z < 0.35) often consist of complex struc- +tures, including spiral arms, tidal and merger features, +in addition to the main galaxy components (bars, bulges, +and disks) (Kim et al. 2017). These complex structures +are extremely challenging to measure even with HST at +higher redshifts. +Due to difficulties in obtaining BH mass and host +properties, many studies are limited to specific sam- +ples that may introduce selection biases. Earlier stud- +ies were often restricted to the bright end of BLAGN, +have small sample sizes and limited dynamical ranges +in BH/host properties. Lauer et al. (2007) showed that +over-massive BHs are favored in flux-limited studies due +to the intrinsic scatter of the scaling relations. For a +“bottom-heavy” galaxy luminosity function, there are +more low-mass hosts than high-mass ones. +However, +more massive BHs are preferentially selected in a flux- +limited sample based on AGN luminosity, resulting in an +average offset in the BH mass−host relations, and a shal- +lower slope than the true underlying relation. Schulze & +Wisotzki (2011, 2014) argued that additional selection +biases could arise from the lack of knowledge in the rel- +evant underlying distribution functions (e.g., the active +fraction of AGNs, bulge properties) and their evolution +with redshift. These biases can account for a large por- +tion of, if not all, the redshift evolution reported in ear- +lier investigations (Schulze & Wisotzki 2011; Shen et al. +2015a). +In this work, we study the BH scaling relations at +0.2 ≲ z ≲ 0.8 using the Sloan Digital Sky Survey Rever- +beration Mapping (SDSS-RM, Shen et al. 2015b) sam- +ple. The SDSS-RM sample has two major advantages +in measuring the redshift evolution of BH−host galaxy +relations: (1) the parent sample is a uniformly selected +flux-limited BLAGN sample, thus the selection effects +can be quantified and corrected, and (2) BH masses are +available from direct RM (Shen et al. 2016; Grier et al. +2017, 2019; Homayouni et al. 2020), rather than from +SE, masses. We have acquired high-resolution imaging +for the SDSS-RM sample with HST to measure the host +galaxy color and luminosity in two bands, tracing young +and old stellar populations, respectively. Our sample in- +cludes 38 sources (10 included in a pilot study in Li, J. I. +et al. 2021), which is comparable in size to the local RM +AGN sample used to calibrate the R−L relation (Bentz + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Redshift +42 +43 +44 +45 +46 +Log L5100, QSO [erg s +1] +SDSS-RM +H Lags +This Work +Local RM +Figure 1. Quasar luminosity and redshift distribution of our +sample (open orange circles), and a representative subset of +the local RM sample (grey squares). The parent SDSS-RM +sample (black dots) and those with Hβ lags (blue dots) are +also labeled for reference. +et al. 2013), and has sufficient statistics and dynamic +range in BH mass (and stellar mass) to characterize the +redshift evolution of BH scaling relations over the red- +shift range of 0.2 ≲ z ≲ 0.8. +This paper is organized as follows. We describe our +data and analysis in Section 2. The main results are +presented in Section 3. We discuss our results in Section +4 and conclude in Section 5. +Throughout this paper +we adopt a flat ΛCDM cosmology with ΩM = 0.3 and +H0 = 70 km s−1 Mpc−1. All host-galaxy measurements +refer to the stellar population only. +2. OBSERVATION AND DATA ANALYSIS +2.1. Sample +Our sample consists of 38 SDSS-RM quasars at 0.2 ≲ +z ≲ 0.8 (median redshift zmed = 0.5) with RM-based +BH masses; 37 of these RM masses were based on the +broad Hβ line (Shen et al. 2016; Grier et al. 2017), with +one source (RM767) based on the broad Mg ii line (Shen +et al. 2016). Ten sources in our HST sample were studied +in a pilot program (Li, J. I. et al. 2021); 28 sources +are presented in this work for the first time. +Among +the 44 quasars with Hβ RM BH masses in Grier et al. +(2017), seven sources beyond z ∼ 0.8 were excluded from +the HST programs to ensure more robust host-galaxy +measurements and to avoid unknown selection biases, as +the lag-detection fraction at z ≳ 0.8 is significantly lower +than that at lower redshifts (e.g., see Figure 1). Figure 1 +presents the redshift and luminosity distribution of our +sample, and Table 1 summarizes the physical properties +of these objects. +2.2. Black-Hole Masses +Reverberation mapping determines BH masses by +measuring the time delay in variability between the con- +tinuum and broad emission lines. The time delay corre- +sponds to the light travel time between the continuum- +emitting accretion disk and the BLR. Assuming the BLR +is virialized, a BH mass can be calculated using the av- +erage time lag (τ) and the width of the broad emission +line (∆V ) via the equation: +MBH = f cτ∆V 2 +G +, +(1) +where G is the gravitational constant and f is a di- +mensionless factor of unity order that accounts for BLR +geometry, kinematics, and inclination. The line width +∆V can be estimated from either the full-width half- +maximum (FWHM) or the line dispersion (σline) of the +broad line measured from the mean or RMS spectra +(e.g., Wang et al. 2019). +For the majority of our sources, we adopt the RM +black hole masses from Grier et al. (2017) computed us- +ing a consant virial coefficient of f =4.47 based on σline +measured from the RMS spectra (equivalent to f = 1.12 +when using the FWHM for ∆V ). For two of our sources, +RM316 and RM519, the original σline measurements in +Grier et al. (2017) based on the first-year SDSS-RM +spectroscopy are significantly overestimated; we adopt +updated σline measurements based on the 4-year SDSS- +RM spectroscopy for these two objects. +For RM767, +Shen et al. (2016) identified a lag between the contin- +uum and broad Mg ii line during the first-year monitor- +ing. However, the lag significance is reduced in the more +recent analysis in Homayouni et al. (2020) using 4-year +light curves, as the broad Mg ii line does not display +strong response to the continuum in the following years. +We adopt the Shen et al. (2016) Mg ii lag for RM767, +and use its σline measured from the RMS spectrum to +derive a BH mass. The BH mass uncertainties are cal- +culated by propagating the statistical uncertainties of +the lag and line width measurements, and then adding +a systematic uncertainty of 0.16 dex, which is the scat- +ter estimated from repeated RM measurements in local +RM campaigns (Fausnaugh et al. 2017). However, the +adopted BH mass uncertainty is still an underestima- +tion, as it does not account for the intrinsic scatter in +the virial coefficient for individual systems, which could +lead to additional BH mass uncertainties of as large as +∼ 0.3 dex. The BH masses are tabulated in Table 1 (with +updates from earlier work indicated by an asterisk). +2.3. HST Imaging Analysis +The HST observations for the 28 new objects were +conducted between 2019 December 23 and 2021 June 09 + +4 +Li et al. +Table 1. Target Properties +RMID +R. A. (J2000) +Dec. (J2000) +z +ipsf +L5100,QSO +log(MBH,SE) +log(MBH,RM) +(deg) +(deg) +(mag) +(erg s−1) +(M⊙) +(M⊙) +017 +213.3511 +53.0908 +0.4559 +19.21 +43.9 +8.36±0.04 +8.92+0.24 +−0.19 +033 +213.8848 +52.8183 +0.7147 +20.49 +44.1 +7.60±0.03 +7.23+0.23 +−0.20 +101 +213.0592 +53.4296 +0.4581 +18.84 +44.4 +7.89±0.004 +7.26+0.17 +−0.19 +160 +212.6719 +53.3136 +0.3593 +19.68 +43.8 +8.20±0.007 +7.85+0.18 +−0.17 +177 +214.3525 +52.5069 +0.4818 +19.56 +44.0 +8.43±0.03 +7.57+0.55 +−0.20 +191 +214.1899 +53.7463 +0.4418 +20.45 +43.6 +7.55±0.01 +6.90+0.22 +−0.16 +229 +212.5752 +53.4937 +0.4696 +20.27 +43.6 +8.00±0.07 +7.65+0.17 +−0.20 +265 +215.0995 +53.2681 +0.7343 +20.65 +44.2 +8.31±0.02 +8.58+0.23 +−0.26 +267 +212.8030 +53.7520 +0.5872 +19.62 +44.1 +7.92±0.02 +7.41+0.17 +−0.17 +272 +214.1071 +53.9107 +0.2628 +18.82 +43.9 +7.82±0.02 +7.58+0.18 +−0.21 +300 +214.9213 +53.6138 +0.6457 +19.49 +44.5 +8.19±0.02 +7.60+0.17 +−0.20 +301 +215.0427 +52.6749 +0.5477 +19.76 +44.1 +8.53±0.09 +8.64+0.25 +−0.22 +305 +212.5178 +52.5281 +0.5266 +19.50 +44.2 +7.92±0.01 +8.32+0.16 +−0.16 +316 +215.2185 +52.9396 +0.6760 +18.03 +45.0 +8.50±0.006 +∗7.55+0.17 +−0.17 +320 +215.1605 +53.4046 +0.2647 +19.47 +43.4 +8.06±0.02 +7.67+0.18 +−0.18 +338 +214.9818 +53.6687 +0.4177 +20.08 +43.4 +8.36±0.05 +7.69+0.27 +−0.24 +371 +212.8476 +52.2255 +0.4719 +19.57 +44.1 +8.13±0.02 +7.38+0.16 +−0.16 +377 +215.1814 +52.6032 +0.3368 +19.77 +43.4 +7.90±0.03 +7.20+0.16 +−0.16 +392 +215.3012 +52.6965 +0.8425 +20.44 +44.3 +8.19±0.04 +8.22+0.19 +−0.18 +457 +213.5714 +51.9563 +0.6037 +20.29 +43.4 +8.10±0.1 +8.03+0.18 +−0.21 +519 +214.3012 +51.9460 +0.5538 +21.54 +43.2 +7.36±0.08 +∗7.38+0.18 +−0.19 +551 +212.9461 +51.9388 +0.6802 +21.52 +44.0 +7.66±0.03 +6.95+0.19 +−0.19 +589 +215.2053 +52.1815 +0.7510 +20.74 +44.4 +8.52±0.02 +9.00+0.18 +−0.18 +601 +212.2685 +54.0623 +0.6585 +20.10 +44.1 +9.06±0.05 +8.45+0.36 +−0.24 +622 +212.8133 +51.8692 +0.5716 +19.55 +44.3 +8.22±0.08 +7.94+0.19 +−0.16 +634 +212.8995 +51.8346 +0.6500 +20.76 +44.0 +7.46±0.03 +7.56+0.27 +−0.24 +645 +215.1658 +52.0666 +0.4738 +19.78 +44.1 +8.22±0.01 +7.57+0.16 +−0.18 +694 +214.2778 +51.7278 +0.5324 +19.62 +44.2 +7.59±0.008 +6.70+0.35 +−0.17 +720 +211.3251 +53.2583 +0.4670 +19.03 +44.3 +8.14±0.007 +7.74+0.22 +−0.18 +767 +214.2122 +53.8658 +0.5266 +20.23 +43.9 +7.51±0.04 +∗7.63+0.17 +−0.16 +772 +215.3996 +52.5275 +0.2491 +18.87 +43.4 +7.63±0.02 +6.60+0.22 +−0.22 +775 +211.9961 +53.7999 +0.1725 +17.91 +43.5 +7.93±0.008 +7.67+0.39 +−0.24 +776 +212.0504 +53.8842 +0.1161 +17.98 +43.1 +7.80±0.007 +7.26+0.17 +−0.19 +779 +214.8474 +54.3671 +0.1525 +19.10 +43.1 +7.43±0.01 +7.18+0.17 +−0.17 +781 +215.2647 +51.9721 +0.2634 +19.31 +43.6 +7.77±0.01 +7.89+0.16 +−0.16 +782 +213.3290 +54.5340 +0.3623 +18.89 +43.9 +8.01±0.009 +7.51+0.16 +−0.18 +790 +214.3720 +53.3074 +0.2374 +18.67 +43.3 +8.43±0.01 +8.28+0.48 +−0.23 +840 +214.1881 +54.4280 +0.2439 +18.63 +43.2 +8.29±0.03 +7.93+0.21 +−0.20 +Note—RM black hole masses are based on Hβ lags from Grier et al. (2017), except for RM767, which is based on the Mg ii lag +from Shen et al. (2016). L5100,QSO are from Shen et al. (2015a), and are the host light-subtracted quasar continuum luminosity +at restframe 5100 ˚A. The single-epoch BH mass uncertainties are 1σ measurement errors only, but SE BE masses are typically +dominated by systematic uncertainty of ∼0.5 dex. The RM BH mass uncertainties also include 0.16 dex systematic uncertainty +following Grier et al. (2017). MBH for RM316, RM519, and RM767 (labeled with asterisks) are updated from Grier et al. (2017) +and Li, J. I. et al. (2021) as described in Section 2.2. + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +5 +in Cycle 27 (GO-15849, PI: Shen). Our observational +design is identical to the pilot program (GO-14109, PI: +Shen): each target was observed with two dedicated or- +bits, one in UVIS filters (F606W for z < 0.6 and F814W +for z > 0.6) and one in IR filters (F110W for z < 0.6 +and F140W for z > 0.6), which are chosen to cover sim- +ilar rest-frame wavelengths at different redshifts. Two +additional orbits were used to observe the white dwarf +EGGR-26 to construct the point spread function (PSF) +models in all bands used for this program. All observa- +tions were performed in dithered patterns (three-point +dithering for UVIS filters and four-point dithering for IR +filters) to improve PSF sampling. The data were pro- +cessed using standard HST calibration procedures and +geometrically corrected and dither-combined with as- +trodrizzle. +The final image sampling is 0′′.033 pixel−1 +for the UVIS F606W/F814W images and 0′′.066 pixel−1 +for the IR F110W/F140W images, which correspond to +∼ 0.2 and ∼ 0.4 kpc at z = 0.5. The FWHM of the PSF +is ∼ 2.2 pixels for the IR images and ∼ 1.8 pixels for the +UVIS images. +For RM177, our HST program only covers the IR +band, because this object was observed in UVIS (F606W +and F814W) from a previous HST program (GO-10134, +PI: Davis, Davis et al. 2007). We processed the individ- +ual UVIS exposures from this earlier program following +the same procedures for our HST program, and the final +imaging sampling is 0′′.05 pixel−1 (∼ 0.3 kpc at z = 0.5). +We use a field star in the same field of view as the PSF +model for the UVIS images of RM177. +We then follow the procedures in Li, J. I. et al. (2021) +and perform 2D image decomposition to separate the +quasar and host light using GALFIT (Peng et al. 2010), +but with two modifications. First, we allow each sys- +tem to be fitted by a PSF+disk model (S´ersic index +n = 1), in addition to the PSF+bulge model (n = 4) +and PSF+bulge+disk model. Upon our analysis with +the full sample, we identified several sources whose +hosts best fitted by an exponential disk, rather than +a bulge or bulge+disk model as used in Li, J. I. et al. +(2021). Second, we revise our model-selection criteria +using reduced-χ2 calculated from a small region sur- +rounding the target. By default, GALFIT’s reduced-χ2 +is calculated from the entire image analysis area (e.g., +roughly 10′′×10′′) where > 60% of all the pixels are +background, so the reduced-χ2 can change based on the +chosen image size, and is largely determined by the accu- +racy of the background estimation. To assess the qual- +ity of the fit, we calculate rχ2 +e, the reduced-χ2 within +the best-fit ellipse at 3σ sky background (estimated by +GALFIT) around the source. If ∆(rχ2 +e) > 5 (threshold +chosen by visual inspection of the data) between the +PSF+bulge+disk model and the 2-component models +(PSF+bulge or PSF+disk), we consider there is strong +evidence that the additional component is necessary and +adopt the three-component model; otherwise, we select +the two-component model (PSF+bulge or PSF+disk) +with the smaller rχ2 +e as the best-fit model. +To briefly summarize our fitting procedure, we first fit +the IR images with three different models: PSF+bulge +(S´ersic index n += +4), +PSF+disk (n += +1), +and +PSF+bulge+disk. The fit is considered successful when +the best-fit parameters are within reasonable ranges +(i.e., the effective radius of the S´ersic component Re > 1 +pixel, axis ratio q > 0.01), which is to prevent intro- +ducing additional components fitting for mismatched +PSF or other small-scale features. +While the galaxy +may not be a perfect bulge or disk, Kim et al. (2008) +showed that fixing the S´ersic index results in more accu- +rate flux recovery during host decomposition when the +host galaxy is faint. We use ∆(rχ2 +e) to select the best- +fit model from the successful PSF+bulge, PSF+disk, +and PSF+bulge+disk models. +In addition to the +quasar+host, we fit additional PSF and/or S´ersic mod- +els for nearby objects to ensure the host decomposition +and sky background estimation are not strongly affected +by nearby objects (e.g., see RM033, RM101, RM694, +RM776, etc, for examples). +We visually inspect all the GALFIT images and man- +ually adjust the GALFIT models only when necessary. +Upon visual inspection, the background in RM776 is +high due to a nearby bright object, and adding an- +other component improves the fitting of its surface- +brightness profile significantly, so we adopt a three- +component model for RM776. RM775 and RM790 dis- +play extended truncated ring features in the residual +images of the PSF+bulge+disk model, so a fourth com- +ponent (an inner-truncated disk) was added to ensure +robust flux recovery for the host. The truncated disk +in RM775 is also fitted with Fourier modes to account +for the irregular ellipsoid shape. However, we only in- +clude the main disk component in the PSF+bulge+disk +model, and not the truncated disk, for estimating the +final photometry for the disks. Finally, we fit the flux of +each component in the UVIS images by fixing the shape +and structural parameters (S´ersic index, effective radius, +ellipticity, and position angle) to the best-fit model in +the IR images. For the sources that preferred the three- +component model in the IR image, we check if the three +components in the UVIS image converge on similar rel- +ative positions as in the IR image, which ensures the +model is fitting the same physical structures in the two +bands. The bulge and disk components of two sources, +RM267 and RM316, failed to converge at similar central + +6 +Li et al. +Data +1" +Model +1" +2= 1.26 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +16 +18 +20 +22 +24 + [mag arcsec +2] +PSF +Bulge +Disk +Model +Data +RM017 F606W +Data +1" +Model +1" +2= 1.25 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +18 +20 +22 +24 +26 + [mag arcsec +2] +PSF +Bulge +Disk +Model +Data +RM017 F110W +Data +1" +Model +1" +2= 3.86 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +18 +20 +22 +24 + [mag arcsec +2] +PSF +Disk +Model +Data +RM033 F814W +Data +1" +Model +1" +2= 2.98 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +20 +22 +24 + [mag arcsec +2] +PSF +Disk +Model +Data +RM033 F140W +Data +1" +Model +1" +2= 1.61 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +16 +18 +20 +22 +24 + [mag arcsec +2] +PSF +Bulge +Model +Data +RM160 F606W +Data +1" +Model +1" +2= 1.49 +Residual +10 +1 +100 +Radius [arcsec] +0.5 +0.0 +0.5 +18 +20 +22 +24 +26 + [mag arcsec +2] +PSF +Bulge +Model +Data +RM160 F110W +Figure 2. Examples of surface-brightness decomposition of three quasars with GALFIT, from top to bottom are sources that are +best-fitted by a PSF+bulge+disk, PSF+disk, and PSF+bulge model, respectively. The left panels are the surface-brightness +profiles of the data (black dots), the model (grey solid line) and each modeled component (red solid lines for PSFs, orange +dotted-dash lines for bulges (n=4), blue dash lines for exponential disks (n=1), and purple dotted lines for truncated rings +in RM775 and RM790). The radial profiles are directly-measured from the GALFIT decomposed models and the HST images +with isophote fitting. The leftmost, bottom sub-panel for each object is the residual of the surface-brightness profile, with the +rms along the isophote elliptical plotted in grey. The three images on the right are (from left to right) the HST image, the +GALFIT model, and the residual. The blue ellipse in the HST image (IR only) encloses the area above 3σ sky background in the +best-fitting model. The residual images display the 1st to 99th percentiles (with linear stretch) of the residual values to provide +better visual contrast. The reduced χ2 of the model is labeled in the lower right corner of each residual image. The full figure +set is available online. +positions, so the two-component model (PSF+bulge) is +adopted instead. Fig. 2 presents a few examples of our +GALFIT decomposition, and the GALFIT decomposition +results are tabulated in Table 2. The complete figure +set, data, PSF templates, and GALFIT decomposition +models are available via ftp://quasar.astro.illinois.edu/ +public/sdssrm/paper data/Li 2023 HST host. +During our analysis of the full HST sample, we discov- +ered an error in our GALFIT analysis in the pilot study +(Li, J. I. et al. 2021). The ncombine parameter was in- +put incorrectly, which caused the sigma image produced +by GALFIT to be overestimated by a factor of ∼ 4 in +areas dominated by emission (see GALFIT user manual, +Equation 33). The error mainly affects the estimation +of χ2, but does not change the fitting results, i.e., all +fitted parameters are consistent with the results with +the correct sigma images within the uncertainties. We +include updated measurements for the 10 objects in the +pilot study in Table 2. +GALFIT only accounts for statistical uncertainties be- +tween the data and the model, and does not take into +account PSF mismatches or complex spatial structures. +There are three major sources of flux uncertainties: (1) +the temporal variability of the HST PSF (derived from +the difference between the dedicated PSF observation +and field stars in science observations, ∼ 0.07 mag in +UVIS and ∼ 0.03 mag in IR), (2) the deviation be- +tween the GALFIT model and the image (∼ 0.02 mag +in UVIS and ∼ 0.005 mag in IR), and (3) fixing the +S´ersic index (∼ 0.05 mag for PSF and ∼ 0.2 mag for the +host/bulge/disk). We combine these flux uncertainties +and adopt typical values of 0.1 and 0.25 mag as the fi- +nal uncertainties for the PSF and galaxy (bulge, disk, +or galaxy) flux measurements in all bands, respectively. +These final uncertainties are consistent with those in our +pilot study and similar observations and simulations in +the literature (e.g., Kim et al. 2008; Jahnke et al. 2009; +Park et al. 2015; Bentz & Manne-Nicholas 2018). See + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +7 +Li, J. I. et al. (2021) for additional technical details on +the flux uncertainty budget. +2.4. Host Galaxy/Bulge Masses +Following the approach in Li, J. I. et al. (2021), we +convert the UVIS/IR photometry to rest-frame B and +I/R band and estimate the host/bulge stellar masses +with the color-M∗/L relations (CMLR) from Into & +Portinari (2013) and CIGALE (Boquien et al. 2019). +First, we correct for Galactic extinction using the re- +calibrated Schlegel et al. (1998) dust map and redden- +ing from Schlafly & Finkbeiner (2011). +We then fit +the extinction-corrected HST photometry with CIGALE +to derive k-corrections and color transformations be- +tween the HST filters and the Johnson−Cousins fil- +ters. +CIGALE is a spectral energy distribution (SED) +fitting code that can model galaxy and AGN emission +from multiwavelength photometry. We set up a simple +CIGALE model that includes basic stellar population syn- +thesis models (Maraston 2005), a initial mass function +(Kroupa 2001), a dust attenuation model (Calzetti et al. +2000; Leitherer et al. 2002), and a delayed star forma- +tion history with optional starburst. We do not include +the AGN model for modeling the quasar-subtracted pho- +tometry. The UVIS filters are converted to B-band mag- +nitudes, and the IR filters are converted to I-band and +R-band filters depending on the source redshift (F110W +to I-band at z < 0.4 and R-band z > 0.4; F140W to +I-band at z < 0.7 and R-band at z > 0.7). +We estimate the host and bulge stellar masses with +the CMLR for dusty galaxy models from Into & Porti- +nari (2013) using the rest-frame photometry and their +uncertainties. CIGALE fits provide the k-corrected pho- +tometry, from which we estimate stellar mass with the +CMLR relation, and a stellar mass from the best-fit SED +model. For RM177 (with two UVIS bands from a sep- +arate HST program), we include both the F606W and +F814W bands for the CIGALE fitting but only use the +F606W band (rest-frame R-band) for the CMLR stel- +lar mass estimation. The CMLR stellar-mass uncertain- +ties are propagated directly from the photometry uncer- +tainties, and the CIGALE stellar-mass uncertainties are +estimated from the SED modeling. +Both CMLR and +CIGALE uncertainties are consistently around 0.3 dex, +which is typical for stellar mass estimation from two- +band photometry. +The final Galactic-extinction corrected, k-corrected, +band-converted magnitudes, and the host/bulge stel- +lar masses are tabulated in Table 3. +We adopt the +CMLR stellar masses as our nominal host/bulge stel- +lar masses. +The best-fit stellar masses from CIGALE +are also reported for comparison, which are generally +consistent with those estimated from the CMLR. The +only exception is RM265. The color derived for RM265 +from CIGALE is unusually red, which led to a large, +likely unphysical, host stellar mass (> 1012M⊙) using +the CMLR. However, the typical B − R color is roughly +0.3 < (B − R) < 1.3, derived from all galaxy types in +the Kinney-Calzetti Spectral Atlas (Calzetti et al. 1994; +Kinney et al. 1996). If we assume a red color of 1.3 and +adopt the R-band luminosity for the CMLR, the host +stellar mass for RM265 is log(M∗) = 11.23, which is con- +sistent with the stellar mass derived from CIGALE. We +show both the CMLR and CIGALE masses for RM265 in +Figures 5, 6, and 10, and use the more physical CIGALE +mass (for RM265 only) when fitting the BH scaling re- +lations and their redshift evolution in our analysis. +3. RESULTS +3.1. Host Properties +At z +> +0.2, it becomes challenging to perform +bulge/disk decomposition due to limited spatial resolu- +tion, even with HST. Our GALFIT analysis shows 16 (out +of 38) quasars are best-fitted by the PSF+bulge model, +nine quasars are best-fitted by the PSF+disk model, +and 13 quasars are decomposed into PSF+bulge+disk +models. In addition, 26 hosts are bulge-dominated, i.e., +M∗,bulge > M∗,disk, and 12 hosts are disk-dominated. A +best-fit profile of n = 4 (n = 1) in our analysis does not +necessarily mean the host galaxy is an elliptical (spiral) +galaxy; the S´ersic index is fixed to n = 1 or n = 4 to +ensure the quasar/host decomposition is robust and not +to provide rigorous classifications of host morphology. +In fact, the majority of local elliptical galaxies are not +well-described by single S´ersic components (e.g., Huang +et al. 2013), and exponential profiles do not always in- +dicate the presence of disks. +The structural parameters (ellipticity, S´ersic index, ef- +fective radius) of the bulge/disk-dominated sources in +our sample are broadly consistent with the statistical +distributions from ∼ 2500, i-mag<22 SDSS quasar hosts +observed by the Hyper Suprime-Cam (HSC) on the Sub- +aru telescope (Li et al. 2021b). When we allow the S´ersic +index to vary in the GALFIT fitting, the median (min- +imum, maximum) S´ersic index of our two-component +model is 2.0 (0.6/7.0), similar to the distribution in Li +et al. (2021b). There are more disk-like (n < 2) hosts +in the SDSS-HSC sample, but roughly equal numbers +of bulge-like and disk-like hosts (S´ersic indices above +and below 2) in our sample. The size and ellipticity of +our quasar hosts are also similar to the SDSS-HSC sam- +ple. The median (16%, 84% percentiles) effective radius +is 0′′.68 (0′′.35/0′′.92), and the median (16%/84% per- + +8 +Li et al. +Table 2. Galaxy Decomposition Results +RMID +Comp. +magUVIS +magIR +r (′′) +n +q +P. A. +rχ2 +UVIS +rχ2 +IR +017 +PSF +20.15 +21.19 +1.26 +1.25 +Bulge +19.98 +19.88 +1.50 +4 +0.89 +149.0 +Disk +21.58 +21.11 +0.63 +1 +0.67 +53.2 +033 +PSF +21.69 +22.59 +3.86 +2.98 +Disk +22.36 +23.01 +0.27 +1 +0.69 +-60.0 +101 +PSF +19.41 +20.53 +1.43 +1.17 +Bulge +21.17 +21.14 +0.71 +4 +0.88 +-158.4 +160 +PSF +19.35 +20.52 +1.61 +1.49 +Bulge +21.28 +21.27 +0.67 +4 +0.99 +79.0 +177 +PSF +20.06 +21.60 +1.76 +1.37 +Disk +21.53 +21.53 +0.53 +1 +0.50 +151.3 +191 +PSF +23.02 +24.29 +1.28 +2.12 +Bulge +23.22 +22.26 +0.50 +4 +0.17 +-130.1 +Disk +20.81 +20.91 +1.33 +1 +0.54 +-154.5 +229 +PSF +21.49 +22.49 +1.24 +1.89 +Bulge +24.70 +23.22 +0.31 +4 +0.21 +-118.5 +Disk +21.66 +21.61 +0.78 +1 +0.69 +-134.4 +265 +PSF +21.57 +22.55 +1.24 +1.36 +Disk +20.84 +21.01 +1.36 +1 +0.95 +118.9 +267 +PSF +20.25 +21.64 +1.36 +1.70 +Bulge +21.11 +21.12 +0.37 +4 +0.71 +67.1 +272 +PSF +19.10 +20.31 +1.26 +1.62 +Bulge +21.47 +21.47 +0.31 +4 +0.56 +144.0 +Disk +21.01 +21.43 +0.82 +1 +0.34 +123.0 +300 +PSF +20.39 +21.56 +1.36 +2.32 +Disk +21.75 +22.33 +0.40 +1 +0.97 +-123.1 +301 +PSF +21.02 +22.42 +1.32 +1.64 +Bulge +22.58 +21.66 +0.32 +4 +0.34 +37.0 +Disk +21.37 +21.38 +1.44 +1 +0.69 +47.4 +305 +PSF +20.39 +21.31 +1.48 +1.39 +Bulge +21.17 +20.90 +0.68 +4 +0.88 +71.6 +316 +PSF +19.07 +20.35 +1.41 +2.38 +Bulge +20.88 +21.10 +0.96 +4 +0.68 +27.4 +320 +PSF +20.75 +21.81 +1.27 +2.73 +Bulge +21.80 +21.65 +0.21 +4 +0.66 +-138.5 +Disk +19.88 +20.02 +2.41 +1 +0.34 +154.5 +338 +PSF +21.14 +21.96 +1.39 +1.47 +Disk +20.94 +21.18 +0.68 +1 +0.89 +-50.5 +371 +PSF +20.29 +21.21 +1.35 +2.09 +Bulge +20.88 +20.77 +1.11 +4 +0.86 +45.0 +377 +PSF +22.64 +23.32 +1.15 +1.59 +Bulge +21.06 +20.91 +0.33 +4 +0.52 +100.8 +Disk +21.36 +21.52 +1.12 +1 +0.92 +-174.1 +392 +PSF +22.62 +23.18 +1.29 +1.42 +Bulge +21.94 +22.10 +0.60 +4 +0.71 +151.6 + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +9 +Table 2. (Continued). +RMID +Comp. +magUVIS +magIR +r (′′) +n +q +P. A. +rχ2 +UVIS +rχ2 +IR +457 +PSF +22.71 +23.56 +1.20 +1.31 +Bulge +22.43 +22.21 +0.78 +4 +0.75 +139.4 +519 +PSF +22.76 +23.59 +1.19 +1.37 +Bulge +23.55 +23.51 +0.15 +4 +0.63 +-126.1 +551 +PSF +22.37 +23.71 +1.29 +1.28 +Bulge +22.34 +22.53 +0.15 +4 +0.70 +42.2 +589 +PSF +21.50 +22.33 +1.29 +1.63 +Disk +22.19 +22.20 +0.55 +1 +0.47 +31.6 +601 +PSF +21.33 +22.20 +1.29 +1.62 +Disk +21.38 +21.46 +1.02 +1 +0.58 +56.2 +622 +PSF +20.55 +21.59 +6.73 +1.12 +Bulge +21.65 +21.24 +0.50 +4 +0.70 +118.1 +634 +PSF +21.85 +22.68 +1.32 +1.50 +Disk +22.17 +22.38 +0.76 +1 +0.34 +118.6 +645 +PSF +20.47 +21.36 +1.41 +1.74 +Bulge +21.23 +21.46 +0.31 +4 +0.89 +77.7 +694 +PSF +20.41 +21.57 +1.20 +1.63 +Bulge +23.07 +23.85 +0.78 +4 +0.76 +-144.2 +720 +PSF +20.27 +21.24 +1.40 +2.52 +Bulge +20.69 +21.14 +0.37 +4 +0.87 +50.3 +767 +PSF +21.59 +22.20 +1.18 +1.31 +Bulge +21.16 +21.26 +1.48 +4 +0.73 +169.0 +772 +PSF +21.78 +22.40 +1.17 +1.00 +Bulge +18.69 +18.99 +1.58 +4 +0.88 +88.8 +775 +PSF +19.73 +21.02 +1.19 +2.14 +Bulge +19.58 +19.69 +0.18 +4 +0.71 +-175.7 +Disk +18.87 +19.03 +2.32 +1 +0.78 +139.7 +UVIS Trunc. +21.83 +0.42 +1 +0.6 +156.0 +Fourier +-0.56 +0.14 +0.10 +-0.07 +0.04 +Fourier +4.35 +2.58 +32.19 +19.36 +1.16 +Radial +0.33 +1.08 +0.40 +152.5 +Fourier +0.53 +0.33 +0.17 +0.04 +0.02 +Fourier +-161.59 +4.08 +-43.21 +-3.69 +-2.25 +IR Trunc. +23.89 +0.24 +1 +0.52 +139.2 +Fourier +-0.53 +0.08 +0.06 +-0.02 +0.05 +Fourier +47.96 +10.96 +41.73 +18.83 +14.42 +Radial +0.88 +1.38 +0.52 +136.5 +Fourier +0.56 +0.11 +-0.07 +-0.03 +-0.07 +Fourier +-125.58 +12.36 +5.47 +0.66 +-16.27 +776 +PSF +20.34 +21.40 +0.43 +0.09 +Bulge +19.33 +19.32 +0.46 +4 +0.77 +176.8 +Disk +19.33 +19.49 +2.17 +1 +0.35 +-156.8 + +10 +Li et al. +Table 2. (Continued). +RMID +Comp. +magUVIS +magIR +r (′′) +n +q +P. A. +rχ2 +UVIS +rχ2 +IR +779 +PSF +20.19 +20.98 +1.32 +1.19 +Disk +20.38 +20.84 +0.75 +1 +0.57 +-35.5 +781 +PSF +20.53 +21.81 +1.33 +1.80 +Bulge +20.84 +20.87 +0.33 +4 +0.45 +68.4 +Disk +20.86 +21.27 +1.01 +1 +0.71 +-156.6 +782 +PSF +19.83 +21.18 +1.38 +2.10 +Bulge +22.52 +22.16 +0.52 +4 +0.28 +48.2 +Disk +20.21 +20.32 +1.63 +1 +0.72 +-0.8 +790 +PSF +21.29 +21.92 +1.30 +2.06 +Bulge +19.70 +19.79 +0.53 +4 +0.75 +122.1 +Disk +21.48 +20.95 +0.47 +1 +0.64 +35.5 +UVIS Trunc. +25.47 +0.71 +1 +0.50 +121.7 +Radial +2.60 +4.43 +0.52 +122.4 +IR Trunc. +25.22 +0.91 +1 +0.49 +122.1 +Radial +2.58 +4.10 +0.54 +123.3 +840 +PSF +20.45 +22.37 +1.30 +2.67 +Bulge +20.19 +19.98 +0.26 +4 +0.86 +36.1 +Disk +19.82 +19.97 +2.95 +1 +0.23 +56.1 +Note—r is the effective radius of the S´ersic component, n is the S´ersic index, q is the ratio between the semi-minor axis and the +semi-major axis, and P.A. is the position angle at the semi-major axis in degrees. The reduced χ2 is calculated from the image +residual, as reported by GALFIT. Magnitudes are reported in ST magnitude (magST = −2.5 log(Fλ[erg s−1 cm−2 ˚A−1]) − 21.1), +which is the default output from GALFIT. For RM775 and RM790, we include the best-fit parameters for the truncated disks +in the UVIS and IR images: the magnitudes are the surface brightness at the break radius (mag/arcsec2), and the best-fit +parameters for the truncated radial profiles are listed in the order of the 1% flux radius (softening length, in arcseconds), 99% +flux radius (break radius, in arcseconds), q, and P.A.. The truncated disk in RM775 is fitted with Fourier modes in both the +disks and truncated radial profiles, and the best-fit Fourier amplitudes (first row) and phase angles (second row) are listed in the +order of Fourier mode 1, 3, 4, 5, 6. No extinction corrections are made for these magnitudes. The uncertainties of the GALFIT +results are discussed in Section 2. +centiles) ellipticity (1 − q) is 0.28 (0.11/0.41) for our +sample. +We also examine the offset between the quasar po- +sition and the host centroid in the IR images, where +the centroid of the host galaxy is better constrained +than in the UVIS band. Off-centered AGN/quasars may +indicate on-going galaxy mergers or recoiling SMBHs +from binary SMBH coalescence (Loeb 2007; Comerford +& Greene 2014). Figure 3 shows that most (34/38) of +the quasars are located within < 1 kpc of the host galaxy +center. The four sources with significant offsets (> 1 kpc, +RM265, RM267, RM634, RM645; see the images and +GALFIT models in full Figure 2 figure set online) show +signs of galaxy interaction or mergers, which would com- +plicate the centroid measurements of the host galaxy. +These results suggest z < 1 quasars are well centered +within ∼ 1 kpc of the host centroid, consistent with the +findings using alternative approaches (Shen et al. 2019). +While studies of local AGNs have demonstrated that +BH properties mainly correlate with the bulge and not +the entire host (e.g., Kormendy & Ho 2013), studies at +higher redshift are often limited to the BH−host rela- +tions when bulge/disk decomposition is difficult or im- +possible (e.g., Jahnke et al. 2009; Merloni et al. 2010). +In this work, we present both the BH−bulge and the +BH−host relations in our sample, where M∗,bulge and +M∗,host refer to the bulge-only and total host stellar +mass, respectively. We include all sources in the MBH − +M∗,host relation and exclude the disk-only (PSF+disk) +objects in the MBH−M∗,bulge relation. When comparing +with earlier work, we examine their bulge/disk decompo- +sition assumptions and place the comparison on an equal +footing, i.e., including bulge-dominated or bulge/disk +decomposed sources only in the MBH−M∗,bulge relation, +and including all sources in the MBH − M∗,host relation. +3.2. Comparison with earlier work +Figure 4 shows the MBH−M∗,host and MBH−M∗,bulge +relations of our sample and several local and higher- +redshift samples. High-resolution HST imaging has been +used to investigate the AGN host galaxies at z > 0.2 +(e.g., +Jahnke et al. 2009; Bennert et al. 2011; Bentz + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +11 +Table 3. Final photometry, Color, Luminosity, and Stellar mass +RMID +Bands +Comp +mB +mI/R +Color +log LB +log LI/R +log M∗ +log M∗,CIGALE +(mag) +(mag) +(mag) +(L⊙) +(L⊙) +(M⊙) +(M⊙) +017 +B, R +Host +19.66 +18.78 +0.88 +11.49±0.10 +11.07±0.10 +11.06±0.35 +10.91±0.31 +Bulge +19.86 +19.03 +0.84 +11.40±0.10 +10.98±0.10 +10.92±0.35 +10.77±0.31 +Disk +21.43 +20.36 +1.06 +10.86±0.10 +10.44±0.10 +10.60±0.35 +10.36±0.32 +033 +B, R +Disk +22.13 +21.79 +0.34 +10.76±0.10 +10.34±0.10 +9.83±0.35 +9.88±0.28 +101 +B, R +Bulge +21.05 +20.24 +0.81 +10.92±0.10 +10.50±0.10 +10.42±0.35 +10.25±0.30 +160 +B, I +Bulge +21.36 +20.16 +1.20 +10.70±0.10 +10.15±0.10 +9.94±0.27 +9.95±0.30 +177 +B, R +Disk +20.86 +20.38 +0.48 +10.91±0.10 +10.49±0.10 +10.11±0.35 +9.78±0.46 +191 +B, R +Host +20.61 +19.76 +0.85 +11.07±0.10 +10.65±0.10 +10.61±0.35 +10.43±0.30 +Bulge +23.09 +21.63 +1.46 +10.32±0.10 +9.90±0.10 +10.43±0.35 +10.00±0.36 +Disk +20.79 +20.02 +0.77 +10.97±0.10 +10.55±0.10 +10.43±0.35 +10.28±0.30 +229 +B, R +Host +21.44 +20.55 +0.90 +10.82±0.10 +10.40±0.10 +10.41±0.35 +10.23±0.31 +Bulge +24.40 +22.67 +1.73 +9.97±0.10 +9.55±0.10 +10.33±0.35 +9.85±0.38 +Disk +21.58 +20.77 +0.81 +10.73±0.10 +10.31±0.10 +10.24±0.35 +10.10±0.31 +265 +B, R +Disk +22.84 +20.60 +2.23 +11.27±0.10 +10.85±0.10 +12.10±0.35 +11.39±0.46 +267 +B, R +Bulge +20.81 +20.14 +0.67 +11.22±0.10 +10.80±0.10 +10.59±0.35 +10.49±0.31 +272 +B, I +Host +20.63 +19.58 +1.05 +10.62±0.10 +10.07±0.10 +9.76±0.27 +9.81±0.29 +Bulge +21.67 +20.38 +1.29 +10.30±0.10 +9.75±0.10 +9.60±0.27 +9.55±0.30 +Disk +21.14 +20.28 +0.86 +10.34±0.10 +9.79±0.10 +9.34±0.27 +9.47±0.29 +300 +B, I +Disk +21.55 +20.93 +0.61 +11.00±0.10 +10.45±0.10 +9.83±0.27 +10.06±0.28 +301 +B, R +Host +20.72 +19.82 +0.89 +11.27±0.10 +10.85±0.10 +10.85±0.35 +10.66±0.31 +Bulge +22.19 +20.91 +1.28 +10.84±0.10 +10.42±0.10 +10.78±0.35 +10.44±0.35 +Disk +21.21 +20.47 +0.74 +11.01±0.10 +10.59±0.10 +10.44±0.35 +10.32±0.31 +305 +B, R +Bulge +20.88 +20.01 +0.87 +11.16±0.10 +10.73±0.10 +10.72±0.35 +10.55±0.31 +316 +B, I +Bulge +20.12 +19.68 +0.44 +11.55±0.10 +11.00±0.10 +10.25±0.27 +10.57±0.26 +320 +B, I +Host +19.91 +18.70 +1.21 +10.98±0.10 +10.43±0.10 +10.23±0.27 +10.21±0.30 +Bulge +22.01 +20.61 +1.40 +10.21±0.10 +9.66±0.10 +9.60±0.27 +9.52±0.30 +Disk +20.06 +18.90 +1.16 +10.90±0.10 +10.34±0.10 +10.11±0.27 +10.11±0.29 +338 +B, R +Disk +20.89 +20.27 +0.62 +10.81±0.10 +10.39±0.10 +10.14±0.35 +10.07±0.30 +371 +B, R +Bulge +20.70 +19.87 +0.83 +11.09±0.10 +10.67±0.10 +10.62±0.35 +10.45±0.31 +377 +B, I +Host +20.54 +19.24 +1.30 +11.00±0.10 +10.45±0.10 +10.32±0.27 +10.23±0.30 +Bulge +21.16 +19.80 +1.36 +10.78±0.10 +10.23±0.10 +10.14±0.27 +10.06±0.30 +Disk +21.45 +20.37 +1.08 +10.55±0.10 +10.00±0.10 +9.71±0.27 +9.74±0.30 +392 +B, R +Bulge +21.76 +20.98 +0.77 +11.26±0.10 +10.84±0.10 +10.73±0.35 +10.58±0.32 +457 +B, R +Bulge +22.04 +20.89 +1.15 +10.94±0.10 +10.39±0.10 +10.15±0.27 +10.15±0.31 +519 +B, R +Bulge +23.30 +22.57 +0.73 +10.18±0.10 +9.76±0.10 +9.61±0.35 +9.50±0.31 +551 +B, I +Bulge +22.20 +21.12 +1.08 +10.98±0.10 +10.43±0.10 +10.14±0.27 +10.14±0.31 +589 +B, R +Disk +22.05 +21.13 +0.92 +11.08±0.10 +10.66±0.10 +10.69±0.35 +10.44±0.32 +601 +B, I +Disk +21.15 +20.07 +1.07 +11.36±0.10 +10.81±0.10 +10.52±0.27 +10.52±0.31 +622 +B, R +Bulge +21.25 +20.33 +0.91 +11.11±0.10 +10.69±0.10 +10.71±0.35 +10.54±0.31 +634 +B, I +Disk +21.81 +20.97 +0.84 +10.99±0.10 +10.44±0.10 +9.98±0.27 +10.09±0.30 +645 +B, R +Bulge +21.14 +20.52 +0.62 +10.84±0.10 +10.42±0.10 +10.16±0.35 +10.08±0.30 +694 +B, R +Bulge +23.10 +22.81 +0.29 +10.05±0.10 +9.63±0.10 +9.07±0.35 +9.10±0.25 +720 +B, R +Bulge +20.62 +20.14 +0.48 +10.97±0.10 +10.55±0.10 +10.17±0.35 +10.14±0.29 +767 +B, R +Bulge +21.02 +20.32 +0.71 +11.03±0.10 +10.61±0.10 +10.44±0.35 +10.31±0.31 +772 +B, I +Bulge +18.88 +17.84 +1.04 +11.26±0.10 +10.71±0.10 +10.39±0.27 +10.42±0.29 +775 +B, I +Host +18.72 +17.46 +1.25 +11.05±0.10 +10.50±0.10 +10.34±0.27 +10.27±0.29 +Bulge +19.84 +18.64 +1.20 +10.58±0.10 +10.03±0.10 +9.83±0.27 +9.82±0.29 +Disk +19.10 +17.97 +1.13 +10.85±0.10 +10.30±0.10 +10.04±0.27 +10.08±0.29 + +12 +Li et al. +Table 3. (continued). +RMID +Bands +Comp +mB +mI/R +Color +log LB +log LI/R +log M∗ +log M∗,CIGALE +(mag) +(mag) +(mag) +(L⊙) +(L⊙) +(M⊙) +(M⊙) +776 +B, I +Host +18.93 +17.62 +1.31 +10.61±0.10 +10.06±0.10 +9.94 ±0.27 +9.87 ±0.30 +Bulge +19.73 +18.31 +1.41 +10.34±0.10 +9.79±0.10 +9.73±0.27 +9.62±0.30 +Disk +19.63 +18.44 +1.19 +10.29±0.10 +9.74±0.10 +9.53±0.27 +9.52±0.29 +779 +B, I +Disk +20.64 +19.65 +0.99 +10.06±0.10 +9.51±0.10 +9.16±0.27 +9.18±0.28 +781 +B, I +Host +20.29 +19.14 +1.15 +10.79±0.10 +10.24±0.10 +10.00±0.27 +9.98±0.29 +Bulge +21.05 +19.76 +1.30 +10.55±0.10 +10.00±0.10 +9.86±0.27 +9.79±0.30 +Disk +21.06 +20.06 +1.01 +10.43±0.10 +9.88±0.10 +9.54±0.27 +9.54±0.29 +782 +B, I +Host +20.08 +18.94 +1.14 +11.20±0.10 +10.65±0.10 +10.40±0.27 +10.39±0.30 +Bulge +22.52 +20.98 +1.54 +10.38±0.10 +9.83±0.10 +9.87±0.27 +9.67±0.31 +Disk +20.22 +19.14 +1.08 +11.12±0.10 +10.56±0.10 +10.28±0.27 +10.31±0.30 +790 +B, I +Host +19.76 +18.41 +1.35 +10.99±0.10 +10.43±0.10 +10.34±0.27 +10.26±0.30 +Bulge +19.96 +18.65 +1.31 +10.89±0.10 +10.34±0.10 +10.21±0.27 +10.10±0.30 +Disk +21.86 +19.94 +1.91 +10.37±0.10 +9.82±0.10 +10.12±0.27 +9.76±0.33 +840 +B, I +Host +19.44 +18.14 +1.30 +11.12±0.10 +10.57±0.10 +10.44±0.27 +10.38±0.30 +Bulge +20.41 +18.95 +1.46 +10.79±0.10 +10.24±0.10 +10.22±0.27 +10.12±0.31 +Disk +20.01 +18.86 +1.15 +10.83±0.10 +10.28±0.10 +10.04±0.27 +10.04±0.29 +Note—Magnitudes are reported in AB magnitudes (Oke & Gunn 1982), and color refers to either B − I or B − R. The last +column lists the stellar masses estimated with CIGALE to compare with our fiducial stellar masses. +2 +4 +6 +8 +10 +12 +Effective Radius [kpc] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +QSO/Host Separation [kpc] +694 +634 +267 +265 +Figure 3. Projected physical offset between the quasar po- +sition and host centroid in the IR images as a function of the +host effective radius. The uncertainties are estimated from +GALFIT (quasar position and effective radius) and the quasar- +subtracted host images (centroid position of the host). The +error bars are inflated by a factor of 10 for clarity, as the +GALFIT uncertainties are small and likely underestimated +(median uncertainties are ∼0.02 kpc for the quasar/host sep- +aration and ∼0.005 kpc for the effective radius). +& Manne-Nicholas 2018; Ding et al. 2020). +Similar +to our study, bulge/disk decomposition is only possi- +ble for a small subset of these non-local samples. We +follow the approach of Jahnke et al. (2009); Bennert +et al. (2011); Bentz & Manne-Nicholas (2018) and as- +sume M∗,bulge ≈ M∗,host when there is no evidence of +additional components, which differs from Ding et al. +(2020) that estimated M∗,bulge by assigning bulge/total +ratios depending on the S´ersic indices of the host profile. +At even higher redshift (e.g., z ≳1.5), host stellar +masses can be obtained by SED fitting (e.g., Merloni +et al. 2010; Dong & Wu 2016; Suh et al. 2020) or imag- +ing analysis of lensed quasars (Peng et al. 2006b; Ding +et al. 2021). SED fitting with wide wavelength coverage +can provide better color information for estimating the +stellar masses (compared to using only two HST bands), +and large samples can be studied simultaneously in mul- +tiwavelength fields. However, it is impossible to distin- +guish between the bulge and disk components through +SED fitting. +M∗,host can also be measured from the +reconstructed images of strongly lensed quasars up to +z ∼ 3. +Our sample is generally consistent with the +MBH − M∗,bulge and MBH − M∗,host relations of these +intermediate-to-high redshift samples. +Our sample is the only uniformly-selected (i.e., se- +lected based on a flux limit) AGN sample with RM- +based BH masses to study the BH scaling relations be- +yond the local Universe (z > 0.1). +The RM masses +in Grier et al. (2017) are consistently calibrated to the +BH−host relations in quiescent local galaxies in Kor- + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +13 +108 +109 +1010 +1011 +1012 +M * , Bulge [M +] +106 +107 +108 +109 +1010 +MBH [M +] +KH13 +Bennert+21 +Bentz+18 +Bentz+18 +Bennert+11 +Ding+20 +Jahnke+09 +108 +109 +1010 +1011 +1012 +M * , Host [M +] +106 +107 +108 +109 +1010 +MBH [M +] +KH13 +Bennert+21 +Ding+21 +Dong+16 +Suh+20 +Figure 4. Comparison with literature samples. Our sample is shown in black circles (bulge-dominated) and gray squares +(disk-dominated), and the open gray square is the CIGALE stellar mass of RM265. The blue points are the local-RM sample +from Bentz & Manne-Nicholas (2018). The triangle symbols show the high redshift samples with HST imaging, orange: Bennert +et al. (2011, median redshift zmed = 1.2), green: Jahnke et al. (2009, zmed = 1.3), and red: Ding et al. (2020, zmed = 1.5), and +the dot symbols indicate the high redshift samples with SED fitting, purple: Ding et al. (2021, zmed = 1.7), brown: Dong & Wu +(2016, zmed = 1.1), pink: Suh et al. (2020, zmed = 1.6). The solid lines show the best-fit relations of the local quiescent galaxy +(red, Kormendy & Ho 2013), active galaxy (orange, Bennert et al. 2021), and RM AGN (blue, Bentz & Manne-Nicholas 2018). +mendy & Ho (2013) using the virial factor from Woo +et al. (2015). BH masses in all the comparison samples, +except for Bentz & Manne-Nicholas (2018), are derived +from the SE method, which is less reliable than RM +masses. The SE method relies on a “tight” R − L rela- +tion to estimate BLR sizes based on quasar luminosities; +however, recent studies (Du et al. 2016; Fonseca Alvarez +et al. 2020) have shown the local R − L relation is bi- +ased towards the local AGN sample and could be over- +estimating SE BH masses by as much as ∼ 0.3 dex when +applying to the general quasar population. Finally, the +SE method is calibrated to local quiescent galaxies or lo- +cal RM AGNs, and different virial factors may be used +for different samples or broad-line species. When com- +paring to samples from the literature, we rescale all MBH +values using f = 4.47 as in Grier et al. (2017), even for +the SE masses. +We also compare our results with local baseline sam- +ples from the literature, including the quiescent galax- +ies (mainly ellipticals, Kormendy & Ho 2013), active +galaxies (Bennert et al. 2021), and RM AGNs (Bentz +& Manne-Nicholas 2018). +The MBH − M∗,bulge and +MBH − M∗,host relations of the three local samples and +our best-fit relations are consistent in slope and inter- +cepts within uncertainties. However, Reines & Volon- +teri (2015) find AGN hosts follow a similar slope as lo- +cal quiescent galaxies but are an order of magnitude +lower in normalization for the MBH − M∗,host relation. +They suggested the difference in normalization may be +due to AGN activity or galaxy morphology (which is +also shown in Greene et al. 2020). These results may +appear contradictory at first glance; however, it is diffi- +cult to provide a straightforward comparison since these +studies adopt different stellar-mass estimation methods. +Bentz & Manne-Nicholas (2018) also observed a differ- +ence in normalization using the Bell & de Jong (2001) +CMLR, but not when they use the Into & Portinari +(2013) CMLR, which is the same CMLR adopted in this +work. Due to different assumptions in the CMLR rela- +tions, we do not compare the Reines & Volonteri (2015) +relation, which uses the Zibetti et al. (2009) CMLR, +with our results directly. +Sijacki et al. (2015) and Mutlu-Pakdil et al. (2018) +studied the MBH − M∗,bulge and MBH − M∗,host rela- +tions in the Illustris simulation. +Sijacki et al. (2015) +found that, at z ∼ 0, the MBH − M∗,bulge relation is +tight at the high-BH/galaxy mass end, but scatter in- +creases below MBH ∼ 108M⊙, similar to the general +trend of our sample. Their bulge mass is defined by the +total stellar mass within the stellar half-mass radius, +and not by morphology or kinematics. The difference +in scatter in the high/low mass end might suggest dif- + +14 +Li et al. +109 +1010 +1011 +1012 +M * , Bulge [M +] +106 +107 +108 +109 +MBH [M +] += 0.39+0.11 +0.11 +r = 0.46 +p-val= 1.1E-02 +KH13 +Bulge-dominated +Disk-dominated + 191 + 267 + 371 + 551 + 772 + 781 + 101 + 272 + 320 +109 +1010 +1011 +1012 +M * , Host [M +] +106 +107 +108 +109 +MBH [M +] += 0.34+0.11 +0.11 +r = 0.59 +p-val= 8.6E-05 +265 +(265) +KH13 +Bulge-dominated +Disk-dominated + 160 + 191 + 267 + 371 + 772 + 781 + 589 + 779 + 101 + 377 + 519 +Figure 5. BH mass as functions of bulge stellar mass (left) and total stellar mass (right) of our sample (black circles for +bulge-dominated sources, gray squares for disk-dominated sources, and the open gray square for the CIGALE stellar mass of +RM265). The blue dashed lines (and the gray shaded areas) are the best-fit relation (1σ range) of our sample. The red solid +lines are the Kormendy & Ho (2013) local MBH − M∗,bulge relation. The Pearson-r coefficient, p-value, and intrinsic scatter of +the relations are labeled in the top-left corner of each panel. +ferent evolutionary paths or feedback mechanisms for +establishing the BH scaling relations. Local studies of +BH scaling relations also found that late-type galax- +ies follow a similar slope in the BH scaling relations, +but at a lower normalization than early-type galaxies +(Reines & Volonteri 2015; Greene et al. 2020; Zhao +et al. 2021). In addition, there is no strong evolution +in the MBH − M∗,bulge relation up to z ∼ 1 in the Illus- +tris simulation. Mutlu-Pakdil et al. (2018) studied the +MBH−M∗,host relations in the Illustris simulation to pro- +vide a better comparison for high redshift observations, +and reported that the MBH−M∗,bulge and MBH−M∗,host +relations are generally consistent with each other up to +z ∼ 1. Volonteri et al. (2016) studied the MBH −M∗,host +and MBH−M∗,bulge relations in the Horizon-AGN simu- +lation. By identifying classical bulges in their simulation +through kinematics and bulge/disk decomposition, they +reproduced the tight MBH − M∗,bulge relation of clas- +sic bulges from Kormendy & Ho (2013). Other simula- +tions, e.g., MassiveBlack-II (Khandai et al. 2015), gen- +erally produce similar trends in BH scaling relations at +z < 1. Habouzit et al. (2021) performed a systematic +analysis on the evolution of MBH − M∗ relations in cos- +mological simulations of Illustris, TNG 100, TNG 300, +Horizon-AGN, EAGLE and SIMBA. They find that the +median/mean MBH − M∗ relations at 0 < z < 1 are in +general agreement with observational data and there is +little evolution with redshift. The observed tight corre- +lation between BH accretion rate and star formation at +0.5 < z < 3 indicates the growth of BH and host galaxies +are in sync, and the BH scaling relations should not have +strong redshift dependence (Yang et al. 2019). However, +the scatter in MBH − M∗ relations differs in these simu- +lations, which mainly depends on the implemented sub- +grid physics in the simulations, e.g., the strength and +efficiency of supernova and AGN feedback. +3.3. MBH − M∗,host and MBH − M∗,bulge Relations of +our sample +Our sample consists of 38 sources spanning more +than two orders of magnitude in MBH and M∗,host/bulge, +which is sufficient for statistical analysis. The Pearson +correlation coefficient r between MBH and M∗,host/bulge +are r ∼ 0.5 with low p-values (<0.05), suggesting the BH +and galaxy/bulge masses are positively correlated and +the correlation is statistically significant. +We use the +LINMIX ERR algorithm (Kelly 2007) to perform linear re- +gression fitting on the MBH−M∗,bulge and MBH−M∗,host +relations. +LINMIX ERR is a Bayesian fitting algorithm +that accounts for uncertainties in both axes and intrin- +sic scatter in the relations. We fit for the equation: +logMBH +M⊙ += a + b × log +� +M∗ +1010M⊙ +� +, +(2) +and tabulate the best-fit parameters in Table 4. +The regression fits to the observed sample do not ac- +count for selection effects. In §4.2 we use a more robust +fitting code to constrain the intrinsic BH-host scaling +relations, and we include the bias-corrected best-fit pa- +rameters in Table 4.. However, given the large dynamic + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f* (This Work) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f* (Shen+15) +109 +1010 +1011 +1012 +M* (This Work) [M +] +109 +1010 +1011 +1012 +M* (Matsuoka+15) [M +] +Figure 6. Left: Comparison of the derived host light fraction from this work and Shen et al. (2015a), both measured in the +UVIS (F606W or F814W) bandpass. The black dashed line shows the 1:1 ratio line. Right: Comparison of the total stellar +masses derived from this work and Matsuoka et al. (2015) for overlapping objects. +Table 4. Best-fit Parameters of the Scaling Relations +Scaling Relations +a +b +σ +Bulge (Original) +7.44+0.13 +−0.16 +1.18+0.76 +−0.52 +0.39+0.11 +−0.11 +Host (Original) +7.34+0.14 +−0.17 +1.36+0.59 +−0.40 +0.34+0.11 +−0.11 +Bulge (Bias-Corrected) +7.03+0.26 +−0.41 +1.67+0.83 +−0.72 +0.59+0.23 +−0.21 +Host (Bias-Corrected) +7.01+0.23 +−0.33 +1.74+0.64 +−0.64 +0.47+0.24 +−0.17 +range of our sample, selection effects do not appear to +impact the results significantly, as we will show in §4.2. +3.4. Comparison with spectral decomposition +We compare the host-light fraction and total stellar +mass derived from HST imaging decomposition in this +work and the spectral decompositions in Shen et al. +(2015a) and Matsuoka et al. (2015). Both of these earlier +studies measured host-galaxy properties using the high- +S/N coadded spectra from the first-year SDSS-RM spec- +tra (Shen et al. 2015b). Shen et al. (2015a) used a prin- +cipal component analysis method to decompose coadded +spectra into quasar and galaxy spectra, and measured +host-galaxy properties directly from the galaxy spectra, +including stellar velocity dispersion and host-free AGN +luminosity. Matsuoka et al. (2015) performed spectral +decomposition on the coadded spectra using models of +AGN and galaxy spectra, and measured host galaxy +properties by fitting the decomposed galaxy spectra with +stellar population models. +To compare the host-light fraction (f∗, the fractional +contribution of the host stellar component to the total +flux), we calculate the host fraction in the HST imaging +using the decomposed GALFIT models within the 2′′ di- +ameter spectral aperture, and the host fraction in spec- +tral decomposition by computing the expected flux in +the F606W and F814W bandpass in the decomposed +spectra from Shen et al. (2015a). Figure 6 (left panel) +reveals that the host fraction from image decomposition +is systematically higher than that derived from spectral +decomposition, similar to our finding in the pilot study +(Li, J. I. et al. 2021) and in Yue et al. (2018). Figure +6 (right panel) compares the host stellar mass derived +from this work and from Matsuoka et al. (2015). Our +stellar mass is systematically smaller by ∼0.5 dex. The +cause of the stellar mass offset is currently unclear, but it +might be partially due to different choices of initial mass +functions and stellar population models in Matsuoka +et al. (2015) and CIGALE, or the fiber-loss correction +applied in Matsuoka et al. (2015), which assumes the +mass-to-luminosity ratio in the central region (within +the 2′′-diameter aperture) represents that for the entire +galaxy. +4. DISCUSSION +4.1. Biases in the Observed BH Scaling Relations +Since our sample is based on RM BH masses, it avoids +the large statistical biases associated with the system- +atic uncertainties of SE masses (Shen & Kelly 2010). +To illustrate selection biases in our flux-limited sam- + +16 +Li et al. +108 +109 +1010 +1011 +1012 +M * , Bulge [M +] +105 +106 +107 +108 +109 +1010 +MBH [M +] +i-mag < 16 +i-mag < 19 +i-mag < 22 +7 +8 +a +0 +1 +2 +b +16 +19 +22 +i-mag +0.0 +0.5 +Figure 7. +Results of the Lauer et al. (2007) bias simulation described in Section 4.1. +Left: grey contours represent the +underlying quasar sample, and the grey points are one realization of the mock sample at the i = 22 flux limit, after being +perturbed with measurement uncertainties. The black solid line is the unbiased Kormendy & Ho (2013) relation. The colored +lines indicate the results of the mock sample at different flux limits. Right: the best-fit MBH − M∗,bulge parameters at each +flux threshold and the best-fit MBH − M∗,host parameters of our sample (plotted in red at i = 21.7). The dotted lines show the +parameters and intrinsic scatter from the Kormendy & Ho (2013) relation. +ple due to intrinsic scatter in BH-host scaling relations +(Lauer et al. 2007), we perform a forward-modeling sim- +ulation following the procedures in Shen et al. (2015a). +We first simulate a parent quasar sample following the +local M∗ distribution from Bernardi et al. (2010) and +the MBH − M∗,bulge relation from Kormendy & Ho +(2013), with an intrinsic scatter of 0.29 dex. +Using +the true MBH, we assign a quasar bolometric luminos- +ity by assuming a lognormal Eddington ratio distribu- +tion (λ ≡ Lbol/LEdd) and the Eddington luminosity is +LEdd = 1.26 × 1038(MBH/M⊙) erg s−1. +We choose a +mean Eddington ratio of ⟨logλ⟩ = −1 and a scatter +of 0.3 dex (Shen et al. 2008; Shen & Kelly 2012). We +include measurement uncertainties of 0.35 dex for M∗ +and 0.2 dex for MBH to mimic the uncertainty levels of +our CMLR-based M∗ and RM MBH measurements. Fi- +nally, for 100 bootstrap iterations, we randomly draw 38 +sources to perform LINMIX ERR fitting with at different +i-mag < 16, 19, and 22 (similar to the flux limit of the +SDSS-RM sample). +Figure 7 shows how the flux limit biases the observed +scaling relations. When the flux limit increases, over- +massive BHs are preferentially selected, the slope of +the best-fit relation becomes shallower, and the nor- +malization increases. +The best-fit intrinsic scatter re- +mains roughly the same in our simulations. However, +our simulation does not include the outlier population +with under-massive BHs seen in observations (e.g., see +Figure 4). Missing the outlier population could lead to +an underestimation of the intrinsic scatter and selection +bias based on the flux limit, since under-massive BHs are +less likely to be selected in flux-limited surveys. Given +the relatively faint flux limit of our SDSS-RM sample, +selection biases do not play an important role in the +measured MBH − M∗ relations (see the right panel of +Figure 7), and we found similar relations at zmed = 0.5 +as the local relations. Our results are consistent with +other studies that properly account for selection biases +(e.g., Sexton et al. 2019; Suh et al. 2020; Li et al. 2021a). +We note that the selection of our sample also depends on +successful RM lag measurements, which may depend on +BH mass and Eddington ratio, etc. However, since the +lag-detection fraction in the Grier et al. (2017) sample +is nearly uniform up to z ∼ 0.8, we assume the sample +selection is not strongly affected by additional selection +biases based on the quasar properties. +4.2. Quantifying the Selection Effects +To quantitatively account for the underlying galaxy +properties (i.e., the galaxy mass function) and selection +effects, we follow the framework of Kelly (2007) to per- +form a Markov chain Monte Carlo (MCMC) fitting for +the intrinsic scaling relations and scatters. Specifically, +we wrote a MCMC fitting code based on the Metropolis- +Hastings algorithm (Metropolis et al. 1953; Hastings + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +17 +109 +1010 +1011 +1012 +M * , Bulge [M +] +106 +107 +108 +109 +MBH [M +] +Bias-Corrected +Original +KH13 +a = 7.03+0.26 +0.41 +0.6 +1.2 +1.8 +2.4 +b +b = 1.67+0.83 +0.72 +5.5 +6.0 +6.5 +7.0 +7.5 +a +0.2 +0.4 +0.6 +0.8 +0.6 +1.2 +1.8 +2.4 +b +0.2 +0.4 +0.6 +0.8 + = 0.59+0.23 +0.21 +Figure 8. MCMC fitting for the MBH −M∗,bulge relation when considering the selection bias. Left: the MBH −M∗,bulge relation +with the 1σ range drawn from the posterior (black dashed line and shaded area). The blue dashed lines and shaded area show +the original best-fit relations from LINMIX ERR (previously shown in Figure 5), and the red solid line is the local MBH − M∗,bulge +from Kormendy & Ho (2013). Right: the posterior distribution of the parameters a, b, and σ. The best-fit parameters from +LINMIX ERR and the MCMC fitting are indicated by the blue and white squares, respectively. The MCMC fitting recovers a +similar BH-galaxy relation to the original linear regression, indicating that the selection effects are minimal. +1970) and the statistical derivation in Section 4.1 and +5.1 of Kelly (2007), which allows us to account for the +selection effect based on the dependent variable. Here, +we briefly summarize the fitting algorithm and param- +eter setup and refer the readers to Kelly (2007) for the +full, detailed mathematical derivation. Our custom fit- +ting code is available via ftp://quasar.astro.illinois.edu/ +public/sdssrm/paper data/Li 2023 HST host. +As shown in Kelly (2007), when the intrinsic scatter +and the uncertainties are comparable to the dynamical +range of the data, the best-fit slope becomes shallower +when the underlying distribution is not considered as a +prior in the fitting procedure. One solution is to incorpo- +rate empirical distributions into the likelihood function, +e.g., the observed local stellar mass function Φ(x) from +literature (e.g., Bernardi et al. 2010). However, due to +the limited dynamical range in M∗ and the small sample +size of our data, the local galaxy mass function is not a +good prior for our sample. Alternatively, Kelly (2007) +suggests using a series of Gaussian functions to model +the underlying distribution, which provides a flexible +and empirical solution even when the underlying dis- +tribution is unknown. This is the method implemented +in the original LINMIX ERR fitting algorithm, which we +continue to adopt in our MCMC fitting for consistency. +When the sample selection is based on the dependent +variable (i.e., MBH), the posterior distribution and likeli- +hood depend on an additional term (P(I = 1|θ), where +θ are the model parameters) that describes the likeli- +hood of including each data point in the observed sam- +ple based on the model parameters (for more details, +see Section 5.1 in Kelly 2007). Following the same pro- +cedure as described in Paragraph 1 of this section, we +estimate the expected i-band magnitude by assigning a +random Eddington ratio and an Eddington luminosity +based on the redshift and a range of “true” MBH for +each data point. +The probability of including a data +point is 1 if i-mag< 21.7 and 0 if i-mag> 21.7. Finally, +we calculate P(I = 1|θ) by integrating the probability +of including each data point and their likelihood over a +range of “true” MBH and M∗ given the model parame- +ters. We adopt uninformative, flat priors for all param- +eters (5 < a < 10, 0 < b < 3, and 0.0001 < σ2 < 1) +and minimize the product of the likelihood and prior to +compute the posterior distribution of the parameters a, +b, and σ. +Figure 8 and 9 present the posterior distribution of a, +b, and σ, and the best-fit values are tabulated in Table +4. As expected, the slope of the intrinsic scaling rela- +tions becomes steeper, and the normalization decreases, +after correcting for the selection biases. The best-fit pa- + +18 +Li et al. +109 +1010 +1011 +1012 +M * , Host [M +] +106 +107 +108 +109 +MBH [M +] +Bias-Corrected +Original +KH13 +a = 7.01+0.23 +0.33 +0.6 +1.2 +1.8 +2.4 +b +b = 1.74+0.64 +0.64 +5.5 +6.0 +6.5 +7.0 +7.5 +a +0.2 +0.4 +0.6 +0.8 +0.6 +1.2 +1.8 +2.4 +b +0.2 +0.4 +0.6 +0.8 + = 0.47+0.24 +0.17 +Figure 9. Same format as Figure 8 but for the MBH − M∗,host relation. +rameters are within uncertainties as the LINMIX ERR fit +(without considering selection bias) discussed in Section +3, demonstrating that our results are not strongly af- +fected by selection biases. +Intrinsic +scatter +of +the +MBH − M∗,bulge +and +MBH − M∗,host relations is an important indicator for +BH−galaxy co-evolution, as it might be related to the +galaxy/AGN properties and their evolutionary path. +The local samples of Kormendy & Ho (2013) and Ben- +nert et al. (2021) only include classical bulges and +pseudo-bulges and have a smaller intrinsic scatter of +0.28 − 0.39 dex. However, when including all morpho- +logical types and active/inactive galaxies, the intrinsic +scatter increases to ∼0.5 dex (Reines & Volonteri 2015; +Bentz & Manne-Nicholas 2018) for the MBH − M∗,bulge +relation, and becomes even slightly larger for the MBH− +M∗,host relation. In addition, the BH accretion rate is +found to be correlated with other host properties, e.g., +compactness of the central ∼1 kpc region (Ni et al. 2019, +2021), which can introduce additional scatter in the BH +scaling relations. +For our quasar sample, the intrinsic scatter of the +MBH − M∗,host and MBH − M∗,bulge relations are +0.47+0.24 +−0.17 dex and 0.59+0.23 +−0.21 dex, respectively, after ac- +counting for the selection effects, which are compara- +ble to the scatter in the local relations. The intrinsic +scatter of our MBH − M∗,host relation is slightly smaller +(≲ 0.5σ of difference) than our MBH −M∗,bulge relation, +which we will further discuss in Section 4.4. Because we +have neglected the systematic uncertainty in our RM +BH masses due to the scatter in individual virial coeffi- +cients, the actual intrinsic scatter in the BH-host stellar +mass relations for 0.2 < z < 0.8 quasars might be even +smaller. +4.3. Evolution of BH Scaling Relations +Earlier works on MBH − M∗,bulge and MBH − M∗,host +relations found that the average BH-to-host galaxy +mass ratio evolves positively with redshift (Peng et al. +2006a,b; Merloni et al. 2010; Bernardi et al. 2010). How- +ever, selection biases and measurement uncertainties +could yield false positives of the evolution. +For ex- +ample, Jahnke et al. (2009) reported that there is no +evidence of evolution when they carefully choose their +sample to avoid selection biases. Similarly, Suh et al. +(2020) found the redshift evolution seen in Merloni et al. +(2010) can be explained by the Lauer bias (Lauer et al. +2007), and there is no trend of evolution in their X- +ray-selected, lower luminosity sample. Using over 500 +uniformly-selected 0.2 < z < 0.8 SDSS quasars, Li et al. +(2021a) found a redshift evolution of the offset in the +MBH−M∗,host relation that is within ±0.2 dex from zero, +consistent with no significant evolution since z ∼ 0.8. +Figure 10 presents the deviation of MBH +from +the Kormendy & Ho (2013) MBH − M∗,bulge rela- +tion of our sample. +We fit the deviation as a +function of log(z), +and the slopes and intercepts +are consistent with zero, suggesting there is no red- +shift evolution from the local relations. +The median +(16th/84th percentiles) black hole/host galaxy mass ra- +tios are MBH/M∗,bulge = 0.0037(0.0007/0.0088) and +MBH/M∗,host = 0.0030(0.0007/0.0076), within 1σ un- + +Black Hole Scaling Relations at 0.2 ≲ z ≲ 0.8 +19 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +log(1 + z) +3 +2 +1 +0 +1 +2 +3 +log(MBH) [M * , Bulge] +Bulge-dominated +Disk-dominated +Bentz+18 +Bennert+11 +Ding+20 +Jahnke+09 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +z +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +log(1 + z) +3 +2 +1 +0 +1 +2 +3 +log(MBH) [M * , Host] +Bulge-dominated +Disk-dominated +Bentz+18 +Bennert+11 +Ding+20 +Jahnke+09 +Ding+21 +Dong+16 +Suh+20 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +z +Figure 10. Evolution of ∆log(MBH) with redshift, with baselines adopted from the best-fit relations of MBH − MBulge and +MBH −LBulge from the Kormendy & Ho (2013) sample. Our work is the only sample with RM-based BH masses beyond z > 0.3. +Vertical error bars are from uncertainties in BH mass only. +certainty of the local value of MBH/M∗,bulge ∼ 0.005 +(Kormendy & Ho 2013). +Our results agree with recent observational studies +that there is limited evolution in the MBH −M∗ relation +(Suh et al. 2020; Li et al. 2021a). The limited redshift +evolution in the MBH−M∗ relations can be explained by +the tight correlation between the BH accretion rates and +star formation rates found in bulge-dominated galaxies +at z = 0.5 − 3 (Yang et al. 2019), which suggests the +growth of SMBH and their host galaxies and bulges are +in sync since z ∼ 3. +4.4. BH−Bulge versus BH−Host Relations +In the local Universe, SMBH masses are tightly cor- +related with the properties of classical bulges, but not +with disks or total mass of the host galaxy. However, the +intrinsic scatter in the MBH−M∗,host relation of our sam- +ple is slightly smaller than that in the MBH − M∗,bulge +relation. There are a few extra sources of uncertainties +for the bulge-mass estimate than for the total mass es- +timate, which will contribute to the intrinsic scatter. +First of all, bulge-disk decomposition could add sig- +nificant uncertainties to the bulge mass. Most of our +host galaxies are far less luminous than the quasar, and +compact hosts could be near the limit of HST imaging +resolution. +Bulge-disk decomposition is reliable when +both the bulge and disk are sufficiently bright (com- +pared to the central quasar) and there is a distinct dif- +ference in their effective radii. +Furthermore, we can- +not distinguish classical bulges from pseudo-bulges or +other bulge-like structures from surface-brightness de- +composition, nor could we model complex bar, spiral, + +20 +Li et al. +and merger structures in the bulge-disk decomposition, +which increases the uncertainties in bulge identification +and mass estimation. Gao & Ho (2017) found that rig- +orous modeling of bars and innermost structures (e.g., +rings and disk breaks near the bulge) is crucial to re- +covering bulge properties, while the modeling of spiral +arms and extended disks have negligible effects. We note +that some host galaxies in our sample show clear ev- +idence of bars (e.g., RM320, RM634, etc), which are +modeled as bulges (n = 4) or disks (n = 1) in our +analysis, without additional bar structures. Moreover, +some host galaxies show clear spiral arm features (e.g., +RM371, RM772, etc), indicating the presence of disks, +but are modeled as “bulges”(n = 4). Previous works +(e.g., Zhao et al. 2021; Greene et al. 2020) showed late- +type quasar hosts preferentially scatter below early-type +hosts in the MBH−M∗,bulge relation, which is not seen in +our data, suggesting our bulge-disk decomposition is not +as reliable in measuring galaxy morphology. A detailed +simulation of bulge-disk decomposition for AGNs with +similar host and quasar properties (e.g., AGN/host flux +ratio, host effective radius, S´ersic indices, and complex +structures) is needed to provide quantitative uncertainty +estimation, which is beyond the scope of this work. +Another possible source of uncertainty is the CMLR +estimation for bulges. Recent studies have reported that +compact regions around the SMBH may have denser in- +terstellar medium, boosted star formation, and complex +stellar populations (e.g., Ni et al. 2019; Kim & Ho 2019; +Zhuang & Ho 2020; Shangguan et al. 2020; Yesuf & Ho +2020; Molina et al. 2021). Two-band color and the use of +empirical M/L relation may not be sufficient to produce +reliable estimates for the bulge stellar mass. +5. CONCLUSIONS +We present the MBH −M∗,bulge and MBH −M∗,host re- +lations of 38 sources with RM-based BH masses (Grier +et al. 2017) and 0.2 ≲ z ≲ 0.8 (median redshift zmed = +0.5). Our sample is the first uniformly-selected sample +with RM-based BH masses at z > 0.3 for studying BH- +host relations, and covers two orders of magnitude in +BH mass and host stellar mass. The reliable RM-based +BH masses and host mass estimates from HST imaging +decomposition, combined with the large sample size and +dynamic range in mass, allow one to alleviate selection +biases in studying the potential evolution of the BH- +host scaling relations. Our scaling relations are consis- +tent with those for local AGNs, quiescent galaxies, and +other high-redshift samples, with negligible redshift evo- +lution up to z ≲ 1. As shown in Table 4, the best-fitting +intrinsic MBH − M∗,host relation is: log(MBH/M⊙) = +7.01+0.23 +−0.33 + 1.74+0.64 +−0.64 log(M∗,host/1010M⊙) after correct- +ing for the underlying sample distribution and selection +effects. We estimate an intrinsic scatter of 0.59+0.23 +−0.21 dex +and 0.47+0.24 +−0.17 dex in the MBH − M∗,bulge and MBH − +M∗,host relations, respectively, which is again consistent +with the local BH scaling relations. +With our ap- +proved Cycle 1 JWST proposal (GO-2057, PI: Shen), +we will continue to explore BH−host relations and their +redshift evolution up to z ∼ 2 using quasars with direct +RM-based BH masses (Grier et al. 2019). +JIL acknowledges support from the Government Schol- +arship to Study Abroad (GSSA) from the Ministry of +Education of Taiwan and support from the Illinois Space +Grant Consortium (ISGC) Graduate Fellowship. +YS +acknowledges support from NSF grants AST-1715579 +and AST-2009947. LCH was supported by the National +Science Foundation of China (11721303, +11991052, +12233001, 12011540375) and the China Manned Space +Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06). +WNB acknowledges support from NSF grant AST- +2106990. +PBH is supported by NSERC grant 2017- +05983. +Based on observations with the NASA/ESA +Hubble Space Telescope obtained from the Data Archive +at the Space Telescope Science Institute, which is oper- +ated by the Association of Universities for Research in +Astronomy, Incorporated, under NASA contract NAS5- +26555. +Support for Program number HST-GO-15849 +was provided through a grant from the STScI under +NASA contract NAS5-26555. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +Software: +AstroDrizzle, Astropy (Astropy Collab- +oration et al. 2013; Price-Whelan et al. 2018), CIGALE +(Boquien et al. 2019), GALFIT (Peng et al. 2010), +LINMIX ERR (Kelly 2007), matplotlib (Hunter 2007), +Numpy (Oliphant 2006), photutils (Bradley et al. +2019), pysynphot (Lim et al. 2015), seaborn (Waskom +et al. 2017). +Facilities: HST (WFC3/UVIS, WFC3/IR) +REFERENCES +Astropy Collaboration, Robitaille, T. 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+page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' People’s Republic of China 5Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' People’s Republic of China 6Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' USA 7Institute for Gravitation and the Cosmos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' USA 8Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' USA 9Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' University of Wisconsin-Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Madison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' WI 53706,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' USA 10Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 4700 Keele St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Toronto, ON M3J 1P3, Canada 11Space Telescope Science Institute, 3700 San Martin Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Baltimore, MD 21218, USA 12Department of Physics, University of Connecticut, 2152 Hillside Rd Unit 3046, Storrs, CT 06269, USA ABSTRACT We measure the correlation between black-hole mass MBH and host stellar mass M∗ for a sample of 38 broad-line quasars at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 (median redshift zmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The black-hole masses are derived from a dedicated reverberation mapping program for distant quasars, and the stellar masses are estimated from two-band optical+IR HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Most of these quasars are well centered within ≲ 1 kpc from the host galaxy centroid, with only a few cases in merging/disturbed systems showing larger spatial offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample spans two orders of magnitude in stellar mass (∼ 109 −1011 M⊙) and black-hole mass (∼ 107 − 109 M⊙), and reveals a significant correlation between the two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We find a best-fit intrinsic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', selection effects corrected) MBH −M∗,host relation of log(MBH/M⊙) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 log(M∗,host/1010M⊙), with an intrinsic scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Decomposing our quasar hosts into bulges and disks, there is a similar MBH−M∗,bulge relation with slightly larger scatter, likely caused by systematic uncertainties in the bulge-disk decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The MBH −M∗,host relation at zmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 is similar to that in local quiescent galaxies, with negligible evolution over the redshift range probed by our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' With direct black-hole masses from reverberation mapping and a large dynamical range of the sample, selection biases do not appear to affect our conclusions significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our results, along with other samples in the literature, suggest that the locally-measured black-hole mass−host stellar mass relation is already in place at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Keywords: black hole physics – galaxies: active – quasars: general – surveys 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' INTRODUCTION The observed scaling relations between supermassive black hole (BH) masses and the properties of their host galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', stellar mass and stellar velocity disper- sion) in the local Universe are the foundation of mod- ern BH−galaxy co-evolution models (Magorrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ferrarese & Merritt 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' H¨aring & Rix 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' G¨ultekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' McConnell & Ma 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Kormendy & Ho 2013, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The tight correlations suggest that active galactic nuclei (AGNs) may play important roles in regulating star for- mation in the host galaxies via self-regulated BH growth and feedback processes (Silk & Rees 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Studying BH scal- ing relations beyond the local Universe is a key to un- derstanding BH and galaxy (co-)evolution over cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Over the past two decades, various investigations have built an inventory of BH and host measurements to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04177v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='GA] 10 Jan 2023 ID2 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' study the redshift evolution of BH−host relations up to z ∼ 3, including the BH mass−stellar velocity disper- sion (MBH − σ∗) relation (Treu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2006, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Sex- ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019), the BH mass−bulge/host luminosity (MBH − L∗,bulge/host) relation (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2006a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' De- carli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010), the BH mass−bulge/host stellar mass (MBH − M∗,bulge/host) relation (Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ben- nert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Dong & Wu 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021), as well as expanding the local baselines to include AGNs of different host properties and lower BH masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2011a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Reines & Volonteri 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bentz & Manne-Nicholas 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Some groups found deviations from local scaling rela- tions as a function of z (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2006a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Sexton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019) while others found similar BH−host rela- tions as in the local Universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Silver- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2022), which is also supported by the tight correlation between the BH accretion rate and star for- mation rate in bulge-dominated galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 − 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The measurements of BH mass−host relations can be challenging beyond z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' First, direct BH mass measurements based on resolved stel- lar/gas dynamics are difficult to obtain beyond the lo- cal Universe where the BH sphere of influence can- not be readily resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Reverberation mapping (RM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Blandford & McKee 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Peterson 2014) is the primary method of measuring BH masses for broad-line (BL) AGN beyond the local Universe, but RM is resource- intensive and only available for a small number of ob- jects beyond z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A sec- ondary BH mass recipe, the single-epoch (SE) virial estimator, is based on the broad-line region (BLR) radius−luminosity relation (the R−L relation) and can be easily adapted for large samples of BLAGNs at higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, Shen & Kelly (2010) demonstrated that there is a statistical bias in SE BH masses for flux-limited samples from the uncertainties in these BH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In addition, the applicability of SE masses to the high-redshift and high-luminosity regime is not well understood, primarily because the local RM AGNs used to derive the R − L relation is not representative of the general quasar population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Fon- seca Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020), and the extrapolated R − L relations for broad Mg ii and C iv used for high-redshift BLAGNs are not as well-studied as the local R − L re- lation based on broad Hβ (Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Host-galaxy properties are also difficult to measure as the unobscured AGN (where virial BH masses are feasi- ble) usually far outshines the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For imaging studies, high-resolution images, such as those from the Hubble Space Telescope (HST), are often necessary to robustly decompose the quasar and host light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' How- ever, rigorous image analysis reveals that host galaxies of local AGNs (z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='35) often consist of complex struc- tures, including spiral arms, tidal and merger features, in addition to the main galaxy components (bars, bulges, and disks) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' These complex structures are extremely challenging to measure even with HST at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Due to difficulties in obtaining BH mass and host properties, many studies are limited to specific sam- ples that may introduce selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Earlier stud- ies were often restricted to the bright end of BLAGN, have small sample sizes and limited dynamical ranges in BH/host properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Lauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2007) showed that over-massive BHs are favored in flux-limited studies due to the intrinsic scatter of the scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For a “bottom-heavy” galaxy luminosity function, there are more low-mass hosts than high-mass ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, more massive BHs are preferentially selected in a flux- limited sample based on AGN luminosity, resulting in an average offset in the BH mass−host relations, and a shal- lower slope than the true underlying relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Schulze & Wisotzki (2011, 2014) argued that additional selection biases could arise from the lack of knowledge in the rel- evant underlying distribution functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', the active fraction of AGNs, bulge properties) and their evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' These biases can account for a large por- tion of, if not all, the redshift evolution reported in ear- lier investigations (Schulze & Wisotzki 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In this work, we study the BH scaling relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 using the Sloan Digital Sky Survey Rever- beration Mapping (SDSS-RM, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015b) sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The SDSS-RM sample has two major advantages in measuring the redshift evolution of BH−host galaxy relations: (1) the parent sample is a uniformly selected flux-limited BLAGN sample, thus the selection effects can be quantified and corrected, and (2) BH masses are available from direct RM (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Homayouni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020), rather than from SE, masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We have acquired high-resolution imaging for the SDSS-RM sample with HST to measure the host galaxy color and luminosity in two bands, tracing young and old stellar populations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample in- cludes 38 sources (10 included in a pilot study in Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021), which is comparable in size to the local RM AGN sample used to calibrate the R−L relation (Bentz Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 Redshift 42 43 44 45 46 Log L5100, QSO [erg s 1] SDSS-RM H Lags This Work Local RM Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Quasar luminosity and redshift distribution of our sample (open orange circles), and a representative subset of the local RM sample (grey squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The parent SDSS-RM sample (black dots) and those with Hβ lags (blue dots) are also labeled for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2013), and has sufficient statistics and dynamic range in BH mass (and stellar mass) to characterize the redshift evolution of BH scaling relations over the red- shift range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We describe our data and analysis in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The main results are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We discuss our results in Section 4 and conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Throughout this paper we adopt a flat ΛCDM cosmology with ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 and H0 = 70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' All host-galaxy measurements refer to the stellar population only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' OBSERVATION AND DATA ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Sample Our sample consists of 38 SDSS-RM quasars at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 (median redshift zmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5) with RM-based BH masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 37 of these RM masses were based on the broad Hβ line (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2017), with one source (RM767) based on the broad Mg ii line (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ten sources in our HST sample were studied in a pilot program (Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 28 sources are presented in this work for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Among the 44 quasars with Hβ RM BH masses in Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017), seven sources beyond z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 were excluded from the HST programs to ensure more robust host-galaxy measurements and to avoid unknown selection biases, as the lag-detection fraction at z ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 is significantly lower than that at lower redshifts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 1 presents the redshift and luminosity distribution of our sample, and Table 1 summarizes the physical properties of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Black-Hole Masses Reverberation mapping determines BH masses by measuring the time delay in variability between the con- tinuum and broad emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The time delay corre- sponds to the light travel time between the continuum- emitting accretion disk and the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Assuming the BLR is virialized, a BH mass can be calculated using the av- erage time lag (τ) and the width of the broad emission line (∆V ) via the equation: MBH = f cτ∆V 2 G , (1) where G is the gravitational constant and f is a di- mensionless factor of unity order that accounts for BLR geometry, kinematics, and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The line width ∆V can be estimated from either the full-width half- maximum (FWHM) or the line dispersion (σline) of the broad line measured from the mean or RMS spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For the majority of our sources, we adopt the RM black hole masses from Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017) computed us- ing a consant virial coefficient of f =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47 based on σline measured from the RMS spectra (equivalent to f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='12 when using the FWHM for ∆V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For two of our sources, RM316 and RM519, the original σline measurements in Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017) based on the first-year SDSS-RM spectroscopy are significantly overestimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' we adopt updated σline measurements based on the 4-year SDSS- RM spectroscopy for these two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For RM767, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2016) identified a lag between the contin- uum and broad Mg ii line during the first-year monitor- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, the lag significance is reduced in the more recent analysis in Homayouni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2020) using 4-year light curves, as the broad Mg ii line does not display strong response to the continuum in the following years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We adopt the Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2016) Mg ii lag for RM767, and use its σline measured from the RMS spectrum to derive a BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The BH mass uncertainties are cal- culated by propagating the statistical uncertainties of the lag and line width measurements, and then adding a systematic uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='16 dex, which is the scat- ter estimated from repeated RM measurements in local RM campaigns (Fausnaugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, the adopted BH mass uncertainty is still an underestima- tion, as it does not account for the intrinsic scatter in the virial coefficient for individual systems, which could lead to additional BH mass uncertainties of as large as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The BH masses are tabulated in Table 1 (with updates from earlier work indicated by an asterisk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' HST Imaging Analysis The HST observations for the 28 new objects were conducted between 2019 December 23 and 2021 June 09 4 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Target Properties RMID R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (J2000) Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (J2000) z ipsf L5100,QSO log(MBH,SE) log(MBH,RM) (deg) (deg) (mag) (erg s−1) (M⊙) (M⊙) 017 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3511 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4559 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='20 Note—RM black hole masses are based on Hβ lags from Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017), except for RM767, which is based on the Mg ii lag from Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' L5100,QSO are from Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a), and are the host light-subtracted quasar continuum luminosity at restframe 5100 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The single-epoch BH mass uncertainties are 1σ measurement errors only, but SE BE masses are typically dominated by systematic uncertainty of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The RM BH mass uncertainties also include 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='16 dex systematic uncertainty following Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' MBH for RM316, RM519, and RM767 (labeled with asterisks) are updated from Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017) and Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021) as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 5 in Cycle 27 (GO-15849, PI: Shen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our observational design is identical to the pilot program (GO-14109, PI: Shen): each target was observed with two dedicated or- bits, one in UVIS filters (F606W for z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 and F814W for z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6) and one in IR filters (F110W for z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 and F140W for z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6), which are chosen to cover sim- ilar rest-frame wavelengths at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Two additional orbits were used to observe the white dwarf EGGR-26 to construct the point spread function (PSF) models in all bands used for this program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' All observa- tions were performed in dithered patterns (three-point dithering for UVIS filters and four-point dithering for IR filters) to improve PSF sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The data were pro- cessed using standard HST calibration procedures and geometrically corrected and dither-combined with as- trodrizzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The final image sampling is 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='033 pixel−1 for the UVIS F606W/F814W images and 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='066 pixel−1 for the IR F110W/F140W images, which correspond to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 kpc at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The FWHM of the PSF is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 pixels for the IR images and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 pixels for the UVIS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For RM177, our HST program only covers the IR band, because this object was observed in UVIS (F606W and F814W) from a previous HST program (GO-10134, PI: Davis, Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We processed the individ- ual UVIS exposures from this earlier program following the same procedures for our HST program, and the final imaging sampling is 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='05 pixel−1 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 kpc at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We use a field star in the same field of view as the PSF model for the UVIS images of RM177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We then follow the procedures in Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021) and perform 2D image decomposition to separate the quasar and host light using GALFIT (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010), but with two modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' First, we allow each sys- tem to be fitted by a PSF+disk model (S´ersic index n = 1), in addition to the PSF+bulge model (n = 4) and PSF+bulge+disk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Upon our analysis with the full sample, we identified several sources whose hosts best fitted by an exponential disk, rather than a bulge or bulge+disk model as used in Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Second, we revise our model-selection criteria using reduced-χ2 calculated from a small region sur- rounding the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' By default, GALFIT’s reduced-χ2 is calculated from the entire image analysis area (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', roughly 10′′×10′′) where > 60% of all the pixels are background, so the reduced-χ2 can change based on the chosen image size, and is largely determined by the accu- racy of the background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' To assess the qual- ity of the fit, we calculate rχ2 e, the reduced-χ2 within the best-fit ellipse at 3σ sky background (estimated by GALFIT) around the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' If ∆(rχ2 e) > 5 (threshold chosen by visual inspection of the data) between the PSF+bulge+disk model and the 2-component models (PSF+bulge or PSF+disk), we consider there is strong evidence that the additional component is necessary and adopt the three-component model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' otherwise, we select the two-component model (PSF+bulge or PSF+disk) with the smaller rχ2 e as the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' To briefly summarize our fitting procedure, we first fit the IR images with three different models: PSF+bulge (S´ersic index n = 4), PSF+disk (n = 1), and PSF+bulge+disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The fit is considered successful when the best-fit parameters are within reasonable ranges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', the effective radius of the S´ersic component Re > 1 pixel, axis ratio q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01), which is to prevent intro- ducing additional components fitting for mismatched PSF or other small-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' While the galaxy may not be a perfect bulge or disk, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2008) showed that fixing the S´ersic index results in more accu- rate flux recovery during host decomposition when the host galaxy is faint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We use ∆(rχ2 e) to select the best- fit model from the successful PSF+bulge, PSF+disk, and PSF+bulge+disk models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In addition to the quasar+host, we fit additional PSF and/or S´ersic mod- els for nearby objects to ensure the host decomposition and sky background estimation are not strongly affected by nearby objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', see RM033, RM101, RM694, RM776, etc, for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We visually inspect all the GALFIT images and man- ually adjust the GALFIT models only when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Upon visual inspection, the background in RM776 is high due to a nearby bright object, and adding an- other component improves the fitting of its surface- brightness profile significantly, so we adopt a three- component model for RM776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' RM775 and RM790 dis- play extended truncated ring features in the residual images of the PSF+bulge+disk model, so a fourth com- ponent (an inner-truncated disk) was added to ensure robust flux recovery for the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The truncated disk in RM775 is also fitted with Fourier modes to account for the irregular ellipsoid shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, we only in- clude the main disk component in the PSF+bulge+disk model, and not the truncated disk, for estimating the final photometry for the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Finally, we fit the flux of each component in the UVIS images by fixing the shape and structural parameters (S´ersic index, effective radius, ellipticity, and position angle) to the best-fit model in the IR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For the sources that preferred the three- component model in the IR image, we check if the three components in the UVIS image converge on similar rel- ative positions as in the IR image, which ensures the model is fitting the same physical structures in the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The bulge and disk components of two sources, RM267 and RM316, failed to converge at similar central 6 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Data 1" Model 1" 2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='26 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 16 18 20 22 24 [mag arcsec 2] PSF Bulge Disk Model Data RM017 F606W Data 1" Model 1" 2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='25 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 18 20 22 24 26 [mag arcsec 2] PSF Bulge Disk Model Data RM017 F110W Data 1" Model 1" 2= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='86 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 18 20 22 24 [mag arcsec 2] PSF Disk Model Data RM033 F814W Data 1" Model 1" 2= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='98 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 20 22 24 [mag arcsec 2] PSF Disk Model Data RM033 F140W Data 1" Model 1" 2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='61 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 16 18 20 22 24 [mag arcsec 2] PSF Bulge Model Data RM160 F606W Data 1" Model 1" 2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='49 Residual 10 1 100 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 18 20 22 24 26 [mag arcsec 2] PSF Bulge Model Data RM160 F110W Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Examples of surface-brightness decomposition of three quasars with GALFIT, from top to bottom are sources that are best-fitted by a PSF+bulge+disk, PSF+disk, and PSF+bulge model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The left panels are the surface-brightness profiles of the data (black dots), the model (grey solid line) and each modeled component (red solid lines for PSFs, orange dotted-dash lines for bulges (n=4), blue dash lines for exponential disks (n=1), and purple dotted lines for truncated rings in RM775 and RM790).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The radial profiles are directly-measured from the GALFIT decomposed models and the HST images with isophote fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The leftmost, bottom sub-panel for each object is the residual of the surface-brightness profile, with the rms along the isophote elliptical plotted in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The three images on the right are (from left to right) the HST image, the GALFIT model, and the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The blue ellipse in the HST image (IR only) encloses the area above 3σ sky background in the best-fitting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The residual images display the 1st to 99th percentiles (with linear stretch) of the residual values to provide better visual contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The reduced χ2 of the model is labeled in the lower right corner of each residual image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The full figure set is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' positions, so the two-component model (PSF+bulge) is adopted instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2 presents a few examples of our GALFIT decomposition, and the GALFIT decomposition results are tabulated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The complete figure set, data, PSF templates, and GALFIT decomposition models are available via ftp://quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='edu/ public/sdssrm/paper data/Li 2023 HST host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' During our analysis of the full HST sample, we discov- ered an error in our GALFIT analysis in the pilot study (Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The ncombine parameter was in- put incorrectly, which caused the sigma image produced by GALFIT to be overestimated by a factor of ∼ 4 in areas dominated by emission (see GALFIT user manual, Equation 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The error mainly affects the estimation of χ2, but does not change the fitting results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', all fitted parameters are consistent with the results with the correct sigma images within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We include updated measurements for the 10 objects in the pilot study in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' GALFIT only accounts for statistical uncertainties be- tween the data and the model, and does not take into account PSF mismatches or complex spatial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' There are three major sources of flux uncertainties: (1) the temporal variability of the HST PSF (derived from the difference between the dedicated PSF observation and field stars in science observations, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='07 mag in UVIS and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03 mag in IR), (2) the deviation be- tween the GALFIT model and the image (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='02 mag in UVIS and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='005 mag in IR), and (3) fixing the S´ersic index (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='05 mag for PSF and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 mag for the host/bulge/disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We combine these flux uncertainties and adopt typical values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='25 mag as the fi- nal uncertainties for the PSF and galaxy (bulge, disk, or galaxy) flux measurements in all bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' These final uncertainties are consistent with those in our pilot study and similar observations and simulations in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bentz & Manne-Nicholas 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' See Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 7 Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021) for additional technical details on the flux uncertainty budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Host Galaxy/Bulge Masses Following the approach in Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021), we convert the UVIS/IR photometry to rest-frame B and I/R band and estimate the host/bulge stellar masses with the color-M∗/L relations (CMLR) from Into & Portinari (2013) and CIGALE (Boquien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' First, we correct for Galactic extinction using the re- calibrated Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (1998) dust map and redden- ing from Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We then fit the extinction-corrected HST photometry with CIGALE to derive k-corrections and color transformations be- tween the HST filters and the Johnson−Cousins fil- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' CIGALE is a spectral energy distribution (SED) fitting code that can model galaxy and AGN emission from multiwavelength photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We set up a simple CIGALE model that includes basic stellar population syn- thesis models (Maraston 2005), a initial mass function (Kroupa 2001), a dust attenuation model (Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2002), and a delayed star forma- tion history with optional starburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We do not include the AGN model for modeling the quasar-subtracted pho- tometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The UVIS filters are converted to B-band mag- nitudes, and the IR filters are converted to I-band and R-band filters depending on the source redshift (F110W to I-band at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 and R-band z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' F140W to I-band at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 and R-band at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We estimate the host and bulge stellar masses with the CMLR for dusty galaxy models from Into & Porti- nari (2013) using the rest-frame photometry and their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' CIGALE fits provide the k-corrected pho- tometry, from which we estimate stellar mass with the CMLR relation, and a stellar mass from the best-fit SED model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For RM177 (with two UVIS bands from a sep- arate HST program), we include both the F606W and F814W bands for the CIGALE fitting but only use the F606W band (rest-frame R-band) for the CMLR stel- lar mass estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The CMLR stellar-mass uncertain- ties are propagated directly from the photometry uncer- tainties, and the CIGALE stellar-mass uncertainties are estimated from the SED modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Both CMLR and CIGALE uncertainties are consistently around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 dex, which is typical for stellar mass estimation from two- band photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The final Galactic-extinction corrected, k-corrected, band-converted magnitudes, and the host/bulge stel- lar masses are tabulated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We adopt the CMLR stellar masses as our nominal host/bulge stel- lar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The best-fit stellar masses from CIGALE are also reported for comparison, which are generally consistent with those estimated from the CMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The only exception is RM265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The color derived for RM265 from CIGALE is unusually red, which led to a large, likely unphysical, host stellar mass (> 1012M⊙) using the CMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, the typical B − R color is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 < (B − R) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3, derived from all galaxy types in the Kinney-Calzetti Spectral Atlas (Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Kinney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' If we assume a red color of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 and adopt the R-band luminosity for the CMLR, the host stellar mass for RM265 is log(M∗) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23, which is con- sistent with the stellar mass derived from CIGALE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We show both the CMLR and CIGALE masses for RM265 in Figures 5, 6, and 10, and use the more physical CIGALE mass (for RM265 only) when fitting the BH scaling re- lations and their redshift evolution in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Host Properties At z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2, it becomes challenging to perform bulge/disk decomposition due to limited spatial resolu- tion, even with HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our GALFIT analysis shows 16 (out of 38) quasars are best-fitted by the PSF+bulge model, nine quasars are best-fitted by the PSF+disk model, and 13 quasars are decomposed into PSF+bulge+disk models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In addition, 26 hosts are bulge-dominated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', M∗,bulge > M∗,disk, and 12 hosts are disk-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A best-fit profile of n = 4 (n = 1) in our analysis does not necessarily mean the host galaxy is an elliptical (spiral) galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' the S´ersic index is fixed to n = 1 or n = 4 to ensure the quasar/host decomposition is robust and not to provide rigorous classifications of host morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In fact, the majority of local elliptical galaxies are not well-described by single S´ersic components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2013), and exponential profiles do not always in- dicate the presence of disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The structural parameters (ellipticity, S´ersic index, ef- fective radius) of the bulge/disk-dominated sources in our sample are broadly consistent with the statistical distributions from ∼ 2500, i-mag<22 SDSS quasar hosts observed by the Hyper Suprime-Cam (HSC) on the Sub- aru telescope (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' When we allow the S´ersic index to vary in the GALFIT fitting, the median (min- imum, maximum) S´ersic index of our two-component model is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0), similar to the distribution in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' There are more disk-like (n < 2) hosts in the SDSS-HSC sample, but roughly equal numbers of bulge-like and disk-like hosts (S´ersic indices above and below 2) in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The size and ellipticity of our quasar hosts are also similar to the SDSS-HSC sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The median (16%, 84% percentiles) effective radius is 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='68 (0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='35/0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='92), and the median (16%/84% per- 8 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Galaxy Decomposition Results RMID Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' magUVIS magIR r (′′) n q P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' rχ2 UVIS rχ2 IR 017 PSF 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='15 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='19 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' RMID Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' magUVIS magIR r (′′) n q P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='14 Bulge 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='58 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='18 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='71 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 Disk 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='87 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='32 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='78 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 UVIS Trunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='42 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 Fourier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04 Fourier 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='58 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='19 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='16 Radial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='40 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 Fourier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='02 Fourier 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='08 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='25 IR Trunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='52 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 Fourier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='05 Fourier 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='96 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='96 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='73 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='83 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='42 Radial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='52 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 Fourier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='07 Fourier 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='58 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='66 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 776 PSF 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='09 Bulge 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='46 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='77 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 Disk 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='35 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 10 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (Continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' RMID Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' magUVIS magIR r (′′) n q P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' rχ2 UVIS rχ2 IR 779 PSF 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='19 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='19 Disk 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='38 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='75 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 Disk 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='82 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='95 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 Note—r is the effective radius of the S´ersic component, n is the S´ersic index, q is the ratio between the semi-minor axis and the semi-major axis, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' is the position angle at the semi-major axis in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The reduced χ2 is calculated from the image residual, as reported by GALFIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Magnitudes are reported in ST magnitude (magST = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 log(Fλ[erg s−1 cm−2 ˚A−1]) − 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1), which is the default output from GALFIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For RM775 and RM790, we include the best-fit parameters for the truncated disks in the UVIS and IR images: the magnitudes are the surface brightness at the break radius (mag/arcsec2), and the best-fit parameters for the truncated radial profiles are listed in the order of the 1% flux radius (softening length, in arcseconds), 99% flux radius (break radius, in arcseconds), q, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='. The truncated disk in RM775 is fitted with Fourier modes in both the disks and truncated radial profiles, and the best-fit Fourier amplitudes (first row) and phase angles (second row) are listed in the order of Fourier mode 1, 3, 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' No extinction corrections are made for these magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The uncertainties of the GALFIT results are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' centiles) ellipticity (1 − q) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='41) for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We also examine the offset between the quasar po- sition and the host centroid in the IR images, where the centroid of the host galaxy is better constrained than in the UVIS band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Off-centered AGN/quasars may indicate on-going galaxy mergers or recoiling SMBHs from binary SMBH coalescence (Loeb 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Comerford & Greene 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 3 shows that most (34/38) of the quasars are located within < 1 kpc of the host galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The four sources with significant offsets (> 1 kpc, RM265, RM267, RM634, RM645;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' see the images and GALFIT models in full Figure 2 figure set online) show signs of galaxy interaction or mergers, which would com- plicate the centroid measurements of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' These results suggest z < 1 quasars are well centered within ∼ 1 kpc of the host centroid, consistent with the findings using alternative approaches (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' While studies of local AGNs have demonstrated that BH properties mainly correlate with the bulge and not the entire host (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Kormendy & Ho 2013), studies at higher redshift are often limited to the BH−host rela- tions when bulge/disk decomposition is difficult or im- possible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In this work, we present both the BH−bulge and the BH−host relations in our sample, where M∗,bulge and M∗,host refer to the bulge-only and total host stellar mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We include all sources in the MBH − M∗,host relation and exclude the disk-only (PSF+disk) objects in the MBH−M∗,bulge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' When comparing with earlier work, we examine their bulge/disk decompo- sition assumptions and place the comparison on an equal footing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', including bulge-dominated or bulge/disk decomposed sources only in the MBH−M∗,bulge relation, and including all sources in the MBH − M∗,host relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Comparison with earlier work Figure 4 shows the MBH−M∗,host and MBH−M∗,bulge relations of our sample and several local and higher- redshift samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' High-resolution HST imaging has been used to investigate the AGN host galaxies at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 (e.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='29 12 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' RMID Bands Comp mB mI/R Color log LB log LI/R log M∗ log M∗,CIGALE (mag) (mag) (mag) (L⊙) (L⊙) (M⊙) (M⊙) 776 B, I Host 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='93 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='94 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='87 ±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 840 B, I Host 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='30 Bulge 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='41 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='46 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='31 Disk 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='29 Note—Magnitudes are reported in AB magnitudes (Oke & Gunn 1982), and color refers to either B − I or B − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The last column lists the stellar masses estimated with CIGALE to compare with our fiducial stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2 4 6 8 10 12 Effective Radius [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 QSO/Host Separation [kpc] 694 634 267 265 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Projected physical offset between the quasar po- sition and host centroid in the IR images as a function of the host effective radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The uncertainties are estimated from GALFIT (quasar position and effective radius) and the quasar- subtracted host images (centroid position of the host).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The error bars are inflated by a factor of 10 for clarity, as the GALFIT uncertainties are small and likely underestimated (median uncertainties are ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='02 kpc for the quasar/host sep- aration and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='005 kpc for the effective radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' & Manne-Nicholas 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Similar to our study, bulge/disk decomposition is only possi- ble for a small subset of these non-local samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We follow the approach of Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bentz & Manne-Nicholas (2018) and as- sume M∗,bulge ≈ M∗,host when there is no evidence of additional components, which differs from Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2020) that estimated M∗,bulge by assigning bulge/total ratios depending on the S´ersic indices of the host profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' At even higher redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', z ≳1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5), host stellar masses can be obtained by SED fitting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Dong & Wu 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020) or imag- ing analysis of lensed quasars (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' SED fitting with wide wavelength coverage can provide better color information for estimating the stellar masses (compared to using only two HST bands), and large samples can be studied simultaneously in mul- tiwavelength fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, it is impossible to distin- guish between the bulge and disk components through SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' M∗,host can also be measured from the reconstructed images of strongly lensed quasars up to z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample is generally consistent with the MBH − M∗,bulge and MBH − M∗,host relations of these intermediate-to-high redshift samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample is the only uniformly-selected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', se- lected based on a flux limit) AGN sample with RM- based BH masses to study the BH scaling relations be- yond the local Universe (z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The RM masses in Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017) are consistently calibrated to the BH−host relations in quiescent local galaxies in Kor- Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 13 108 109 1010 1011 1012 M * , Bulge [M ] 106 107 108 109 1010 MBH [M ] KH13 Bennert+21 Bentz+18 Bentz+18 Bennert+11 Ding+20 Jahnke+09 108 109 1010 1011 1012 M * , Host [M ] 106 107 108 109 1010 MBH [M ] KH13 Bennert+21 Ding+21 Dong+16 Suh+20 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Comparison with literature samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample is shown in black circles (bulge-dominated) and gray squares (disk-dominated), and the open gray square is the CIGALE stellar mass of RM265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The blue points are the local-RM sample from Bentz & Manne-Nicholas (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The triangle symbols show the high redshift samples with HST imaging, orange: Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2011, median redshift zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2), green: Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2009, zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3), and red: Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2020, zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5), and the dot symbols indicate the high redshift samples with SED fitting, purple: Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021, zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7), brown: Dong & Wu (2016, zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1), pink: Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2020, zmed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The solid lines show the best-fit relations of the local quiescent galaxy (red, Kormendy & Ho 2013), active galaxy (orange, Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021), and RM AGN (blue, Bentz & Manne-Nicholas 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' mendy & Ho (2013) using the virial factor from Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' BH masses in all the comparison samples, except for Bentz & Manne-Nicholas (2018), are derived from the SE method, which is less reliable than RM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The SE method relies on a “tight” R − L rela- tion to estimate BLR sizes based on quasar luminosities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' however, recent studies (Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Fonseca Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020) have shown the local R − L relation is bi- ased towards the local AGN sample and could be over- estimating SE BH masses by as much as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 dex when applying to the general quasar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Finally, the SE method is calibrated to local quiescent galaxies or lo- cal RM AGNs, and different virial factors may be used for different samples or broad-line species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' When com- paring to samples from the literature, we rescale all MBH values using f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47 as in Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017), even for the SE masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We also compare our results with local baseline sam- ples from the literature, including the quiescent galax- ies (mainly ellipticals, Kormendy & Ho 2013), active galaxies (Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021), and RM AGNs (Bentz & Manne-Nicholas 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The MBH − M∗,bulge and MBH − M∗,host relations of the three local samples and our best-fit relations are consistent in slope and inter- cepts within uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, Reines & Volon- teri (2015) find AGN hosts follow a similar slope as lo- cal quiescent galaxies but are an order of magnitude lower in normalization for the MBH − M∗,host relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' They suggested the difference in normalization may be due to AGN activity or galaxy morphology (which is also shown in Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' These results may appear contradictory at first glance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' however, it is diffi- cult to provide a straightforward comparison since these studies adopt different stellar-mass estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bentz & Manne-Nicholas (2018) also observed a differ- ence in normalization using the Bell & de Jong (2001) CMLR, but not when they use the Into & Portinari (2013) CMLR, which is the same CMLR adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Due to different assumptions in the CMLR rela- tions, we do not compare the Reines & Volonteri (2015) relation, which uses the Zibetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2009) CMLR, with our results directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015) and Mutlu-Pakdil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2018) studied the MBH − M∗,bulge and MBH − M∗,host rela- tions in the Illustris simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015) found that, at z ∼ 0, the MBH − M∗,bulge relation is tight at the high-BH/galaxy mass end, but scatter in- creases below MBH ∼ 108M⊙, similar to the general trend of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Their bulge mass is defined by the total stellar mass within the stellar half-mass radius, and not by morphology or kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The difference in scatter in the high/low mass end might suggest dif- 14 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 109 1010 1011 1012 M * , Bulge [M ] 106 107 108 109 MBH [M ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='46 p-val= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1E-02 KH13 Bulge-dominated Disk-dominated 191 267 371 551 772 781 101 272 320 109 1010 1011 1012 M * , Host [M ] 106 107 108 109 MBH [M ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59 p-val= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6E-05 265 (265) KH13 Bulge-dominated Disk-dominated 160 191 267 371 772 781 589 779 101 377 519 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' BH mass as functions of bulge stellar mass (left) and total stellar mass (right) of our sample (black circles for bulge-dominated sources, gray squares for disk-dominated sources, and the open gray square for the CIGALE stellar mass of RM265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The blue dashed lines (and the gray shaded areas) are the best-fit relation (1σ range) of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The red solid lines are the Kormendy & Ho (2013) local MBH − M∗,bulge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The Pearson-r coefficient, p-value, and intrinsic scatter of the relations are labeled in the top-left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' ferent evolutionary paths or feedback mechanisms for establishing the BH scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Local studies of BH scaling relations also found that late-type galax- ies follow a similar slope in the BH scaling relations, but at a lower normalization than early-type galaxies (Reines & Volonteri 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In addition, there is no strong evolution in the MBH − M∗,bulge relation up to z ∼ 1 in the Illus- tris simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Mutlu-Pakdil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2018) studied the MBH−M∗,host relations in the Illustris simulation to pro- vide a better comparison for high redshift observations, and reported that the MBH−M∗,bulge and MBH−M∗,host relations are generally consistent with each other up to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2016) studied the MBH −M∗,host and MBH−M∗,bulge relations in the Horizon-AGN simu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' By identifying classical bulges in their simulation through kinematics and bulge/disk decomposition, they reproduced the tight MBH − M∗,bulge relation of clas- sic bulges from Kormendy & Ho (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Other simula- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', MassiveBlack-II (Khandai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015), gen- erally produce similar trends in BH scaling relations at z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Habouzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021) performed a systematic analysis on the evolution of MBH − M∗ relations in cos- mological simulations of Illustris, TNG 100, TNG 300, Horizon-AGN, EAGLE and SIMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' They find that the median/mean MBH − M∗ relations at 0 < z < 1 are in general agreement with observational data and there is little evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The observed tight corre- lation between BH accretion rate and star formation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 < z < 3 indicates the growth of BH and host galaxies are in sync, and the BH scaling relations should not have strong redshift dependence (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, the scatter in MBH − M∗ relations differs in these simu- lations, which mainly depends on the implemented sub- grid physics in the simulations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', the strength and efficiency of supernova and AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' MBH − M∗,host and MBH − M∗,bulge Relations of our sample Our sample consists of 38 sources spanning more than two orders of magnitude in MBH and M∗,host/bulge, which is sufficient for statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The Pearson correlation coefficient r between MBH and M∗,host/bulge are r ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 with low p-values (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='05), suggesting the BH and galaxy/bulge masses are positively correlated and the correlation is statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We use the LINMIX ERR algorithm (Kelly 2007) to perform linear re- gression fitting on the MBH−M∗,bulge and MBH−M∗,host relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' LINMIX ERR is a Bayesian fitting algorithm that accounts for uncertainties in both axes and intrin- sic scatter in the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We fit for the equation: logMBH M⊙ = a + b × log � M∗ 1010M⊙ � , (2) and tabulate the best-fit parameters in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The regression fits to the observed sample do not ac- count for selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 we use a more robust fitting code to constrain the intrinsic BH-host scaling relations, and we include the bias-corrected best-fit pa- rameters in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='. However, given the large dynamic Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 f* (This Work) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 f* (Shen+15) 109 1010 1011 1012 M* (This Work) [M ] 109 1010 1011 1012 M* (Matsuoka+15) [M ] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Left: Comparison of the derived host light fraction from this work and Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a), both measured in the UVIS (F606W or F814W) bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The black dashed line shows the 1:1 ratio line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Right: Comparison of the total stellar masses derived from this work and Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015) for overlapping objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Best-fit Parameters of the Scaling Relations Scaling Relations a b σ Bulge (Original) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 Host (Original) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='11 Bulge (Bias-Corrected) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 Host (Bias-Corrected) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 range of our sample, selection effects do not appear to impact the results significantly, as we will show in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Comparison with spectral decomposition We compare the host-light fraction and total stellar mass derived from HST imaging decomposition in this work and the spectral decompositions in Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a) and Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Both of these earlier studies measured host-galaxy properties using the high- S/N coadded spectra from the first-year SDSS-RM spec- tra (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a) used a prin- cipal component analysis method to decompose coadded spectra into quasar and galaxy spectra, and measured host-galaxy properties directly from the galaxy spectra, including stellar velocity dispersion and host-free AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015) performed spectral decomposition on the coadded spectra using models of AGN and galaxy spectra, and measured host galaxy properties by fitting the decomposed galaxy spectra with stellar population models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' To compare the host-light fraction (f∗, the fractional contribution of the host stellar component to the total flux), we calculate the host fraction in the HST imaging using the decomposed GALFIT models within the 2′′ di- ameter spectral aperture, and the host fraction in spec- tral decomposition by computing the expected flux in the F606W and F814W bandpass in the decomposed spectra from Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 6 (left panel) reveals that the host fraction from image decomposition is systematically higher than that derived from spectral decomposition, similar to our finding in the pilot study (Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021) and in Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 6 (right panel) compares the host stellar mass derived from this work and from Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our stellar mass is systematically smaller by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The cause of the stellar mass offset is currently unclear, but it might be partially due to different choices of initial mass functions and stellar population models in Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015) and CIGALE, or the fiber-loss correction applied in Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015), which assumes the mass-to-luminosity ratio in the central region (within the 2′′-diameter aperture) represents that for the entire galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Biases in the Observed BH Scaling Relations Since our sample is based on RM BH masses, it avoids the large statistical biases associated with the system- atic uncertainties of SE masses (Shen & Kelly 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' To illustrate selection biases in our flux-limited sam- 16 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 108 109 1010 1011 1012 M * , Bulge [M ] 105 106 107 108 109 1010 MBH [M ] i-mag < 16 i-mag < 19 i-mag < 22 7 8 a 0 1 2 b 16 19 22 i-mag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Results of the Lauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2007) bias simulation described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Left: grey contours represent the underlying quasar sample, and the grey points are one realization of the mock sample at the i = 22 flux limit, after being perturbed with measurement uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The black solid line is the unbiased Kormendy & Ho (2013) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The colored lines indicate the results of the mock sample at different flux limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Right: the best-fit MBH − M∗,bulge parameters at each flux threshold and the best-fit MBH − M∗,host parameters of our sample (plotted in red at i = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The dotted lines show the parameters and intrinsic scatter from the Kormendy & Ho (2013) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' ple due to intrinsic scatter in BH-host scaling relations (Lauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2007), we perform a forward-modeling sim- ulation following the procedures in Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We first simulate a parent quasar sample following the local M∗ distribution from Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2010) and the MBH − M∗,bulge relation from Kormendy & Ho (2013), with an intrinsic scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='29 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Using the true MBH, we assign a quasar bolometric luminos- ity by assuming a lognormal Eddington ratio distribu- tion (λ ≡ Lbol/LEdd) and the Eddington luminosity is LEdd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='26 × 1038(MBH/M⊙) erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We choose a mean Eddington ratio of ⟨logλ⟩ = −1 and a scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 dex (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Shen & Kelly 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We include measurement uncertainties of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='35 dex for M∗ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 dex for MBH to mimic the uncertainty levels of our CMLR-based M∗ and RM MBH measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Fi- nally, for 100 bootstrap iterations, we randomly draw 38 sources to perform LINMIX ERR fitting with at different i-mag < 16, 19, and 22 (similar to the flux limit of the SDSS-RM sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 7 shows how the flux limit biases the observed scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' When the flux limit increases, over- massive BHs are preferentially selected, the slope of the best-fit relation becomes shallower, and the nor- malization increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The best-fit intrinsic scatter re- mains roughly the same in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, our simulation does not include the outlier population with under-massive BHs seen in observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Missing the outlier population could lead to an underestimation of the intrinsic scatter and selection bias based on the flux limit, since under-massive BHs are less likely to be selected in flux-limited surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Given the relatively faint flux limit of our SDSS-RM sample, selection biases do not play an important role in the measured MBH − M∗ relations (see the right panel of Figure 7), and we found similar relations at zmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 as the local relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our results are consistent with other studies that properly account for selection biases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Sexton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We note that the selection of our sample also depends on successful RM lag measurements, which may depend on BH mass and Eddington ratio, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, since the lag-detection fraction in the Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2017) sample is nearly uniform up to z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8, we assume the sample selection is not strongly affected by additional selection biases based on the quasar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Quantifying the Selection Effects To quantitatively account for the underlying galaxy properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', the galaxy mass function) and selection effects, we follow the framework of Kelly (2007) to per- form a Markov chain Monte Carlo (MCMC) fitting for the intrinsic scaling relations and scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Specifically, we wrote a MCMC fitting code based on the Metropolis- Hastings algorithm (Metropolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Hastings Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 17 109 1010 1011 1012 M * , Bulge [M ] 106 107 108 109 MBH [M ] Bias-Corrected Original KH13 a = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 b b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' MCMC fitting for the MBH −M∗,bulge relation when considering the selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Left: the MBH −M∗,bulge relation with the 1σ range drawn from the posterior (black dashed line and shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The blue dashed lines and shaded area show the original best-fit relations from LINMIX ERR (previously shown in Figure 5), and the red solid line is the local MBH − M∗,bulge from Kormendy & Ho (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Right: the posterior distribution of the parameters a, b, and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The best-fit parameters from LINMIX ERR and the MCMC fitting are indicated by the blue and white squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The MCMC fitting recovers a similar BH-galaxy relation to the original linear regression, indicating that the selection effects are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1970) and the statistical derivation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 of Kelly (2007), which allows us to account for the selection effect based on the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Here, we briefly summarize the fitting algorithm and param- eter setup and refer the readers to Kelly (2007) for the full, detailed mathematical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our custom fit- ting code is available via ftp://quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='edu/ public/sdssrm/paper data/Li 2023 HST host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' As shown in Kelly (2007), when the intrinsic scatter and the uncertainties are comparable to the dynamical range of the data, the best-fit slope becomes shallower when the underlying distribution is not considered as a prior in the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' One solution is to incorpo- rate empirical distributions into the likelihood function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', the observed local stellar mass function Φ(x) from literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, due to the limited dynamical range in M∗ and the small sample size of our data, the local galaxy mass function is not a good prior for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Alternatively, Kelly (2007) suggests using a series of Gaussian functions to model the underlying distribution, which provides a flexible and empirical solution even when the underlying dis- tribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' This is the method implemented in the original LINMIX ERR fitting algorithm, which we continue to adopt in our MCMC fitting for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' When the sample selection is based on the dependent variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', MBH), the posterior distribution and likeli- hood depend on an additional term (P(I = 1|θ), where θ are the model parameters) that describes the likeli- hood of including each data point in the observed sam- ple based on the model parameters (for more details, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 in Kelly 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Following the same pro- cedure as described in Paragraph 1 of this section, we estimate the expected i-band magnitude by assigning a random Eddington ratio and an Eddington luminosity based on the redshift and a range of “true” MBH for each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The probability of including a data point is 1 if i-mag< 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 and 0 if i-mag> 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Finally, we calculate P(I = 1|θ) by integrating the probability of including each data point and their likelihood over a range of “true” MBH and M∗ given the model parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We adopt uninformative, flat priors for all param- eters (5 < a < 10, 0 < b < 3, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0001 < σ2 < 1) and minimize the product of the likelihood and prior to compute the posterior distribution of the parameters a, b, and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 8 and 9 present the posterior distribution of a, b, and σ, and the best-fit values are tabulated in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' As expected, the slope of the intrinsic scaling rela- tions becomes steeper, and the normalization decreases, after correcting for the selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The best-fit pa- 18 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 109 1010 1011 1012 M * , Host [M ] 106 107 108 109 MBH [M ] Bias-Corrected Original KH13 a = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 b b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Same format as Figure 8 but for the MBH − M∗,host relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' rameters are within uncertainties as the LINMIX ERR fit (without considering selection bias) discussed in Section 3, demonstrating that our results are not strongly af- fected by selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Intrinsic scatter of the MBH − M∗,bulge and MBH − M∗,host relations is an important indicator for BH−galaxy co-evolution, as it might be related to the galaxy/AGN properties and their evolutionary path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The local samples of Kormendy & Ho (2013) and Ben- nert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021) only include classical bulges and pseudo-bulges and have a smaller intrinsic scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='28 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='39 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, when including all morpho- logical types and active/inactive galaxies, the intrinsic scatter increases to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 dex (Reines & Volonteri 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bentz & Manne-Nicholas 2018) for the MBH − M∗,bulge relation, and becomes even slightly larger for the MBH− M∗,host relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' In addition, the BH accretion rate is found to be correlated with other host properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', compactness of the central ∼1 kpc region (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019, 2021), which can introduce additional scatter in the BH scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For our quasar sample, the intrinsic scatter of the MBH − M∗,host and MBH − M∗,bulge relations are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 dex and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 dex, respectively, after ac- counting for the selection effects, which are compara- ble to the scatter in the local relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The intrinsic scatter of our MBH − M∗,host relation is slightly smaller (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5σ of difference) than our MBH −M∗,bulge relation, which we will further discuss in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Because we have neglected the systematic uncertainty in our RM BH masses due to the scatter in individual virial coeffi- cients, the actual intrinsic scatter in the BH-host stellar mass relations for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 quasars might be even smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Evolution of BH Scaling Relations Earlier works on MBH − M∗,bulge and MBH − M∗,host relations found that the average BH-to-host galaxy mass ratio evolves positively with redshift (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2006a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' How- ever, selection biases and measurement uncertainties could yield false positives of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' For ex- ample, Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2009) reported that there is no evidence of evolution when they carefully choose their sample to avoid selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Similarly, Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2020) found the redshift evolution seen in Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2010) can be explained by the Lauer bias (Lauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2007), and there is no trend of evolution in their X- ray-selected, lower luminosity sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Using over 500 uniformly-selected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 SDSS quasars, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' (2021a) found a redshift evolution of the offset in the MBH−M∗,host relation that is within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 dex from zero, consistent with no significant evolution since z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Figure 10 presents the deviation of MBH from the Kormendy & Ho (2013) MBH − M∗,bulge rela- tion of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We fit the deviation as a function of log(z), and the slopes and intercepts are consistent with zero, suggesting there is no red- shift evolution from the local relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The median (16th/84th percentiles) black hole/host galaxy mass ra- tios are MBH/M∗,bulge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0037(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0007/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0088) and MBH/M∗,host = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0030(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0007/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0076), within 1σ un- Black Hole Scaling Relations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 log(1 + z) 3 2 1 0 1 2 3 log(MBH) [M * , Bulge] Bulge-dominated Disk-dominated Bentz+18 Bennert+11 Ding+20 Jahnke+09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='7 log(1 + z) 3 2 1 0 1 2 3 log(MBH) [M * , Host] Bulge-dominated Disk-dominated Bentz+18 Bennert+11 Ding+20 Jahnke+09 Ding+21 Dong+16 Suh+20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 z Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Evolution of ∆log(MBH) with redshift, with baselines adopted from the best-fit relations of MBH − MBulge and MBH −LBulge from the Kormendy & Ho (2013) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our work is the only sample with RM-based BH masses beyond z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Vertical error bars are from uncertainties in BH mass only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' certainty of the local value of MBH/M∗,bulge ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='005 (Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our results agree with recent observational studies that there is limited evolution in the MBH −M∗ relation (Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The limited redshift evolution in the MBH−M∗ relations can be explained by the tight correlation between the BH accretion rates and star formation rates found in bulge-dominated galaxies at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5 − 3 (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019), which suggests the growth of SMBH and their host galaxies and bulges are in sync since z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' BH−Bulge versus BH−Host Relations In the local Universe, SMBH masses are tightly cor- related with the properties of classical bulges, but not with disks or total mass of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' However, the intrinsic scatter in the MBH−M∗,host relation of our sam- ple is slightly smaller than that in the MBH − M∗,bulge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' There are a few extra sources of uncertainties for the bulge-mass estimate than for the total mass es- timate, which will contribute to the intrinsic scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' First of all, bulge-disk decomposition could add sig- nificant uncertainties to the bulge mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Most of our host galaxies are far less luminous than the quasar, and compact hosts could be near the limit of HST imaging resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Bulge-disk decomposition is reliable when both the bulge and disk are sufficiently bright (com- pared to the central quasar) and there is a distinct dif- ference in their effective radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Furthermore, we can- not distinguish classical bulges from pseudo-bulges or other bulge-like structures from surface-brightness de- composition, nor could we model complex bar, spiral, 20 Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' and merger structures in the bulge-disk decomposition, which increases the uncertainties in bulge identification and mass estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Gao & Ho (2017) found that rig- orous modeling of bars and innermost structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', rings and disk breaks near the bulge) is crucial to re- covering bulge properties, while the modeling of spiral arms and extended disks have negligible effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We note that some host galaxies in our sample show clear ev- idence of bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', RM320, RM634, etc), which are modeled as bulges (n = 4) or disks (n = 1) in our analysis, without additional bar structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Moreover, some host galaxies show clear spiral arm features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', RM371, RM772, etc), indicating the presence of disks, but are modeled as “bulges”(n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020) showed late- type quasar hosts preferentially scatter below early-type hosts in the MBH−M∗,bulge relation, which is not seen in our data, suggesting our bulge-disk decomposition is not as reliable in measuring galaxy morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' A detailed simulation of bulge-disk decomposition for AGNs with similar host and quasar properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', AGN/host flux ratio, host effective radius, S´ersic indices, and complex structures) is needed to provide quantitative uncertainty estimation, which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Another possible source of uncertainty is the CMLR estimation for bulges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Recent studies have reported that compact regions around the SMBH may have denser in- terstellar medium, boosted star formation, and complex stellar populations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=', Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Kim & Ho 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Zhuang & Ho 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Shangguan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Yesuf & Ho 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Molina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Two-band color and the use of empirical M/L relation may not be sufficient to produce reliable estimates for the bulge stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' CONCLUSIONS We present the MBH −M∗,bulge and MBH −M∗,host re- lations of 38 sources with RM-based BH masses (Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2017) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='2 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='8 (median redshift zmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our sample is the first uniformly-selected sample with RM-based BH masses at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='3 for studying BH- host relations, and covers two orders of magnitude in BH mass and host stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' The reliable RM-based BH masses and host mass estimates from HST imaging decomposition, combined with the large sample size and dynamic range in mass, allow one to alleviate selection biases in studying the potential evolution of the BH- host scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Our scaling relations are consis- tent with those for local AGNs, quiescent galaxies, and other high-redshift samples, with negligible redshift evo- lution up to z ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' As shown in Table 4, the best-fitting intrinsic MBH − M∗,host relation is: log(MBH/M⊙) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='33 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='64 log(M∗,host/1010M⊙) after correct- ing for the underlying sample distribution and selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' We estimate an intrinsic scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='21 dex and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content='17 dex in the MBH − M∗,bulge and MBH − M∗,host relations, respectively, which is again consistent with the local BH scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' With our ap- proved Cycle 1 JWST proposal (GO-2057, PI: Shen), we will continue to explore BH−host relations and their redshift evolution up to z ∼ 2 using quasars with direct RM-based BH masses (Grier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' JIL acknowledges support from the Government Schol- arship to Study Abroad (GSSA) from the Ministry of Education of Taiwan and support from the Illinois Space Grant Consortium (ISGC) Graduate Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' YS acknowledges support from NSF grants AST-1715579 and AST-2009947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' LCH was supported by the National Science Foundation of China (11721303, 11991052, 12233001, 12011540375) and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' WNB acknowledges support from NSF grant AST- 2106990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' PBH is supported by NSERC grant 2017- 05983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Based on observations with the NASA/ESA Hubble Space Telescope obtained from the Data Archive at the Space Telescope Science Institute, which is oper- ated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5- 26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' Support for Program number HST-GO-15849 was provided through a grant from the STScI under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E2T4oBgHgl3EQf4Ain/content/2301.04177v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Software: 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